Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
SSHEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONSAND ERGODIC THEORY OF THE EARTHQUAKE FLOW
AARON CALDERON AND JAMES FARRE
Abstract.
We extend Mirzakhani’s conjugacy between the earthquake and horocycle flows to a bijection,demonstrating conjugacies between these flows on all strata and exhibiting an abundance of new ergodicmeasures for the earthquake flow. The structure of our map indicates a natural extension of the earthquakeflow to an action of the the upper-triangular subgroup
P < SL R and we classify the ergodic measuresfor this action as pullbacks of affine measures on the bundle of quadratic differentials. Our main tool is ageneralization of the shear coordinates of Bonahon and Thurston to arbitrary measured laminations. Main results
Conjugating earthquake and horocycle flow.
This paper deals with two notions of unipotent flowover the moduli space M g of Riemann surfaces. The first is the Teichm¨uller horocycle flow , defined on thebundle Q M g of unit area quadratic differentials q by postcomposing the charts of the flat metric | q | by theparabolic transformation (cid:18) s (cid:19) . This flow is ergodic with respect to a finite measure induced by Lebesguein local period coordinates [Mas82, Vee82] and is a fundamental object of study in Teichm¨uller dynamics.The second is the earthquake flow on the bundle P M g , whose fiber is the sphere of unit-length measuredgeodesic laminations on a hyperbolic surface. The earthquake flow is defined as a generalization of twistingabout simple closed curves, or by postcomposing hyperbolic charts by certain piecewise-isometric transfor-mations. While this flow is more mysterious, earthquakes are a familiar tool in Teichm¨uller theory, playinga central role in Kerckhoff’s proof of the Nielsen realization conjecture [Ker83], for example.These two flows are both assembled from families of Hamiltonian flows (extremal length for horocycle[Pap86] and hyperbolic length for earthquake [Ker83, Wol83, SB01]) and exhibit similar non-divergenceproperties [MW02], but the horocycle flow belongs properly to the flat-geometric viewpoint and the earth-quake flow to the hyperbolic one. All the same, in [Mir08, Theorem 1.1] Mirzakhani established a bridgebetween the two worlds, demonstrating a measurable conjugacy between the earthquake and horocycle flows.Consequently, the earthquake flow is ergodic with respect to the measure class of Lebesgue on P M g .In this article, we deepen this connection between flat and hyperbolic geometry, proving that the corre-spondence can be further upgraded to yield new results on both the ergodic theory of the earthquake flowand the structure of Teichm¨uller space. Theorem A.
Mirzakhani’s conjugacy extends to a bijection O : P M g ↔ Q M g that conjugates earthquake flow to horocycle flow.The moduli space of quadratic differentials is naturally partitioned into strata Q M g ( κ ), disjoint subsetsparametrizing unit-area differentials with zeros of order κ = ( κ , . . . , κ n ). Similarly, for any κ we maydefine the regular locus P M reg g ( κ ) to be the set of ( X, λ ) where λ cuts X into regular ideal polygons with( κ + 2 , . . . , κ n + 2) many sides.With this notation, Mirzakhani’s conjugacy can more precisely be stated as the existence of a bijection P M reg g (1 g − ) ↔ Q M nsc g (1 g − )taking earthquake flow to horocycle flow, where the superscript nsc specifies the (full-measure) sublocus ofthe stratum consisting of those differentials with no horizontal saddle connections.One of our main applications of Theorem A is to produce an analogue of Mirzakhani’s conjugacy forcomponents of strata (even those coming from global squares of Abelian differentials), confirming a conjectureof Alex Wright [Wri18, Remark 5.6] (see also [Wri20, Problems 12.5 and 12.6]). Theorem B.
For every κ , the map O restricts to a bijection P M reg g ( κ ) ↔ Q M nsc g ( κ )that takes earthquake to horocycle flow and (generalized) stretch rays to Teichm¨uller geodesics. Date : March 1, 2021. a r X i v : . [ m a t h . G T ] F e b AARON CALDERON AND JAMES FARRE
While strata of holomorphic quadratic differentials are generally not connected, for g (cid:54) = 4 their connectedcomponents are classified by whether or not they consist of squares of abelian differentials and the parity ofthe induced spin structure (both of which depend only on the horizontal foliation when there are no horizontalsaddles), as well as hyperellipticity [KZ03, Lan08]. The bijection O respects both the horizontal directionand the Mod( S ) action, so Theorem B can be refined to describe the preimages of these components.As an immediate consequence of Theorem B, the earthquake flow is ergodic with respect to the pushfor-ward by O − of the Masur-Veech measure on any component of any stratum of quadratic differentials.1.2. Geodesic flows and P -invariant measures. Pulling back the Teichm¨uller geodesic flow via O allowsus to specify a family of “dilation rays” which serve as a geodesic flow for the earthquake flow’s parabolicaction and in many cases project to geodesics for Thurston’s Lipschitz asymmetric metric. Combiningdilation rays and the earthquake flow therefore gives a action of the upper triangular subgroup P < SL R on P M g by “stretchquakes.” See Section 15.3.Due in part to the failure of O to be continuous, the stretchquake action on P M g is not by homeomor-phisms but rather by measurable bijections. More precisely, it preserves the σ -algebra obtained by pullingback the Borel σ -algebra of Q M ( S ) along O . In a forthcoming sequel [CF], the authors show that O isactually a measurable isomorphism with respect to the Borel σ -algebra on P M g and that the stretchquakeaction restricted to each P M reg g ( κ ) is by homeomorphisms; see also Remark 2.2.In their foundational work on SL R -invariant ergodic measures on the moduli space of flat surfaces, Eskinand Mirzakhani [EM18, Theorem 1.4] proved that the support of any P -invariant ergodic measure on Q M g is locally an affine manifold cut out by linear equations in period coordinates. Our conjugacy translates thisclassification into a classification of ergodic measures for the extension of the earthquake flow defined above: Theorem C.
Every stretchquake-invariant ergodic measure is the pullback of an affine measure.
Proof. If ν is a stretchquake-invariant ergodic measure on P M g , then O ∗ ν is a P -invariant ergodic measureon Q M g , which is affine by [EM18, Theorem 1.4]. (cid:3) Using this correspondence we obtain a geometric rigidity phenomenon for stretchquake-invariant ergodicmeasures on P M g : the generic point is made out of a fixed collection of regular ideal polygons. Corollary 1.1.
For any stretchquake-invariant ergodic probability measure ν on P M g , there is some κ sothat ν -almost every ( X, λ ) lies in P M reg g ( κ ).This in particular implies that the dynamics of the stretchquake action with respect to any ergodicprobability measure are measurably the same as its restriction to a stratum, on which we can identify dilationrays as (directed, unit-speed) geodesics for the Lipschitz asymmetric metric on T ( S ) (see Proposition 15.12). Remark 1.2.
We note that general ergodic measures for the stretchquake action can look quite differentthan the Lebesgue measure class on P M g , even when pushed down to M g .For example, if ν gives full measure to P M reg g (4 g −
4) then a ν -generic point is obtained by gluingtogether a single regular ideal (4 g − g → ∞ . This implies that ν gives zero mass to (the restriction of P M g to) sufficiently thin parts of moduli space, as any ( X, λ ) where X has a very short pants decomposition hasinjectivity radius uniformly bounded above. Remark 1.3.
While an important result of [EM18] is that any P -invariant ergodic measure on Q M g isactually SL R -invariant, the circle action on Q M g (corresponding to rotating a quadratic differential) doesnot have an obvious geometric interpretation on P M g . See also [Wri20, Problems 12.3 and 12.4]1.3. Dual foliations from hyperbolic structures.
A foundational result of Gardiner and Masur (The-orem 2.1 below) states that quadratic differentials are parametrized by their real and imaginary parts;equivalently, their vertical and horizontal foliations (or laminations). In particular, the set F uu ( λ ) of allquadratic differentials with horizontal lamination λ is the same as the space MF ( λ ) of foliations that bindtogether with λ . As the horocycle flow preserves the horizontal foliation, it induces a flow on MF ( λ ).Mirzakhani’s conjugacy and our extension therefore both follow from the construction of flow-equivariantmaps that assign to a hyperbolic surface X and a measured lamination λ a “dual” measured foliation. In genus 4, there are certain strata whose components have only been characterized via algebraic geometry [CM14].
HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 3
For maximal laminations λ , this dual is the horocyclic foliation F λ ( X ) introduced by Thurston [Thu98],obtained by foliating the spikes of each triangle of X \ λ by horocycles and extending across the leaves of λ .The measure of an arc transverse to F λ ( X ) is then the total distance along λ between horocycles meetingthe arc at endpoints. As F λ ( X ) necessarily binds S together with λ , this defines a map F λ : T ( S ) → MF ( λ ) , where MF ( λ ) is the space of measured foliations that bind together with λ .The main engine of Mirzakhani’s conjugacy is the following theorem of Bonahon [Bon96] and Thurston[Thu98]; see also Section 2.1 for a discussion of her interpretation of this result. Theorem 1.4 (Bonahon, Thurston) . For any maximal λ , the horocyclic foliation map F λ is a real-analytichomeomorphism which conjugates earthquake flow to horocycle flow. Moreover, the family { F λ } is equivari-ant with respect to the Mod( S ) action. That is, F gλ ( gX ) = gF λ ( X ) for all g ∈ Mod( S ).When λ is not maximal the horocyclic foliation is no longer defined. The first thing one might try isto simply choose a completion of λ , but this approach is too na¨ıve. Indeed, this would require choosing acompletion of every lamination, which necessarily destroys Mod( S )–equivariance because laminations (anddifferentials) can have symmetries. Such a map will not descend to moduli space and is therefore unsuitablefor our applications. Besides, for our purposes it is important that the geometry of the subsurfaces of X \ λ predicts the singularity structure of the corresponding differential.If one restricts their attention to the case when λ is filling and cuts X into regular ideal polygons thenthere is a canonical notion of horocyclic foliation. While this construction is equivalent on the regular locusto the more general procedure we describe just below, any attempt to prove Theorem B with this restrictedviewpoint would necessarily rely on (Mod( S )–equivariant) descriptions of the loci of surfaces built fromregular polygons, as well as the intersection of F uu ( λ ) with strata, results which (to the knowledge of theauthors) were heretofore unknown. Compare Corollary 2.6 and Section 2.2.We therefore place no restrictions on the topological type or the complementary geometry of λ . Followinga suggestion of Yi Huang (communicated to us by Alex Wright), we prove that the correct analogue of thehorocyclic foliation for non-maximal λ is the orthogeodesic foliation O λ ( X ), whose leaves are the fibers ofthe closest point projection to λ and whose measure is given by length of the projection to λ . As in themaximal case, the orthogeodesic foliation binds together with λ , inducing a map O λ : T ( S ) → MF ( λ ) . See Section 5 for a more detailed discussion of this construction.
Theorem D.
For any λ ∈ ML ( S ), the orthogeodesic foliation map O λ is a homeomorphism which con-jugates earthquake flow to horocycle flow. Moreover, the family {O λ } is equivariant with respect to theMod( S ) action. That is, O gλ ( gX ) = g O λ ( X ) for all g ∈ Mod( S ).Although MF ( λ ) does not have an obvious smooth structure, the map O λ still exhibits a surprisingamount of regularity; see Theorem E.The proof of Theorem D requires generalizing Bonahon’s machinery of transverse cocycles to new com-binatorial objects called “shear-shape cocycles” which capture the essential structure of the orthogeodesicfoliation; see Section 2.1 just below. The space of shear-shape cocycles forms a common coordinatization ofboth T ( S ) and MF ( λ ) that is compatible with the map O λ and reveals an abundance of structure encodedin the orthogeodesic foliation: • When λ cuts X into regular ideal polygons, the orthogeodesic and horocyclic foliations agree. • The locus of points of X which are closest to at least two leaves of λ forms a piecewise geodesicspine for X \ λ which captures the geometry and topology of the complementary subsurfaces (seeTheorem 6.4). Moreover, this spine is exactly the diagram of horizontal separatrices for the quadraticdifferential with horizontal foliation λ and vertical foliation O λ ( X ). • For every measure µ on λ , the intersection of µ and O λ ( X ) is the hyperbolic length of µ on X . • The pullback of Teichm¨uller geodesics with no horizontal saddle connections are geodesics withrespect to Thurston’s Lipschitz (asymmetric) metric (Proposition 15.12). For example, take γ to be a simple closed curve; completions of γ correspond to triangulations of X \ γ where the boundariesare shrunk to cusps (up to a choice of spiraling about each side of γ ). The space of such triangulations carries a rich Stab( γ )action, and a computation shows that the horocyclic foliations for two completions in the same Stab( γ ) orbit need not be equal. AARON CALDERON AND JAMES FARRE
A statement similar to Theorem D is probably also true for any (unmeasured) geodesic lamination, butfor technical reasons regarding compatibility of complementary subsurfaces and the spiraling behavior of λ we have restricted ourselves to the measured setting. See Remark 7.10.The orthogeodesic foliation map can also be thought of as relating the hyperbolic and extremal lengthfunctions (cid:96) λ ( · ) and Ext λ ( · ) for any fixed λ . Indeed, a seminal theorem of Hubbard and Masur [HM79] statesthat the natural projection π : F uu ( λ ) → T ( S )that records only the complex structure underlying a differential is a homeomorphism. Combining this withthe fact that the extremal length of λ on Y is exactly the area of the differential π − ( Y ), we deduce that Corollary 1.5.
For every λ ∈ ML ( S ), the map π ◦ O λ is a Stab( λ )–equivariant self-homeomorphism of T ( S ) that takes the hyperbolic length function (cid:96) λ ( · ) to the extremal length function Ext λ ( · ).1.4. Acknowledgments.
The authors would firstly like to thank Alex Wright for providing the germ ofthis project and useful suggestions, as well as for helpful comments on a preliminary draft of this paper. Weare grateful to Francis Bonahon, Feng Luo, Howie Masur, and Jing Tao for lending their expertise and forenlightening discussions.The authors would also like to thank those who contributed helpful comments or with whom we hadclarifying conversations, including Daniele Alessandrini, Francisco Arana-Herrera, Mladen Bestvina, JonChaika, Vincent Delecroix, Valentina Disarlo, Spencer Dowdall, Ben Dozier, Aaron Fenyes, Ser-Wei Fu,Curt McMullen, Mareike Pfeil, Beatrice Pozzetti, John Smillie, and Sam Taylor.Finally, the authors are indebted to Yair Minsky for his dedicated guidance and mentorship throughoutall stages of this project, as well as his generous listening and insightful comments.The first author gratefully acknowledges support from NSF grants DGE-1122492, DMS-161087, and DMS-2005328, and travel support from NSF grants DMS-1107452, -1107263, and -1107367 “RNMS: GeometricStructures and Representation Varieties” (the GEAR Network). The second author gratefully acknowledgessupport from NSF grants DMS-1246989, DMS-1509171, DMS-1902896, DMS-161087, and DMS-2005328.Portions of this work were accomplished while the authors were visiting MSRI for the Fall 2019 program“Holomorphic Differentials in Mathematics and Physics,” and the authors would like to thank the venue forits hospitality and excellent working environment. Part of this material is based upon work supported bythe National Science Foundation under Grant No. DMS-1928930 while the second author participated ina program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall2020 semester on “Random and Arithmetic Structures in Topology.” The second author would also like tothank the both the Department of Mathematics at the University of Utah and the Mathematics Institute atUniversit¨at Heidelberg for their hospitality and rich working environments.
Contents
1. Main results 12. About the proof 53. Outline of the paper 104. Crowned hyperbolic surfaces 125. The orthogeodesic foliation 146. Cellulating crowned Teichm¨uller spaces 187. Transverse and shear-shape cocycles 248. The structure of shear-shape space 319. Train track coordinates for shear-shape space 3510. Shear-shape cooordinates for transverse foliations 3911. Flat deformations in shear-shape coordinates 4712. Shear-shape coordinates for hyperbolic metrics 4813. Measuring hyperbolic shears and shapes 5014. Shape-shifting cocycles 5815. Shear-shape coordinates are a homeomorphism 7616. Future and ongoing work 83References 85
HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 5 About the proof
Given Theorem D which associates to (
X, λ ) a dual foliation O λ ( X ) describing the geometry of the pair, itis not difficult to prove Theorems A and B. First, we recall the relationship between differentials, foliations,and laminations in a little more detail.The space of measured foliations (up to equivalence) on a closed surface S of genus g ≥ MF ( S ). There is a canonical identification [Lev83] between MF ( S ) and ML ( S ), the space of measuredlaminations on S ; throughout this paper we will implicitly pass between the two notions at will, dependingon our situation. By QT g and Q T g we mean the bundle of holomorphic quadratic differentials over theTeichm¨uller space and the locus of unit area quadratic differentials, respectively. We similarly denote PT g = T ( S ) × ML ( S ) and P T g the locus of pairs ( X, λ ) where λ has unit length on X .To every q ∈ QT g one may associate the real measured foliation | Re( q ) | which measures the totalvariation of the real part of the holonomy of an arc; the imaginary foliation | Im( q ) | is defined similarly.These foliations have vertical, respectively horizontal, trajectories, and so we will also refer to them as thevertical and horizontal foliations (or laminations) of q and write q = q ( | Re( q ) | , | Im( q ) | ) . A foundational theorem of Gardiner and Masur implies that the real and imaginary foliations completelydetermine q , and that given any two foliations which “fill up” the surface, one can integrate against theirmeasures to recover a quadratic differential.A pair of measured foliations/laminations ( η, λ ) is said to bind S if for every γ ∈ ML ( S ), i ( γ, η ) + i ( γ, λ ) > , where i ( · , · ) is the geometric intersection pairing. Theorem 2.1 ([GM91, Thereom 3.1]) . There is a Mod( S )–equivariant homemomorphism QT ( S ) ∼ = MF ( S ) × MF ( S ) \ ∆where ∆ is the set of all non-binding pairs ( η, λ ). In particular, the set F uu ( λ ) of all quadratic differentialswith | Im( q ) | = λ may be identified with MF ( λ ), the set of foliations which together bind with λ . Proof of Theorems A and B.
By definition, there is a Mod( S )–equivariant projection PT g → ML ( S ) withfiber T ( S ). Theorem 2.1 implies there is a Mod( S )–equivariant projection QT g → ML ( S ) whose fiber over λ may be identified with MF ( λ ). Applying Theorem D on the fibers therefore yields an equivariant bijection O : PT g ↔ QT g which takes unit-length laminations to unit-area differentials (Corollary 13.14), and quotienting by theMod( S ) action proves Theorem A.Furthermore, we observe that the spine of the orthogeodesic foliation of a regular ideal ( k + 2)-gon is justa star with k + 2 edges, which corresponds to the separatrix diagram of a zero of order k when there are nohorizontal saddle connections. Thus O restricts to the promised conjugacy on strata (Theorem B). (cid:3) Remark 2.2.
Mirzakhani’s conjugacy is defined on a Borel subset PT reg g (1 g − ) ⊂ PT g of full Lebesguemeasure and is moreover Borel measurable on its domain of definition. The latter assertion is a consequenceof a stronger result, namely that PT reg g (1 g − ) → QT g is continuous (with respect to the subspace topologyon PT reg g (1 g − )).While convergence of measured laminations (in measure) does not typically imply Hausdorff convergence ofthe supports, whenever a sequence { λ n } of maximal measured laminations converges to a maximal measuredlamination λ , then λ n is eventually carried (snugly) on a maximal train track also carrying λ . From here,it’s not difficult to deduce that λ n → λ in the Hausdorff topology [ZB04]. Intuitively, the leaves of λ n intersect the leaves of λ with small angle (depending on the specific surface on which they are realized), sothe orthogonal directions become more parallel, and the dual horocylic foliations converge as n → ∞ .In forthcoming work [CF], the authors extend these ideas and prove that O is (everywhere) Borel measur-able with Borel measurable inverse by identifying a countable partition of PT g and QT g into Borel subsetson which O is homeomorphic. See also Section 16. AARON CALDERON AND JAMES FARRE
In general, the compact edges of the spine of a pair (
X, λ ) correspond exactly to horizontal saddle connec-tions in the differential O ( X, λ ). This observation allows us to prove that the generic point for a P -invariantergodic probability measure on P M g consists of pairs ( X, λ ) where λ cuts X into a fixed set of regular idealpolygons. Proof of Corollary 1.1.
Using our conjugacy, the desired statement is equivalent to the fact that any P -invariant ergodic probability measure on Q M g is (a) supported in a single stratum and (b) gives 0 measureto the set of differentials with horizontal saddle connections.The first statement is implied by ergodicity, while the second follows from the fact that the support L ofthe measure is cut out by affine equations in period coordinates with real coefficients. Indeed, in any fixedperiod chart the set of differentials with some horizontal saddle connection cuts out a countable union ofhyperplanes (since the set of possible saddle connections is countable). Since their defining equations are ofthe form Re( z ) = 0, these hyperplanes are all transverse to the subspace corresponding to L . Thus the setof differentials in L with no horizontal saddle connections has full measure; pulling back along the conjugacyfinishes the proof of the corollary. (cid:3) Refining the proof by considering connected components of strata, we see that we can also conclude that ν -almost every pair has the same orientability, spin, and hyperellipticity properties.2.1. Shear-shape coordinates.
Our strategy to prove Theorem D follows Mirzakhani’s intepretation ofTheorem 1.4, in which she clarifies the relationship between Thurston’s geometric perspective on the horo-cyclic foliation and Bonahon’s powerful analytic approach in terms of transverse cocycles. Namely, she showsthat the horocyclic foliation map F λ is compatible with shearing coordinates for both hyperbolic structuresand measured foliations. To motivate our construction, we give a brief outline of Mirzakhani’s proof below.A (real-valued) transverse cocycle for λ is a finitely additive signed measure on arcs transverse to λ that is invariant under isotopy transverse to λ ; observe that transverse measures are themselves transversecocycles. These objects equivalently manifest as transverse H¨older distributions, cohomology classes, orweight systems on snug train tracks [Bon97a, Bon96, Bon97b]. The space H ( λ ) of transverse cocycles formsa finite dimensional vector space which carries a natural homological intersection pairing which is non-degenerate when λ is maximal. The intersection pairing then identifies a “positive locus” H + ( λ ) ⊂ H ( λ ) cutout by finitely many geometrically meaningful linear inequalities. See also Section 7.1.In [Bon96, Theorem A], Bonahon proved that for any maximal geodesic lamination λ there is a real-analytic homeomorphism σ λ : T ( S ) → H + ( λ ) that takes a hyperbolic metric to its “shearing cocycle,” whichessentially records the signed distance along λ between the centers of ideal triangles in the complement of λ .Mirzakhani then constructed a homeomorphism I λ (essentially by a well-chosen system of period coordinates)that coordinatizes MF ( λ ) by H + ( λ ) and for which the following diagram commutes [Mir08, §§ T ( S ) MF ( λ ) H + ( λ ) F λ σ λ I λ Since F λ = I λ − ◦ σ λ is a composition of homeomorphisms, it is itself a homeomorphism. As the constructionof the horocyclic foliation requires no choices, the family { F λ } is necessarily Mod( S )–equivariant. Finally,a direct computation shows that σ λ transports the earthquake in λ to translation in H + ( λ ) by λ , and I λ similarly takes horocycle to translation, demonstrating Theorem 1.4. Shear-shape cocycles.
When λ is not maximal, the space of transverse cocycles is no longer suitable tocoordinatize hyperbolic structures (or transverse foliations). Indeed, in this case the vector space H ( λ ) hasdimension less than 6 g − S \ λ now has a rich analytic structure that transverse cocycles cannot see.In order to imitate Diagram (1) and its concomitant arguments for arbitrary λ ∈ ML ( S ), we thereforeintroduce the notion of shear-shape cocycles on λ . Roughly, a shear-shape cocyle consists of finitely additivesigned data on certain arcs transverse to λ together with a weighted arc system that cuts S \ λ into cells;this pair is also required to satisfy a certain compatibility condition mimicking features of the orthogeodesicfoliation (Definition 7.12). Generalizing results of Luo [Luo07, Theorem 1.2 and Corollary 1.4], we showthat such an arc system is equivalent to a hyperbolic structure on S \ λ (Theorem 6.4), so shear-shape HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 7 cocycles may equivalently be thought of as transverse data together with a compatible hyperbolic structureon the complementary subsurface(s). Like transverse cocycles, shear-shape cocycles also admit realizationsas cohomology classes or weight systems on certain train tracks (Definition 7.5 and Proposition 9.5).
Remark 2.3.
We note that only certain classes of arcs admit consistent weights when measured by ashear-shape cocycle, whereas transverse cocycles provide a measure to any arc transverse to λ . While thissubtlety is exactly what allows us to understand how to relate shear-shape cocycles with the geometry ofcomplementary subsurfaces, it is also a technical nuisance throughout the paper.Unlike transverse cocycles, the space SH ( λ ) of shear-shape cocycles is not a vector space, instead forminga principal H ( λ ) bundle over a contractible analytic subvariety of T ( S \ λ ) (Theorem 8.1). All the same, thecohomological realization of shear-shape cocycles equips SH ( λ ) with an intersection form ω SH : SH ( λ ) × H ( λ ) → R that identifies a “positive locus” SH + ( λ ) and equips both SH ( λ ) and SH + ( λ ) with piecewise-integral-linearstructures. The positive locus forms a H + ( λ ) cone-bundle over the same subvariety of T ( S \ λ ) (Proposition8.5) and fits into a familiar-looking commutative diagram:(2) T ( S ) MF ( λ ) SH + ( λ ) O λ σ λ I λ where σ λ and I λ record shearing data along λ as well as shape data in the complementary subsurfaces.These maps can be thought of as a common generalization of Bonahon and Mirzakhani’s shear coordinatesas well as Fenchel–Nielsen and Dehn–Thurston coordinates adapted to a pants decomposition (see Section2.2). In the case when λ is orientable, the map I λ can also be viewed as an extension of Minsky and Weiss’sdescription of the set of Abelian differentials with given horizontal foliation [MW14, Theorem 1.2]. The conjugacy of Theorem D is then a consequence of the following structural theorem, which is anamalgam of the main technical results of the paper (compare Theorems 10.15, 12.1, and 13.13).
Theorem E.
For any measured lamination λ , Diagram (2) commutes and all arrows are Stab( λ )–equivarianthomeomorphisms. Moreover, • σ λ is (stratified) real-analytic and transports the earthquake flow to translation by λ and the hyper-bolic length of λ to ω SH ( · , λ ). • The weighted arc system underlying σ λ ( X ) records the hyperbolic structure X \ λ under the corre-spondence of Theorem 6.4. • I λ is piecewise-integral-linear and transports horocycle flow to translation by λ and intersection with λ to ω SH ( · , λ ). • The weighted arc system underlying I λ ( η ) records the compact horizontal separatrices of q ( η, λ ).In the course of our proof, we also describe new “shape-shifting deformations” of hyperbolic surfaceswhich generalize Bonahon and Thurston’s cataclysms by shearing along a lamination while also varying thehyperbolic structures on complementary pieces. See Section 15.1.One particularly interesting family of deformations is obtained by dilation. The space SH + ( λ ) admitsa natural scaling action by R > , and since both earthquake and horocycle flow are carried to translationin coordinates, this scaling action indicates extensions of each to P actions. A quick computation (Lemma11.1) shows that the pullback of a dilation ray by I λ is (a variant of) the Teichm¨uller geodesic flow, so the P action on the flat side is just the standard P action on QT g .On the hyperbolic side, these dilation rays define our extension of the earthquake flow, and correspondto families of hyperbolic metrics on which the length of λ is scaled by a uniform factor. They are thereforenatural candidates for (directed, unit-speed) geodesics for the Lipschitz asymmetric metric on T ( S ), and insome cases we can identify them as such (see Propositions 15.12 and 15.17, as well as Remark 15.18). Technically, [MW14] investigates the family of Abelian differentials with a fixed horizontal foliation and fixed topologicaltype of horizontal separatrix diagram, whereas our map applies to quadratic differentials (whether or not they are globally thesquare of an Abelian differential) and packages together all possible types of separatrix diagrams.
AARON CALDERON AND JAMES FARRE
Remark 2.4.
Over the course of the paper we formalize the notion that shear-shape coordinates for hyper-bolic structures are essentially the “real part” of period coordinates for PT g . Interpreting σ λ ( X ) + iλ as acomplex weight system on a train track, Theorem C implies that the support of every stretchquake-invariantergodic measure on P M g is locally an affine measure in train track charts. See Lemma 10.10. Coordinatizing horospheres.
Since the Thurston intersection form ω SH captures both the hyperboliclength of and geometric intersection with λ , the coordinate systems of Theorem E also allow us to giveglobal descriptions of the level sets of these functions. In particular, we can recover Gardiner and Masur’sdescription of extremal length horospheres [GM91, p.236] as well as Bonahon’s description of the hyperboliclength ones (which is implicit in the structure of shear coordinates for maximal completions). Corollary 2.5.
Suppose that λ supports k ergodic transverse measures λ , . . . , λ k . Then for all L , . . . , L k ∈ R > , the level sets { X ∈ T ( S ) | (cid:96) X ( λ i ) = L i for all i } and { η ∈ MF ( λ ) | i ( η, λ i ) = L i for all i } are both homeomorphic to R g − − k .Analyzing this coordinatization more closely, we see that in fact both level sets can be described as affinebundles of dimension dim R H + ( λ ) − k over the same subvariety of T ( S \ λ ) as underlies SH ( λ ).From this refinement we are able to describe the intersection of F uu ( λ ) of the Teichm¨uller geodesicflow with strata. The decomposition of period coordinates into real and imaginary parts shows that thisintersection (when not empty) is locally homeomorphic to R d , where d is the complex dimension of thestratum; our work shows that these local homeomorphisms patch together to a global one. Compare [MW14,Theorem 1.2]. Corollary 2.6.
Suppose that λ is a filling measured lamination that cuts a surface into polygons with κ + 2 , . . . , κ n + 2 many sides, and let ε = +1 if λ is orientable and − QT g ( κ ; ε ) denote theunion of the components of the stratum QT g ( κ ) ⊂ QT g that either are ( ε = +1) or are not ( ε = −
1) globalsquares of Abelian differentials. Then { q ∈ QT g ( κ ; ε ) | | Im( q ) | = λ } ∼ = H + ( λ ) ∼ = R d where d is the complex dimension of QT g ( κ ; ε ). Proof.
The first homeomorphism is a consequence of our general structure theorem, while the second is justa dimension count (see Lemmas 4.6 and 7.3 in particular). (cid:3)
In general, we see that F uu ( λ ) ∩QT g ( κ ; ε ) forms a H + ( λ ) bundle over a union of faces of an arc complex of S \ λ . As a consequence we find that the only obstruction to completeness of any such leaf comes from zeroscolliding along a horizontal saddle connection (see also [MW14, Theorem 11.2]). This global descriptionof F uu ( λ ) ∩ QT g ( κ ; ε ) also allows the import of arguments from homogeneous dynamics to investigateequidistribution in both Q M g and P M g and their strata; see the discussion in Section 16.2.2. Generalized Fenchel–Nielsen coordinates.
Our shear-shape coordinates for hyperbolic structurescan be thought of as interpolating between the classical Fenchel–Nielsen coordinates adapted to a pantsdecomposition and Bonahon and Thurston’s shear coordinates. In both cases, one remembers the shapes ofthe complementary subsurfaces (pairs of pants and ideal triangles, respectively) and the space of all hyperbolicstructures with given complementary shapes is parametrized by gluing data (twist/shear parameters).For general λ , there is a map cut λ : T ( S ) → T ( S \ λ )that remembers the induced hyperbolic structure on each complementary subsurface. Theorem 12.1 thenimplies that the image of cut λ is a real-analytic subvariety B ( S \ λ ) of T ( S \ λ ) consisting of those structuressatisfying a “metric residue condition” (see Lemma 13.1). In the case where each component of λ is eithernon-orientable or a simple closed curve, B ( S \ λ ) is just the space of hyperbolic structures for which the twoboundary components of the cut surface corresponding to a simple curve component of λ have equal length.Theorem 12.1 together with the structure of SH + ( λ ) also allows us to identify the fiber cut − λ ( Y ) over any Y ∈ B ( S \ λ ) with the gluing data H + ( λ ) (though not in a canonical way). See the discussion around (18) in regards to the positivity condition for disconnected λ ; in essence, H + ( λ ) is the productof H + ( λ i ) for each non-closed minimal component together with the twisting data around simple closed curves. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 9
We summarize this discussion in the following triptych:(3)
Fenchel–Nielsen Shear-shape Shear R g − T ( S ) R g − > H + ( λ ) T ( S ) B ( S \ λ ) H + ( λ ) T ( S ) { pt } λ a pants decomposition λ arbitrary λ maximalIn each coordinate system, T ( S ) is the total space of a fiber bundle over a base space of allowable shapedata on the subsurface complementary to λ , while the fiber consists of gluing data.A completely analogous picture also holds for foliations transverse to λ , demonstrating I λ as a commongeneralization of both Dehn–Thurston and Mirzakhani’s shear coordinates.2.3. Fenchel–Nielsen and Dehn–Thurston via shears and shapes.
In order to give the reader aconcrete example of shear-shape coordinates, we include here a discussion of our construction for λ = P apants decomposition. In this case, we see that shear-shape coordinates are just a (mild) reformulation of theclassical Fenchel–Nielsen and Dehn–Thurston ones.First we consider a hyperbolic structure X . A pair of pants in X \ P is typically parametrized byits boundary lengths ( a, b, c ), or equivalently, by the alternating side lengths of either of the right angledhexagons coming from cutting along seams. The orthogeodesic foliation on a pair of pants picks out eithera pair or triple of seams (those which are realized as leaves of O P ( X )), each weighted by the length of aboundary arc consisting of endpoints of leaves of O P ( X ) isotopic to the seam. In this case, these lengthsare just simple (piecewise) linear combinations of the boundary lengths and the metric residue conditiondefining B ( S \ P ) just states that the boundaries that are glued together must have the same length. SeeFigure 1. a bc a + c − b b + c − a a + b − c a bc a b c − a − b Figure 1.
The orthogeodesic foliation on pairs of pants. Note that the weight of eachbolded arc is a linear combination of the boundary lengths, hence the correspondence be-tween shear-shape and Fenchel–Nielsen/Dehn–Thurston coordinates.The space H + ( P ) reduces to a sum of the twist spaces for each curve of P , and so Theorem 8.5 impliesthat SH + ( P ) is a principal R g − bundle over B ( S \ P ) ∼ = R g − > . The transverse data recorded by this twistspace then describes the signed distance between certain reference points in pairs of right-angled hexagonsin (cid:101) X that are adjacent to the same curve of ˜ P , which is the same as the twist parameter measured by theappropriate choice of Fenchel–Nielsen coordinates. We can similarly recognize I λ : MF ( P ) → SH + ( P ) as Dehn–Thurston coordinates. Note first that thereare no essential simple closed curves in the complement of P , so MF ( P ) is simply the space of measuredfoliations not contained in the support of P . Now from any integral point σ ∈ SH + ( P ) we can constructa multicurve α with prescribed intersection and twisting parameters as follows: the weighted arc systemdescribes how strands of α pass between and meet the components of P , while the transverse data recorded Fenchel–Nielson coordinates always involve some choice of section of the space of twists over the length parameters, and sohave only the structure of a principal R g − bundle over R g − . by H + ( P ) ∼ = R P describes the extent that strands of α wrap around components of P . This procedure isclearly reversible and can easily be extended to transverse foliations using a family of standard train trackson each pair of pants (see [PH92, § α going fromone boundary to the other with the total intersection of α with each boundary.3. Outline of the paper
The rest of this paper is roughly divided into four parts, corresponding to the orthogeodesic foliation,shear-shape cocycles, and shear-shape coordinates for flat and hyperbolic structures, as well as a collectionof further directions for investigation (Section 16). While the constructions of I λ and σ λ both rely onfoundational results established in the first two parts, we have attempted to direct the reader eager tounderstand our coordinates to the most important statements of these sections. §§ Cutting along a lamination results in a (possibly disconnected)hyperbolic surface Σ with crown boundary, and in Section 4 we recall some useful information about theTeichm¨uller spaces of such surfaces. One particularly important definition is that of the “metric residue”of a crown end, which is a generalization of boundary length and plays an important role in cohomologicalconstraints on the shape data of shear-shape cocycles (Lemma 7.9).With these preliminaries established, in Section 5 we discuss in more detail the orthogeodesic foliation andthe hyperbolic geometry of X in a neighborhood of λ . In this section we also give a geometric interpretationof the map in Corollary 1.5 that relates hyperbolic and extremal length.The most important result of this part occupies Section 6, in which we show that the orthogeodesic foliationrestricted to Σ completely determines its hyperbolic structure. More explicitly, dual to each compact edgeof the spine of O λ ( X ) is a packet of properly isotopic arcs joining non-asymptotic boundary components ofΣ. By assigning geometric weights to each of these packets we can therefore combinatorialize the restrictionof O λ ( X ) to Σ by a weighted, filling arc system.Using a geometric limit argument, in Theorem 6.4 we prove that the map which associates to a hyperbolicstructure on Σ the associated arc system is a Mod(Σ)–equivariant stratified real-analytic homeomorphismbetween T (Σ) and a certain type of arc complex for Σ, generalizing a theorem of Luo [Luo07] for surfaces withtotally geodesic boundary (see also [Mon09b, Do08, Ush99]). Moreover, by construction this map recordsboth the combinatorial structure of the spine of O λ ( X ) as well as the metric residue of the crowns of Σ.Theorem 6.4 is used extensively throughout the paper in order to pass between the combinatorial dataof a weighted arc system, the restriction of O λ ( X ) to Σ, and the corresponding hyperbolic structure on Σ.The proof is independent of the main line of argument; as such, the reader is encouraged to understand thestatement, but may wish only to skim the proof. §§ The second part of the paper is devoted to our constructionof shear-shape cocycles for a given λ and an analysis of the space SH ( λ ) of all shear-shape cocycles. Uponreaching this section, the reader may find it useful to glance ahead to either Section 10 or 13 to instantiateour definitions.After reviewing structural results on transverse cocycles, in Section 7 we give both cohomological andaxiomatic definitions of shear-shape cocycles (Definitions 7.5 and 7.12, respectively), both predicated onsome underlying weighted arc system on Σ. In Proposition 7.14 we prove these definitions agree. Usingthe cohomological description, we observe a constraint on the weighted arc systems that can underlie ashear-shape cocycle coming from metric residue conditions (Lemma 7.9); this can also be thought of as ageneralization of the fact that one can only glue together totally geodesic boundary components of the samelength (compare Lemma 13.1).Letting B ( S \ λ ) denote the subvariety of the filling arc complex of Σ cut out by the aforementionedresidue conditions, we show in Section 8 that the space SH ( λ ) of shear-shape cocycles forms a bundle oftransverse cocycles over B ( S \ λ ) with some additional structure (Theorem 8.1) whose total space is a cellof dimension 6 g − SH ( λ ) (Section 8.2) and prove that the positive locus SH + ( λ ) it defines is itself a bundle over B ( S \ λ )(Proposition 8.5).Finally, in Section 9 we give train track coordinates for the space of shear-shape cocycles. The traintracks we use give a preferred decomposition of arcs on S into pieces that are measurable by shear-shape HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 11 cocycles and as such give a useful way of specifying shear-shape cocycles by a finite amount of data. Theweight space for a train track is also a natural model in which to consider local deformations of a shear-shapecocycle, a feature which we exploit in Section 14. In Section 9.3 we discuss how the PIL structure inducedby train track charts endows SH + ( λ ) with a well defined integer lattice and preferred measure in the classof Lebesgue.The reader willing to accept the structure theorems can adequately navigate the remaining two partsof the paper using weight systems on (augmented) train tracks as a local description of the structure ofshear-shape space. §§
10 and 11: Coordinates for transverse foliations.
At this point, we have established the structurenecessary to coordinatize foliations transverse to λ by shear-shape cocycles.A measured foliation η ∈ MF ( λ ) determines a holomorphic quadratic differential q = q ( η, λ ) ∈ F uu ( λ )via Theorem 2.1, and we begin by specifying an arc system α ( q ) that records the horizontal separatricesof q . We then build a train track τ carrying λ from a triangulation by saddle connections (Construction10.4); augmenting τ by the arc system α ( q ) then allows us to realize the periods of the triangulation asa (cohomological) shear-shape cocycle I λ ( η ). This identification also gives a useful formula for I λ ( η ) as aweight system on the augmented train track τ (Lemma 10.10).We then show that one can rebuild q just from the train track weights defined by I λ ( η ); a similar (butmore technical) argument then gives that I λ ( η ) ∈ SH + ( λ ) (Proposition 10.12). This reconstruction techniquetogether with the structure of shear-shape space therefore allows to deduce that I λ is a homeomorphism ontoits image. At the end of this section, we explain how the work done in the fourth and final part of the paperimplies that I λ surjects onto SH + ( λ ) (Theorem 10.15), and why we choose to prove surjectivity this way.See Remark 10.16 in particular.Since I λ essentially yields period coordinates, it is not surprising that (a variant of) Teichm¨uller geodesicflow is given in coordinates by dilation (Lemma 11.1), while the Teichm¨uller horocycle flow is translation by λ (Lemma 11.2). We also naturally recover the “tremor deformations” introduced in [CSW20] as translationby measures µ supported on λ that are not necessarily absolutely continuous with respect to λ (Definition11.3). Figure 15 details a dictionary between the language of [CSW20] and our own. §§ In the final part of the paper, we use the geometryof the orthogeodesic foliation to coordinatize hyperbolic structures via shear-shape cocycles.From Theorem 6.4, we know that the combinatorialization of O λ ( X ) on each subsurface S \ λ by a weightedarc system completely encodes the geometry of the pieces. Cutting X \ λ further along the orthogeodesicrealization of each such arc, we obtain a family of (partially ideal) right-angled polygons. The orthogeodesicfoliation equips each polygon with natural a family of basepoints, one on each of its sides adjacent to λ ,that vary analytically in T ( S \ λ ). We are thus able to define a “shear” parameter between (some pairs of)degenerate polygons, and this shear data assembles together with the “shape” data on each subsurface togive instructions for gluing the polygonal pieces back together to obtain X .In Section 12 we state the main Theorem 12.1, that the shear-shape coordinate map σ λ : T ( S ) → SH + ( λ )is a homeomorphism, supply an outline of its proof, and derive some immediate corollaries. The constructionof σ λ is given in Section 13, where we formalize the discussion from the previous paragraph. We also provethat the central Diagram (2) commutes (Theorem 13.13), which then implies that σ λ takes hyperbolic lengthto the Thurston intersection form (Corollary 13.14).Section 14 is the most technical part of the paper. In it, we define the “shape-shifting” cocycles (Propo-sition 14.26) along which a hyperbolic structure can be deformed (Theorem 15.1); these deformations aregeneralizations of Thurston’s cataclysms or Bonahon’s shear deformations. Although the construction ofa shape-shifting deformation is rather involved, we attempt to keep the reader informed of the geometricintuition that guides the construction throughout. Finally, in Section 15 we assemble all of the necessaryingredients to prove Theorem 12.1. That the earthquake along λ is given by translation by λ in SH + ( λ )(Corollary 15.2) is an immediate consequence of the construction of shape-shifting deformations as general-izations of cataclysms. We then discuss how action of dilation in coordinates can sometimes be identifiedwith directed geodesics in Thurston’s asymmetric metric (Propositions 15.12 and 15.17). Crowned hyperbolic surfaces
When a hyperbolic surface is cut along a geodesic multicurve, the (completion of the) resulting space isa compact hyperbolic surface with compact, totally geodesic boundary. When the same surface is cut alonga geodesic lamination, the (completion of the) complementary subsurface can have non-compact “crownedboundaries.” This section collects results about hyperbolic structures on such “crowned surfaces” as well asthe relationship between properties of the lamination and the topology of its complementary subsurfaces.
Remark 4.1.
Throughout this section and the following, we reserve S to denote a closed surface. If λ is ageodesic lamination, then S \ λ denotes the metric completion of the complementary subsurfaces to λ (withrespect to some auxiliary hyperbolic metric); we will refer to the topological type of a component of S \ λ by Σ. Hyperbolic metrics on S and Σ will be denoted by X and Y , respectively. Hyperbolic crowns.
While less familiar than surfaces with boundary, crowned hyperbolic surfaces natu-rally arise by uniformizing surfaces with boundary and marked points on the boundary. They are also intri-cately related to meromorphic differentials on Riemann surfaces with high order poles (see, e.g., [Gup17]).A hyperbolic crown with c k spikes is a complete, finite-area hyperbolic surface with geodesic boundary thatis homeomorphic to an annulus with c k points removed from one boundary component. In the hyperbolicmetric, the circular boundary component corresponds to a closed geodesic and each interval of the otherboundary becomes a bi-infinite geodesic running between ideal vertices; compare Figure 2.In general, a hyperbolic surface with crowned boundary is a complete, finite-area hyperbolic surface withtotally geodesic boundary; the boundary components are either compact or hyperbolic crowns. We recordthe topological type of a crowned surface of genus g with b closed boundary components and k crowns with c , . . . , c k many spikes as Σ { c } g,b , where { c } = { c , . . . , c k } . Remark 4.2.
Ideal polygons may be considered as crowned surfaces of genus 0 with a single (crowned)boundary component. All of the results in this section hold for both crowned surfaces with nontrivialtopology as well as for ideal polygons, but their proofs are slightly different. Our citations of [Gup17] are allfor the case when Σ is not an ideal polygon; for the corresponding statements for ideal polygons, see [Gup17,Section 3.3] or [HTTW95].Every crowned surface Y with non-cyclic (and non-trivial) fundamental group contains a “convex core”obtained by cutting off its crowns along a geodesic multicurve [CB88, Lemma 4.4]. When Y has type Σ { c } g,b ,this core is a subsurface of genus g with b + k closed boundary components. Since each crown with c i spikesmay be decomposed into c i ideal hyperbolic triangles, we have the following expression for the area:(4) 1 π Area ( Y ) = 4 g − b + k (cid:88) i =1 ( c i + 2) . Note that one can triangulate an ideal polygon of c sides into ( c −
2) ideal triangles, and so the above formulaalso holds for ideal polygons.
The metric residue.
While crown ends (and ideal polygons) do not have well-defined boundary lengths,one can define a natural generalization when there are an even number of spikes. This turns out to be afundamental invariant that controls when crowns can be glued together along a lamination (Lemma 13.1).Let C be a hyperbolic crown or an ideal polygon with c spikes, where c is even. One can then orient C , that is, pick an orientation of the boundary leaves so that the orientations of asymptotic leaves agree.Truncating each spike of C along a horocycle based at the tip of the spike yields a surface with a boundarymade up of horocyclic segments h , . . . , h c and geodesic segments g , . . . , g c . See Figure 2. Definition 4.3 (Definition 2.9 of [Gup17]) . Let C be either an oriented hyperbolic crown or an orientedideal polygon with an even number of spikes. Then its metric residue res( C ) isres( C ) = c (cid:88) i =1 ε i (cid:96) ( g i )where ε i is positive if C lies on the left of g i and negative if C lies on the right. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 13 g h g h Figure 2.
Truncating an (oriented) crown to compute its metric residue.Since changing the truncation depth of a spike increases the length of two adjacent sides, the metricresidue evidently does not depend on the choice of truncation [Gup17, Lemma 2.10]. Observe also thatflipping the orientation of C flips the sign of its metric residue.Similarly, define the metric residue of an oriented totally geodesic boundary component β of Y to be ± (cid:96) ( β ), where the sign depends on whether Y lies to the left of β (positive) or right (negative). Deformation spaces of crowned surfaces.
We now record some useful facts about the Teichm¨ullerspaces of crowned hyperbolic surfaces.Given any crowned hyperbolic surface Y , one can obtain a natural compactification (cid:98) Y by adding on anideal vertex at the end of each spike of each crown. The corresponding (topological) surface (cid:98) Σ { c } g,b then has b + k boundary components with c i marked points on the ( b + i ) th boundary component. A marking of acrowned hyperbolic surface Y is a homeomorphism f : (cid:98) Σ { c } g,b → (cid:98) Y which takes boundary marked points to ideal vertices. We think of the boundary marked points as havingdistinct labels, so different identifications of the boundary points of (cid:98) Σ { c } g,b with the spikes of Y yield differentmarkings. The Teichm¨uller space of a crowned hyperbolic surface Σ { c } g,b is then defined to be the space ofall marked hyperbolic metrics on Σ { c } g,b , up to isotopies which fix the totally geodesic boundary componentspointwise and fix each ideal vertex of each crown.As noted above, any crowned hyperbolic surface Σ { c } g,b contains an uncrowned subsurface which serves asits convex core. Therefore, the Teichm¨uller space of a crowned hyperbolic surface may be parametrizedby the Teichm¨uller space of its convex core together with parameters describing each crown and how it isattached. A precise version of this dimension count is recorded below. Lemma 4.4 (Lemma 2.16 of [Gup17]) . Let Σ = Σ { c } g,b be a crowned hyperbolic surface or an ideal polygon.Then T (Σ) ∼ = R d , where(5) d = 6 g − b + k (cid:88) i =1 ( c i + 3) . Fixing the length of any closed boundary component of Σ { c } g,b cuts out a codimension 1 subvariety of T (Σ).Similarly, the subspace of surfaces with fixed metric residues at an even–spiked crown has codimension one.The following proposition ensures that the intersections of the level sets of length and metric residue aretopologically just cells of the proper dimension: Proposition 4.5 (Corollary 2.17 in [Gup17]) . Let Σ = Σ { c } g,b be a crowned surface or an ideal polygon.Let β , . . . , β b denote the closed boundary components of Σ and let C , . . . , C e denote the crown ends whichhave an even number of spikes. Fix an orientation of each crown end. Then for any ( L i ) ∈ R b> and any ( R j ) ∈ R e , { ( Y, f ) ∈ T (Σ) | (cid:96) ( β i ) = L i and res( C j ) = R j for all i, j } ∼ = R d − b − e where d is as in (5). Topology.
When a crowned surface Σ comes from cutting a closed surface S along a geodesic lamination λ , we can relate the topology of λ to the topology of Σ.Recall that the Euler characteristic of a lamination is defined to be alternating sum of the ranks of its ˇCechcohomology groups, viewing λ as a subset of S . Below, we compute the Euler characteristic of a geodesiclamination in terms of the topological type of its complementary subsurfaces. Lemma 4.6.
Let λ be a geodesic lamination on S . Then the total number of spikes of S \ λ equals − χ ( λ ).We also record the corresponding formula for later use. Suppose that S \ λ = Σ ∪ . . . ∪ Σ m ; then(6) χ ( λ ) = − m (cid:88) j =1 k j (cid:88) i =1 c ji where { c j , . . . , c jk j } denotes the crown type of Σ j . Proof.
Fix some train track τ which snugly carries λ . Lemma 13 of [Bon97b] states that χ ( λ ) = χ ( τ ), andso it suffices to compute the Euler characteristic of τ .Splitting the switches of τ if necessary, we may assume that τ is trivalent (observe that this operationpreserves the Euler characteristic). Then each spike of S \ λ corresponds to a unique switch of τ , and eachswitch corresponds to three half–edges, so S \ λ ) = τ ) = 23 · τ ) . Plugging this into the formula χ ( τ ) = τ ) − τ ) proves the claim. (cid:3) In general, the relationship between the boundary components of S \ λ and λ can be rather involved.For example, one can construct a lamination on a closed surface of genus g ≥ λ and totally geodesic boundarycomponents of its complementary subsurface.When λ also supports a measure of full support, however, things become much nicer. In particular, eachcomponent of λ is minimal (in that every leaf is dense in the component) and so the closed leaves of λ areall isolated. In this case, there is a natural 1-to-2 correspondence between closed leaves of λ and totallygeodesic boundary components of S \ λ . N.B.
So that we do not have to deal with possible spiraling behavior of λ , we henceforth restrict ourdiscussion to those laminations that support a measure. See also Remark 7.10.5. The orthogeodesic foliation
In this section we construct the orthogeodesic foliation O λ ( X ) ∈ MF ( λ ) of a hyperbolic surface X withrespect to λ and describe some of its basic properties.5.1. The spine of a hyperbolic surface.
We begin by describing the othogeodesic foliation restricted tosubsurfaces Y complementary to λ . Let Y be a finite area hyperbolic surface with totally geodesic boundary,possibly with crowned boundary. As we are most interested in the Y coming from cutting a closed surfacealong a lamination, we also assume that Y has no annular cusps. Definition 5.1.
The orthogeodesic foliation O ∂Y ( Y ) of Y is the (singular, piecewise-geodesic) foliation of Y whose leaves are fibers of the closest point projection to ∂Y .Near ∂Y , the leaves of O ∂Y ( Y ) are geodesic arcs meeting ∂Y orthogonally. To understand the globalstructure of the foliation, however, we need to determine how the leaves extend into the interior of Y . Inparticular, we must understand the locus of points that are closest to multiple points of ∂Y .To that end, for any point x ∈ Y , define the valence of x to beval( x ) := { y ∈ ∂Y : d ( x, y ) = d ( x, ∂Y ) } . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 15
The (geometric) spine Sp ( Y ) of Y is the set of points of Y with valence at least 2, and has a natural partitioninto subsets Sp k ( Y ), where x ∈ Sp k ( Y ) if it is equidistant from exactly k points in ∂Y . For the rest of thesection we fix a hyperbolic surface Y and refer to Sp ( Y ) and Sp k ( Y ) simply as Sp and Sp k .It is not hard to see that Sp is a properly embedded, piecewise geodesic 1-complex with some nodes ofvalence 1 removed (equivalently, a ribbon graph with some half-infinite edges). Indeed, Sp decomposes intoa finite core Sp and a finite collection of open geodesic rays; since we assumed Y had no annular cusps, eachray correspondes with a spike of a crowned boundary component. See [Mon09b, Section 2] for a discussionof the structure of the spine of a compact hyperbolic surface with geodesic boundary in which Sp = Sp .We record below a summary of this discussion. Lemma 5.2.
The finite core Sp is a piecewise geodesically embedded graph, whose edges correspond tothe components of Sp with finite hyperbolic length and vertex set ∪ k ≥ Sp k . Each geodesic ray of Sp \ Sp exits a unique spike of Y .By definition, the orthogeodesic foliation O ∂Y ( Y ) has k -pronged singular leaves emanating from ∪ k ≥ Sp k for k ≥
3. The nonsingular leaves of O ∂Y ( Y ) glue along Sp ( Y ) (usually at an angle) and can be smoothedby an arbitrarily small isotopy supported near Sp . As the geometry of Sp interacts nicely with the leaves of O ∂Y ( Y ), we generally prefer to think about O ∂Y ( Y ) as a piecewise geodesic singular foliation rather than asa smooth one. When convenient, we will pass freely between the orthogeodesic foliation and a smoothing.We observe that there is also an isotopy supported in the ends of the spikes of Y and restricting to theidentity on ∂Y that maps leaves of the orthogeodesic foliation to horocycles joining based at the tip of thespike. This equivalence between the orthogedesic and horocyclic foliations in spikes is of vital importance inSections 13–15 as it allows us to adapt many of Bonahon and Thurston’s arguments to this setting. Remark 5.3.
One can check that, for regular ideal polygons, the isotopy in spikes extends to a globalisotopy between the orthogeodesic foliation and the symmetric partial foliation by horocycles.Following the leaves of the orthogeodesic foliation in the direction of Sp defines a deformation retractionof Y onto Sp ; let r : Y → Sp be the retraction at time 1. For x and y in the same component of Sp , theleaves r − ( x ) and r − ( y ) of O ∂Y ( Y ) are properly isotopic. We may therefore associate to each edge e of Sp the (proper) isotopy class of r − ( x ) for x ∈ e ; we call this the dual arc α e to e .There is a distinguished representative of α e that is geodesic and orthogonal to both ∂Y and e . By abuseof notation, we henceforth identify α e with its orthogeodesic representative and define α ( Y ) := (cid:91) e ⊂ Sp α e . Lemma 5.4.
The metric completion of the surface with corners Y \ α ( Y ) is homeomorphic to a union ofclosed disks and closed disks with finitely many points on the boundary removed. That is, α ( Y ) fills Y . Proof.
Each component of Y \ α ( Y ) deformation retracts onto a component of the metric completion of Sp \ α ( Y ). By the duality of arcs and edges of Sp , each component of Sp \ α ( Y ) is contractible. (cid:3) The orthogeodesic foliation also comes with a natural transverse measure: the measure of an arc k transverse to (a smoothing of) O ∂Y ( Y ) is defined on small enough transverse arcs k first by isotoping the arcinto ∂Y transversely to O ∂Y ( Y ) and then measuring the hyperbolic length there. Extending to all transversearcs by additivity defines a transverse measure on O ∂Y ( Y ).To each component e of Sp we associate the length c e > r − ( e ) ∩ ∂Y ; thetransverse measure of e is exactly c e . Anticipating the contents of the next section (see, e.g., Theorem 6.4),we define the formal sum(7) A ( Y ) := (cid:88) e ⊂ Sp c e α e . The orthogeodesic foliation.
Now that we have described the orthogeodesic foliation on each com-ponent of S \ λ , we can glue these pieces together along the leaves of λ to get a foliation of S . Construction 5.5.
Let X ∈ T ( S ) and λ be a geodesic lamination on X . Cutting X open along λ takingthe metric completion of each component, we obtain a union of hyperbolic surfaces with totally geodesic boundary (possibly with crowned boundary). On each such component Y , we construct the orthogeodesicfoliation O ∂Y ( Y ) as described in Section 5.1 above.A standard fact from hyperbolic geometry [CEG06, Lemma 5.2.6] shows that the line field defined by (asmoothing of) the orthogeodesic foliation forms a Lipschitz line field on X \ λ . Since λ has measure 0, thisline field is integrable near λ , so the partial foliation defined on X \ λ extends across the leaves of λ . Thisdefines a measured foliation O λ ( X ) ∈ MF ( S ), hence a map O λ : T ( S ) → MF ( S )Later, we prove Lemma 5.8 that λ and O λ ( X ) bind, allowing us to restrict the codomain of O λ to MF ( λ ).Ultimately, our goal is to show that O λ is a homeomorphism onto MF ( λ ). Geometric train tracks.
We now consider the geometry of O λ ( X ) in a neighborhood of λ . Construction 5.6.
Let (cid:15) > (cid:15) -neighborhood N (cid:15) ( λ ) is topologically stable.The orthogeodesic foliation O λ ( X ) restricts to a foliation of N (cid:15) ( λ ) without singular points, and collapsingthe leaves yields a quotient map π : N (cid:15) ( λ ) → τ where τ can be C -embedded in N (cid:15) ( λ ) as a train trackcarrying λ in X . By changing (cid:15) , we may assume that τ is trivalent. Then τ = τ ( λ, X, (cid:15) ) is a geometrictrain track .We sometimes refer to N (cid:15) ( λ ) as a train track neighborhood of λ and the leaves of O λ ( X ) | N (cid:15) ( λ ) as ties . Werecall that N (cid:15) ( λ ) is said to be snug (and the corresponding τ snugly carries λ ) if λ meets every tie of τ andthere is no arc in N (cid:15) (() λ ) but disjoint from λ that joins two spikes to each other. Equivalently, λ and τ havethe same topological type. With this definition, it is clear that the geometric train tracks constructed abovealways carry λ snugly.Using the geometry of π : N (cid:15) ( λ ) → τ , the branches of τ admit a well defined notion of length. Indeed,let b ⊂ τ be a branch, and choose a lift (cid:101) b to (cid:101) X . Let (cid:96), (cid:96) (cid:48) ⊂ (cid:101) λ be leaves of λ that meet π − ( (cid:101) b ) ⊂ N (cid:15) ( (cid:101) λ ) insegments g and g (cid:48) . Since O λ ( X ) is equivalent to a horocyclic foliation in N (cid:15) ( λ ), transporting g along theleaves of (cid:94) O λ ( X ) near (cid:101) b onto g (cid:48) is isometric, so (cid:96) X ( g ) = (cid:96) X ( g (cid:48) ). We may therefore define the length of b (along λ ) as (cid:96) X ( b ) := (cid:96) X ( g )for any g as above. Similarly, for any branch b ⊂ τ , the ties of N (cid:15) ( λ ) collapsing to b all have the sameintegral with respect to λ . Define λ ( b ) := λ ( k )for any tie k ⊂ O λ ( X ) | π − ( b ) ; this is equivalently the weight deposited by λ on b in its τ train trackcoordinates. Lemma 5.7.
For any hyperbolic structure X and any measure λ (cid:48) on λ , we have i ( λ (cid:48) , O λ ( X )) = (cid:96) X ( λ (cid:48) ). Proof.
Using Construction 5.6, find a geometric train track π : N (cid:15) ( λ ) → τ snugly carrying λ on X . Bydefinition, the intersection pairing is given by the integral over X of the product measure dλ (cid:48) ⊗ d O λ ( X ),whose support is contained entirely in the train track neighborhood N (cid:15) ( λ ). For each branch b ⊂ τ , theintegral of this measure on π − ( b ) is just λ (cid:48) ( b ) (cid:96) X ( b ), so i ( λ (cid:48) , O λ ( X )) = (cid:90) (cid:90) X dλ (cid:48) ⊗ d O λ ( X )= (cid:90) (cid:90) N (cid:15) ( λ ) dλ (cid:48) ⊗ d O λ ( X ) = (cid:88) b ⊂ τ (cid:90) (cid:90) π − ( b ) dλ (cid:48) ⊗ d O λ ( X )= (cid:88) b ⊂ τ λ (cid:48) ( b ) (cid:96) X ( b ) . On the other hand, (cid:96) X ( λ (cid:48) ) is the integral over X of the measure dλ (cid:48) ⊗ dl λ (cid:48) , locally the product of thetransverse measure λ (cid:48) and 1-dimensional Lebesgue measure l λ (cid:48) on the support of λ (cid:48) . Since λ (cid:48) is supported in λ , the integral of dλ (cid:48) ⊗ dl λ (cid:48) is equal to the integral of dλ (cid:48) ⊗ dl λ , and again the support of the product measureis contained in N (cid:15) ( λ ). On each thickened branch π − ( b ) ⊂ N (cid:15) ( λ ), the integral of dλ (cid:48) ⊗ dl λ is λ (cid:48) ( b ) (cid:96) X ( b ),giving the equality (cid:96) X ( λ (cid:48) ) = (cid:88) b ⊂ τ λ (cid:48) ( b ) (cid:96) X ( b ) . In the literature, trivalent train tracks are also called “generic.”
HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 17
This completes the proof of the lemma. (cid:3)
With this computation, we can now show that λ and O λ ( X ) together bind S . Lemma 5.8.
For any X ∈ T ( S ) and λ ∈ ML ( S ), we have O λ ( X ) ∈ MF ( λ ). Proof.
Suppose that η is an ergodic measured lamination so that i ( η, λ ) = 0. Then one of two things must betrue: either η is supported on λ or its support is disjoint from λ . In the first case, i ( η, O λ ( X )) = (cid:96) X ( η ) > η is disjoint from λ then it is contained in a component Y of X \ λ , and we need only show that i ( η, O λ ( X )) >
0. To do this, we recall that length-normalized simple closed curves are dense in the sphere S ( X ) = { γ ∈ ML ( S ) : (cid:96) X ( γ ) = 1 } . By homogeneity and continuity of the intersection pairing it therefore suffices to find some uniform (cid:15) > i ( γ, O λ ( X )) ≥ (cid:15)(cid:96) X ( γ )for all simple closed curves γ ⊂ Y .Let Y be the convex hull of r − ( Sp ); Y is compact and the inclusion of Y into Y is a homotopyequivalence. Any simple closed geodesic γ in Y is contained in Y , and since Y deformation retracts ontothe component of Sp contained in Y , γ is homotopic to a concatenation of edges in Sp .Give Sp a metric making its edges e have length c e = i ( e, O λ ( X )); then the inclusion Sp → Y with thismetric induces an equivariant quasi-isometry on universal covers. The geodesic lengths of closed curves in Sp and in Y are therefore comparable, so that there is some (cid:15) > i ( λ, γ ) = (cid:96) Sp ( γ ) ≥ (cid:96) X ( γ ) (cid:15), demonstrating the desired uniform bound. (cid:3) Deflation.
For a given pair (
X, λ ) ∈ T ( S ) × ML ( S ), the pair of laminations O λ ( X ) and λ bind byLemma 5.8. By Theorem 2.1, there is a unique quadratic differential q = q ( O λ ( X ) , λ ), holomorphic on someRiemann surface Z , whose real and imaginary foliations are O λ ( X ) and λ , respectively. In this section wedefine a deflation map D λ : X → Z that allows us to make direct comparisons between the hyperbolicgeometry of X and the singular flat geometry q .An informal description of D λ is that it “deflates” the subsurfaces of X \ λ , retracting them to Sp alongthe leaves of O λ ( X ), while it “inflates” along the leaves of λ according to the transverse measure. Theorthogeodesic foliation in a neighborhood of λ assembles into the vertical foliation of the resulting quadraticdifferential metric and D λ maps Sp ⊂ X to the horizontal separatrices; compare Figure 3.( X, λ ) q ( O λ ( X ) , λ ) Figure 3.
Inflating a lamination and deflating its complementary components.
Proposition 5.9.
Given a marked hyperbolic structure [ f : S → X ] ∈ T ( S ) and λ ∈ ML ( S ), let[ g : S → Z ] ∈ T ( S ) be the marked complex structure on which q ( O λ ( X ) , λ ) is holomorphic. There is a map D λ : X → Z that is a homotopy equivalence restricting to an isometry between Sp with its metric induced by integratingthe edges against O λ ( X ) and the graph of horizontal saddle connections of q ( O λ ( X ) , λ ) with the inducedpath metric. Moreover, D λ ◦ f ∼ g and D λ ∗ O λ ( X ) = Re( q ) and D λ ∗ λ = Im( q ) as measured foliations. This heuristic can be made precise by grafting X along λ and then collapsing the hyperbolic pieces of the resulting metricalong the leaves of O λ ( X ). Throughout the paper we suppress markings in our notation, but reintroduce them here to state the proposition precisely.
Proof.
Construction 5.6 supplies us with a geometric train track π : N (cid:15) ( λ ) → τ . On the preimage π − ( b )of each closed branch b of τ we integrate the two measures O λ ( X ) | N (cid:15) ( λ ) and λ giving π − ( b ) the structureof a bi-foliated Euclidean rectangle of length (cid:96) X ( b ) and height λ ( b ) These rectangles glue along their ‘short’sides { π − ( s ) : s is a switch of τ } to give N (cid:15) ( λ ) the structure of a bi-foliated Euclidean band complex.The map π extends to a self–homotopy equivalence of X homotopic to the identity preserving the ortho-geodesic foliation leafwise. This means that the boundary of N (cid:15) ( λ ) admits a natural retraction onto Sp bycollapsing the leaves of the orthogeodesic foliation in the complement of N (cid:15) ( λ ), and we take the quotientgenerated by this equivalence relation to obtain a new surface Y with its complex structure described below.On each rectangle π − ( b ), the bi-foliated Euclidean structure gives local coordinates to C away from thesingular points of O λ ( X ) locally mapping O λ ( X ) to | dx | and λ to | dy | , thought of as measured foliations onthe plane. These coordinate patches glue together along the spine to give local coordinates away from thepoints of valence ≥
3. Moreover, these charts preserve | dx | and | dy | , so the transitions must be of the form z (cid:55)→ ± z + α for some α ∈ C . We have therefore built a Riemann surface Z equipped with a half-translationstructure away from the vertices of Sp , which become cone points of cone angle equal to π · val( v ). Edges of Sp join vertices along horizontal trajectories representing all horizontal saddle connections on q ; their lengthsin the singular flat metric are given by the integral over O λ ( X ). Thus D λ induces an isometry of metricgraphs, as claimed. (cid:3) This explicit description of the quadratic differential associated to the pair (
X, λ ) by the map O from theintroduction will be useful in order to prove in Theorem 13.13 that Diagram (2) commutes.6. Cellulating crowned Teichm¨uller spaces
We now define a certain arc complex which combinatorializes the structure the orthogeodesic foliationon complementary subsurfaces. The main result of this section is Theorem 6.4, which shows that this arccomplex is equivariantly homeomorphic to the Teichm¨uller space of the complementary surface. In particular,this shows that the restriction of the orthogeodesic foliation to each component of S \ λ completely determinesthe hyperbolic structure on that piece.Before stating the theorem, we must first set up our combinatorial analogue for Teichm¨uller space. Thisappears as Definition 6.1 after a series of auxiliary constructions.Suppose that Σ = Σ { c } g,b is a finite-area hyperbolic surface with boundary and without annular cusps. Aproperly embedded arc I → Σ is essential if I cannot be isotoped (through properly embedded arcs) into ∂ Σor into a spike. The arc complex A (Σ , ∂ Σ) of Σ rel boundary is the (simplicial, flag) complex whose verticesare isotopy classes of essential arcs of Σ. Vertices span a simplex in A (Σ , ∂ Σ) if and only if there exist acollection of pairwise disjoint representatives for each isotopy class. The filling arc complex A fill (Σ , ∂ Σ) isthe subset of A (Σ , ∂ Σ) consisting only of those arc systems which cut Σ into a union of topological disks.The geometric realization | A (Σ , ∂ Σ) | of A (Σ , ∂ Σ) is obtained by declaring every simplex to be a regularEuclidean simplex of the proper dimension; note that the topology of | A (Σ , ∂ Σ) | obtained from the metricstructure is in general different from the standard simplicial topology (see, e.g., [BE88]). The geometricrealization | A fill (Σ , ∂ Σ) | is then the subspace of filling arc systems equipped with the subspace topologyinduced by the metric structure. Definition 6.1.
The weighted filling arc complex | A fill (Σ , ∂ Σ) | R of Σ rel boundary is the set of all weightedmulti-arcs of the form A = (cid:88) c i α i where α = (cid:83) α i ∈ A fill (Σ , ∂ Σ) and c i > i .Throughout, we will use α to denote a single arc, and α to denote an (unweighted) multi-arc. The symbol A will be reserved to denote a weighted multi-arc. Note.
If Σ is an ideal hyperbolic polygon, then the empty arc system fills Σ and we consider it as an elementof | A fill (Σ , ∂ Σ) | R . If Σ is not a polygon, then the empty arc system never fills.So long as Σ is not an ideal polygon, | A fill (Σ , ∂ Σ) | R is just | A fill (Σ , ∂ Σ) | × R > . When Σ is an idealpolygon, then | A fill (Σ , ∂ Σ) | R is homeomorphic to the open cone on the filling arc complex: (cid:0) | A fill (Σ , ∂ Σ) | × R ≥ (cid:1) / (cid:0) | A fill (Σ , ∂ Σ) | × { } (cid:1) . See Figure 4a for an example of | A fill (Σ , ∂ Σ) | R in the case when Σ is an ideal pentagon. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 19
Remark 6.2.
The standard duality between arc systems and ribbon graphs (see, e.g., [Mon09a]) assigns toevery A ∈ | A fill (Σ , ∂ Σ) | R a metric ribbon graph spine for Σ (with some infinitely long edges if Σ has crowns).One could of course translate the cell structure of | A fill (Σ , ∂ Σ) | R into a cellulation of an appropriate spaceof marked metric ribbon graphs.While the arc complex definition is more practical for our definition of shear-shape cocycles, the dualribbon graph picture allows us to immediately understand how to record the geometry of the horizontaltrajectories of a quadratic differential (see Section 10). (a) The weighted arc complex of an idealpentagon rel its boundary. c α c α c α c α β C (cid:96) A ( β ) = c + 2 c + c + c res A ( C )= c − c (b) The combinatorial length and residue associated to aweighted filling arc system A . Figure 4.
Arc complexes and combinatorial geometry.
Combinatorial geometry.
Now that we have defined our combinatorial analogue of Teichm¨uller space, wecan also define combinatorial notions of both length and metric residue.Suppose that β is a compact boundary component of Σ and A ∈ | A fill (Σ , ∂ Σ) | R ; then we define the A –length (cid:96) A ( β ) of β to be the sum of the weights of the arcs of A incident to β (counted with multiplicity, sothat if both endpoints of α lie on β then its weight is counted twice).Similarly, let C be an oriented crowned boundary component with an even number of spikes. Then theedges of C are partitioned into those that that have the surface lying on their left and those which havethe surface on their right; call these edges positively and negatively oriented, respectively. The A –residueres A ( C ) of C is then defined to be the sum of the weights of the arcs incident to each positively oriented edgeof C minus the sum of the weights of the arcs incident to the negatively oriented edges (where both sums areagain taken with multiplicity). See Figure 4b for an example calculation.We have now come to the most important object of this section, and a foundational result of this paperthat allows us to pass between hyperbolic metrics, orthogeodesic foliations, and metric graphs embedded inflat structures. Construction 6.3.
Let Y be a crowned hyperbolic surface. As discussed in Section 5.1, the orthogeodesicfoliation determines a spine for Y together with a dual (filling) arc system α ( Y ). Weighting each dual arcby integrating the measure induced by O ∂Y ( Y ) over the corresponding edge of Sp (compare (7)) thereforedefines a map A : T (Σ) → | A fill (Σ , ∂ Σ) | R . When Σ has compact boundary, [Luo07, Theorem 1.2 and Corollary 1.4] states that A ( · ) is a Mod(Σ)-equivariant stratified real-analytic homeomorphism; see also [Mon09b, Do08, Ush99]. Our aim is to generalizeLuo’s theorem to surfaces with crowned boundary. While the arguments of [Luo07] can probably be adaptedto this setting, we prefer to use some elementary hyperbolic geometry to realize | A fill (Σ , ∂ Σ) | R as a subcom-plex sitting “at infinity” of the weighted filling arc complex of a surface with compact boundary. Theorem 6.4.
Let Σ be a crowned hyperbolic surface. Then the map A : T (Σ) → | A fill (Σ , ∂ Σ) | R is a Mod(Σ)–equivariant stratified real analytic homeomorphism. Moreover, let β , . . . , β b denote the closedboundary components of Σ and C , . . . , C e the crown ends which have an even number of spikes. Fix anorientation of each C j . Then the map above identifies the level sets { ( Y, f ) ∈ T (Σ) | (cid:96) ( β i ) = L i , res( C j ) = R j } ∼ = (cid:8) A ∈ | A fill (Σ , ∂ Σ) | R : (cid:96) A ( β i ) = L i , res A ( C j ) = R j (cid:9) for any ( L i ) ∈ R b> and any ( R j ) ∈ R e .The remainder of this section is devoted to deducing Theorem 6.4 from [Luo07, Theorem 1.2, Corollary1.4] and [Mon09b, Section 2.4]. Our plan is to appeal to the aforementioned references to prove that fora given maximal arc system α , the map A ( · ) extends to a real analytic map A α : T (Σ) → R α that agreeswith A ( · ) on the locus of hyperbolic surfaces whose spine has dual arc system contained in α (Lemma 6.9).We show that A ( · ) is a homeomorphism by building a continuous right inverse Y : | A fill (Σ , ∂ Σ) | R → T (Σ); Y ( A ) is obtained as a geometric limit metrics on a larger compact surface with boundary as some arcs arepinched to spikes.Endowing Σ with an auxiliary hyperbolic metric, we take Σ ◦ to be the surface with geodesic and horocyclicboundary components obtained by truncating the tips of the spikes. Let γ be the union of horocyclicboundary components of Σ ◦ and double Σ ◦ along γ to obtain a (topological) surface D Σ and an identificationof Σ ◦ with a subsurface of D Σ taking ∂ Σ ◦ \ γ into ∂D Σ; see Figure 5. Σ D Σ γ γ Figure 5.
The truncation of a crowned surface Σ along γ and its double D Σ.Let A = (cid:80) c i α i be a weighted filling arc system on Σ and let β be the mirror image of α in D Σ, so that α ∪ γ ∪ β is a filling arc system on D Σ. For each t >
0, define B t = (cid:88) c i β i + t (cid:88) γ i + (cid:88) c i α i ∈ | A fill ( D Σ , ∂D Σ) | R . Since D Σ is compact, we can apply [Luo07, Corollary 1.4] which states that there is a unique hyperbolicstructure X t ∈ T ( D Σ) whose natural weighted arc system coincides with B t . Remark 6.5.
It will be convenient to assume that α is maximal, formally adding arcs of weight 0 to A (and B t ) as necessary.Our goal is now to show that that ( X t ) converges as t → ∞ to a surface Y ∈ T (Σ) such that A ( Y ) = A .The convergence is geometric: we take basepoints x t ∈ X t lying outside of the “thin parts” of the subsurfacecorresponding to Σ ◦ and extract a geometric limit of ( X t , x t ) as t → ∞ . The limit metric Y has spikescorresponding to γ and so defines a point in T (Σ). Moreover, Y inherits a filling arc system naturallyidentified with α , which is necessarily realized orthogeodesically.We begin with an estimate on the lengths of orthogeodesic arcs. Lemma 6.6. If X is a hyperbolic metric on a compact surface with totally geodesic boundary and A ( X ) = (cid:80) c i α i , then min (cid:40) log 3 , − (cid:32) tanh(log √ c i / (cid:33)(cid:41) ≤ (cid:96) X ( α i ) ≤ πc i , for each i . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 21
Proof.
Any leaf of the orthogeodesic foliation properly homotopic to α i has hyperbolic length at least (cid:96) X ( α i ).Thus the embedded “collar” about α i consisting of all leaves of the orthogeodesic foliation in the samehomotopy class of α i has area at least c i (cid:96) X ( α i ) (see Figure 6). On the other hand, the Gauss–Bonnettheorem bounds area of the collar above by 2 π , so we get a bound (cid:96) X ( α i ) ≤ πc i . Now we would like to find a lower bound for (cid:96) X ( α i ) in terms of c i ; for notational convenience we fix i and set α = α i and c = c i . Assume that (cid:96) X ( α ) < log 3. Let H be a component of X \ α meeting α ; thenthere is a unique point u ∈ H equidistant from all boundary components of X meeting H . There is also auniversal lower bound to the distance from u to any such boundary component, given by log √
3, the radiusof the circle inscribed in an ideal triangle. Thus the leaf of O ∂X ( X ) through u has length at least log(3).Since (cid:96) X ( α ) < log 3, we know that there is a leaf of the orthogeodesic foliation parallel to α with lengthlog 3. Using a formula relating the lengths of the sides of a hyperbolic tri-rectangle [Bus10, Theorem 2.3.1],the distance c from α and this leaf is given by(8) tanh( (cid:96) X ( α ) /
2) = tanh(log √ c ) . Now this expression is decreasing in c , and x (cid:55)→ tanh − ( x ) is increasing. We have that c > c by definition(see Figure 6), so the lemma follows. (cid:3) αc c α j ≤ log 2 H u Figure 6.
A foliated collar of width c about an orthogeodesic arc α . If the arc is shorterthan log 3, then there are (bold green) leaves of this collar of length equal to log 3. For avery short arc γ i , the distance between the longest leaf of its collar and the leaf of lengthlog 3 is at most log 2. The dashed arc α j has weight 0 and corresponds to one of two possiblechoices of maximal completion of α .For any arc γ i of γ , some elementary estimates similar to those given in the proof of Lemma 6.6 (compareEquation (8)) give (cid:96) t ( γ i ) = O ( e − t/ ). If α i appears in B t with coefficient c i = 0, then Lemma 6.6 providesa lower bound of log 3 for the length (cid:96) t ( α i ) of α i on X t . We also have the following upper bound: Lemma 6.7. If c j = 0 for some j , then for t large enough, we havelog 3 ≤ (cid:96) t ( α j ) ≤ (cid:88) c i + 8 π (cid:88) c i + | γ | log 144 . Proof.
We remove all arcs of α ∪ γ ∪ β with positive weight from X t and let H t be (the metric completionof) the right-angled polygon component that contains α j . Our strategy is to find a path of controlled lengthcontained in ∂H t joining the endpoints of α j .Notice that ∂H t alternates between segments of ∂X t and arcs of α ∪ γ ∪ β with positive weight. FromLemma 6.6, the total length of segments coming from arcs of α ∪ β is at most 8 π (cid:80) /c i , because each arc of α ∪ β can appear at most two times on ∂H t . Similarly, from the construction of our coordinate system, thetotal length of the segments coming from ∂X t that correspond to collars of arcs in α ∪ β is at most 2 (cid:80) c i . Suppose some arc γ i of γ forms a segment of ∂H t . The distance between the leaf of the orthogeodesicfoliation parallel to γ i with length log(3) and the singular, longest leaf parallel to γ i has distance uniformlybounded above by log 2 for large values of t (see Figure 6). Truncate H t by removing the leaves of theorthogeodesic foliation parallel to γ i with length at most log 3 to obtain a new (non-convex) geodesic polygon H ◦ t . An application of the Collar Lemma [Bus10, Theorem 4.1.1] to the double DX t along its boundaryshows that α j does not enter the region of H t that we removed.Each arc γ i of γ contributed at most 2 t + O ( e − t/ ) to the length of ∂H t . However, after truncating, each γ i contributes at most 2(log 2 + log 3 + log 2) = log 144 to the length of ∂H ◦ t . Putting together all of ourestimates completes the proof. (cid:3) For each α i ∈ α with positive coefficient c i in B t , the orthogeodesic length (cid:96) t ( α i ) of α i on X t is boundedabove and below by the positive real numbers independent of t provided by Lemma 6.6. If c i = 0 for some i , then Lemma 6.7 provides bounds on (cid:96) t ( α i ) independent of t . Therefore, there exists a subsequence t k tending to infinity such that ( (cid:96) t k ( α i )) converges to a positive number (cid:96) i for each i , while (cid:96) t ( γ i ) = O ( e − t/ )for each γ i ∈ γ .The metric completion of X t k \ ( α ∪ γ ∪ β ) is a collection of hyperbolic right-angled hexagons, each withthree non-adjacent sides that correspond to arcs of α ∪ γ ∪ β . The lengths of these sides determine uniquelyan isometry class of right-angled hexagons, which we have just proved converge to (degenerate) right-angledhexagons in which the edges corresponding to arcs of γ become spikes in the limit. The (degenerate) right-angled hexagons glue along α to form a complete hyperbolic surface Y homeomorphic to Σ with a maximalfilling arc system labeled by α and realized orthogeodesically on Y . That is, we have constructed a surface Y ( A ) = Y ∈ T (Σ). Lemma 6.8.
We have an equality A ( Y ( A )) = A . Proof.
By construction, the length of the projection of every edge of the spine of X t dual to an arc of α wasconstant along the sequence ( X t k ) converging geometrically to Y ( A ). The lemma follows. (cid:3) In order to show that the inverse Y ( · ) is well-defined, we will need the following statement, which refinesthe relationship between the coefficients of B t and the lengths of its arcs.Let δ = α ∪ γ ∪ β denote the support of B t . According to [Luo07, Theorem 1.2], the projection lengthsof the edges of the spine dual the arcs of δ extend to an analytic local diffeomorphism B δ : T ( D Σ) → R δ whose image is a convex cone with finitely many sides. Now we show that analyticity extends to infinity.
Lemma 6.9.
For each maximal filling arc system α defining a cell of full dimension in A fill (Σ , ∂ Σ), there isan analytic map A α : T (Σ) → R α such that if the spine of Y ∈ T (Σ) has dual arc system contained in α , then A α ( Y ) = A ( Y ). Proof.
The orthogeodesic length functions associated to our maximal arc system δ = α ∪ γ ∪ β on D Σ forman analytic parameterization of T ( D Σ), which we denote by (cid:96) δ : T ( D Σ) → R δ> . We have a commutativediagram of analytic embeddings(9) R δ> R δ T ( D Σ) . B δ ◦ (cid:96) − δ (cid:96) δ B δ An explicit formula for B δ ◦ (cid:96) − can be recovered from [Mon09b, Section 2.4], which produces an analyticmapping G : R α ∪ β> → R α ∪ β that describes how B δ behaves when the arcs corresponding to γ have lengthclose to 0. More precisely, let π α ∪ β : R δ → R α ∪ β be the coordinate projection. Then with (cid:96) δ = ( (cid:96) α , (cid:96) γ , (cid:96) β ),we have(10) π α ∪ β ◦ B δ ( (cid:96) δ ) = G ( (cid:96) α , (cid:96) β ) + O (cid:18) max γ ∈ γ { (cid:96) γ } (cid:19) The “projection length” associated to each arc of δ (called the “radius coordinate” in [Luo07] and the “width” in [Mon09b])is positive when that arc is dual to an edge of the spine of a surface X ∈ T ( D Σ).
HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 23 uniformly on compact subsets of R α ∪ β> × R γ ≥ .Restricting to the locus of symmetric surfaces { X ∈ T ( D Σ) : (cid:96) α i ( X ) = (cid:96) β i ( X ) , ∀ i } , the map G thereforeinduces an analytic map F : R α> → R α . Again, we have an analytic parameterization (cid:96) α : T (Σ) → R α> bylength functions and a diagram(11) R α> R α T (Σ) . F(cid:96) α F ◦ (cid:96) α So take A α = F ◦ (cid:96) α ; it follows from the definitions that if the dual arc system to the spine of a surface Y ∈ T (Σ) is contained in α , then A α ( Y ) = A ( Y ). This completes the proof of the lemma. (cid:3) A priori, Y ( A ) depends on the subsequence X t k converging geometrically to Y ( A ). However, Lemma 6.10.
The limit Y ( A ) does not depend on choice of subsequence X t k , i.e., X t → Y . Moreover, Y : | A fill (Σ , ∂ Σ) | R → T (Σ) is continuous. Proof.
Throughout this proof, we let π := π α ∪ β be the coordinate projection from the proof of Lemma 6.9.Let s > X t,s ∈ T ( D Σ) be the surface obtained from X t by keeping all lengths of arcs of α ∪ β fixedand taking (cid:96) γ i ( X t,s ) := (cid:96) γ i ( X t + s ), for each γ i ∈ γ . Note that (cid:96) γ i ( X t + s ) = O ( e − ( s + t ) / ). By construction of X t,s , the lengths of arcs of α ∪ β agree with those of X t , so (10) gives π ( B δ ( X t )) − π ( B δ ( X t,s )) = O ( e − ( s + t ) ) . Recall that π ( B α ( X t )) = π ( B t ) is constant for all t >
0, so that π ( B δ ( X s + t )) − π ( B δ ( X t,s )) = O ( e − ( s + t ) ) , as well. Since B δ is open, analytic, and { π ( (cid:96) δ ( X t )) : t > } ⊂ R α ∪ β> lies in a compact set (Lemmas 6.6 and6.7), we can adjust the lengths of arcs α i and β i of α ∪ β in X t,s by O ( e − ( s + t ) ) to obtain X s + t . Thus, forany t k → ∞ , the lengths ( (cid:96) t k ( α ∪ β )) form a Cauchy sequence, hence converge. Thus any two subsequentialgeometric limits (with basepoints away from the spikes of the subsurface associated with Σ ◦ ) coincide, whichproves that Y ( A ) is well defined.To see that Y ( · ) is continuous, let A k → A ; by passing to a subsequence, we may assume that A k are inthe closure of the cell associated to a maximal filling arc system α . Let A k and A be the mirror images (withcorresponding weights) of A k and A in D Σ, respectively. We build two families of approximating surfaces X k , X kk ∈ T ( D Σ) corresponding to the weighted arc systems A + k (cid:88) γ i + A and A k + k (cid:88) γ i + A k on D Σ, respectively. By [Luo07, Theorem 1.2] (alternately, the proof of Lemma 6.9), each X kk is close to X k in T ( D Σ), hence X kk and X k have the same geometric limit Y ( A ) ∈ T (Σ), which is what we wanted toshow. (cid:3) We now have all of the pieces in place to complete the proof of Theorem 6.4.
Proof of Theorem 6.4.
By Lemma 6.10, Y ( · ) is well defined and continuous, and by Lemma 6.8, Y ( · ) is aright inverse to A ( · ); in particular, Y ( · ) is injective. For a given maximal arc system α , the open orthant U α = R α> ⊂ R α is identified with the interior of a top dimensional cell of | A fill (Σ , ∂ Σ) | R . Some of thehyperplanes in ∂ R α ≥ are identified with the interior of cells associated with non-maximal filling arc systemscontained in α ; let U α denote the closure of U α in | A fill (Σ , ∂ Σ) | R .Then Y ( · ) defines a continuous bijection U α → Y ( U α ), and this identification is homeomorphic, because A α supplies an analytic inverse on Y ( U α ), by Lemma 6.9. Since these homeomorphisms glue along thecombinatorics of | A fill (Σ , ∂ Σ) | R , the map Y ( · ) is the desired global homeomorphic inverse to A ( · ).Again, by Lemma 6.9, A ( · ) is analytic restricted to the relative interior of the image under Y ( · ) of eachcell of | A fill (Σ , ∂ Σ) | R , demonstrating the stratified real analytic structure. That level sets of the residuefunctions are mapped to one another is an exercise in unpacking the definitions. (cid:3) Transverse and shear-shape cocycles
We now define the main protagonists of this paper, the shear-shape cocycles on a measured lamination.In Section 7.2, we give a first definition of shear-shape cocycles in terms of the cohomology of an augmentedneighborhood of λ , twisted by its local orientation (Definition 7.5). While this definition has technical merit(and exactly parallels the construction of period coordinates for quadratic differentials, a fact which weexploit in Section 10), it is impractical to use. We rectify this deficiency in Section 7.3 by giving a secondformulation which parallels Bonahon’s axiomatic approach to transverse cocycles (compare Definitions 7.4and 7.12). The main result of this section, Proposition 7.14, proves that these two definitions agree.The reader may find it helpful to consult Sections 10 or 13 while digesting these definitions so as to havea concrete model of shear-shape cocycles in mind.7.1. Transverse cocycles.
As shear-shape cocycles generalize Bonahon’s transverse cocycles, we begin byrecalling two equivalent definitions of transverse cocycles for geodesic laminations which we generalize inSections 7.2 and 7.3 below.
Remark 7.1.
We have chosen to present transverse cocycles in a way that anticipates our construction ofshear-shape cocycles. The reader is advised that our treatment is ahistorical, and in particular omits thefascinating (and quite subtle) relationship between transverse cocycles and transverse H¨older distributions.For more on this correspondence, see [Bon97a], [Bon97b], and [Bon96].The first definition we consider is cohomological. Let λ be a measured lamination on S ; then an orientationof λ is a continuous choice of orientation of the leaves of λ . If N is any snug neighborhood of λ , then onemay take a corresponding (snug) neighborhood (cid:98) N of the orientation cover ˆ λ of λ . Let ι be the coveringinvolution of (cid:98) N → N , and let H ( (cid:98) N , ∂ (cid:98) N ; R ) − denote the − ι ∗ , Definition 7.2.
With all notation as above, a transverse cocycle for λ is an element of H ( (cid:98) N , ∂ (cid:98) N ; R ) − . Weuse H ( λ ) to denote the set of all transverse cocycles for λ .With the definition above it is clear that H ( λ ) is a vector space, and if λ is a union of sublaminations λ , . . . , λ L , then the space of transverse cocycles splits as H ( λ ) = L (cid:77) (cid:96) =1 H ( λ (cid:96) ) . We record the dimension of H ( λ ) below. Lemma 7.3 (Theorem 15 of [Bon97b]) . The space of transverse cocycles forms a vector space of realdimension − χ ( λ ) + n ( λ ) , where n ( λ ) is the number of orientable components of λ .When working with individual transverse cocycles, the above definition is rather unwieldy. Instead, it isoften more useful to think of a transverse cocycle as a function on actual arcs instead of on homology classes. Definition 7.4.
Let λ ∈ ML ( S ). A transverse cocycle σ for λ is a function which assigns to every arc k transverse to λ a real number σ ( k ) such that(H0) (support): If k does not intersect λ then σ ( k ) = 0.(H1) (transverse invariance): If k and k (cid:48) are isotopic transverse to λ then σ ( k ) = σ ( k (cid:48) ).(H2) (finite additivity): If k = k ∪ k where k i have disjoint interiors then σ ( k ) = σ ( k ) + σ ( k ).The reader familiar with train tracks will recognize that these rules resemble those governing weightsystems on train tracks; see Section 9 for a continuation of this discussion.We direct the reader to [Bon97b] or [Bon96, §
3] for a proof of the equivalence of Definitions 7.2 and 7.4(our proof of Proposition 7.14, the corresponding statement for shear-shape cocycles, can also be adapted toprove this equivalence).7.2.
Shear-shape cocycles as cohomology classes.
Our first definition of a shear-shape cocycle is as acohomology class on an appropriate augmented orientation cover, paralleling Definition 7.2. This viewpointallows us to deduce global structural results about spaces of shear-shape cocycles (Lemma 7.8) and alsoreveals implicit constraints on the structure of individual shear-shape cocycles (Lemma 7.9).
HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 25
Suppose that α is a filling arc system for S \ λ . For each arc α i ∈ α , choose an arc t i which meets α i exactly once and is disjoint from λ ∪ α \ { α i } . We call such an arc t i a standard transversal to α i . An orientation of λ ∪ α is a continuous orientation of the leaves of λ together with a choice of orientation on each t i such that t i can be isotoped transverse to α i into λ so that the orientations agree. Most pairs λ ∪ α are notorientable, but each has an orientation double cover (cid:98) λ ∪ ˆ α (the reader should have in mind the orientationcover of a quadratic differential). We note that if λ ∪ α is orientable then λ itself must be.Let N α be a snug neighborhood of λ ∪ α ; then the cover (cid:98) λ ∪ ˆ α → λ ∪ α extends to a covering (cid:98) N α → N α with covering involution ι . By definition of the orientation cover, each standard transversal t i lifts to a pairof distinguished homology classes t (1) i , t (2) i ∈ H ( (cid:98) N α , ∂ (cid:98) N α ; R )such that ι ∗ t (1) i = − t (2) i .The odd cocycles H ( (cid:98) N α , ∂ (cid:98) N α ; R ) − for the covering involution ι ∗ now provide a local cohomological modelfor the space of shear-shape cocycles on λ . Observe that for each i and each σ ∈ H ( (cid:98) N α , ∂ (cid:98) N α ; R ) − , we have σ ( t (1) i ) = − ι ∗ σ ( t (1) i ) = − σ ( ι ∗ t (1) i ) = σ ( t (2) i ) . Definition 7.5.
Let λ ∈ ML ( S ). A shear-shape cocycle for λ is a pair ( α, σ ) where α = (cid:80) α i is a fillingarc system on S \ λ and σ ∈ H ( (cid:98) N α , ∂ (cid:98) N α ; R ) − is such that the values σ ( t ( j ) i ) are all positive. Let Σ ∪ . . . ∪ Σ m denote the components of S \ λ ; then we define the weighted arc system underlying σA := (cid:88) σ ( t ( j ) i ) α i ∈ m (cid:89) j =1 | A fill (Σ j , ∂ Σ j ) | R . We denote the set of all shear-shape cocycles for λ by SH ( λ ), the set of all shear-shape cocycles withunderlying arc system α by SH ◦ ( λ ; α ), and the set of all shear-shape cocycles with underlying weighted arcsystem A by SH ( λ ; A ). Often, we will leave the arc system implicit and just say that σ is a shear-shapecocycle for λ . Remark 7.6.
By Theorem 6.4, a filling weighted arc system A is the same data as a marked hyperbolicstructure on each component of S \ λ . In Sections 12–15 below, we prove that (so long as σ satisfies apositivity condition) these metrics glue together to give a complete hyperbolic metric on S .Our definition of shear-shape cocycle a priori depends on the choice of auxiliary neighborhood N α of λ ∪ α . However, it is not hard to see that Lemma 7.7.
The spaces of shear-shape cocycles defined by different snug neighborhoods are linearly iso-morphic. Moreover, any two choices of snug neighborhoods define the same underlying weighted arc system.
Proof.
Given two nested, snug neighborhoods N (cid:48) α ⊂ N α there is a deformation retraction of N α onto N (cid:48) α (this comes from the assumption of snugness). This induces an isomorphism(12) H ( (cid:98) N α , ∂ (cid:98) N α ; R ) ∼ = H ( (cid:98) N (cid:48) α , ∂ (cid:98) N (cid:48) α ; R )which also identifies the − N α with those defined by N (cid:48) α . To see that the weights on α do not depend onthe choice of N α , we note that the deformation retraction of N α onto N (cid:48) α takes standard transversals tostandard transversals, and hence the value of the cocycle on the transversals does not change as we changeneighborhoods.Now given any two snug neighborhoods N α and N (cid:48) α of λ ∪ α , one may take a common refinement N (cid:48)(cid:48) α of N α and N (cid:48) α and apply (12) to deduce that the spaces of shear-shape cocycles defined by N α and N (cid:48) α arelinearly isomorphic and define the same underlying arc system. (cid:3) By Poincar´e–Lefschetz duality, we have a linear isomorphism H ( (cid:98) N α , ∂ (cid:98) N α ; R ) ∼ = H ( (cid:98) N α ; R ) mapping the odd cocyclesfor ι ∗ to the odd cycles for ι ∗ . Compare with [BD17, §§ In view of this lemma, throughout the sequel we will change the neighborhood N α carrying σ at will.As the orientation cover of λ naturally embeds into (cid:98) N α , we may identify H ( λ ) with a subspace of H ( (cid:98) N α , ∂ (cid:98) N α ; R ). Since any element of H ( λ ) evaluates to 0 on each standard transversal, we can add andsubtract transverse cocycles from shear-shape cocyles without changing the underlying weighted arc system.We therefore have the following analogue of Lemma 7.3: Lemma 7.8.
Let A be the weighted arc system underlying some shear-shape cocycle. Then SH ( λ ; A ) is anaffine space modeled on the vector space H ( λ ). In particular, dim R ( SH ( λ ; A )) = − χ ( λ ) + n ( λ ) . Homological constraints on residues.
When λ is orientable (or more generally, contains orientablecomponents), there are homological constraints governing which weighted arc systems may underlie a shear-shape cocycle. Passing between arc systems and hyperbolic strutures on complementary subsurfaces (viaTheorem 6.4), these homological constraints govern when two structures can be glued together along λ .For example, if λ is a simple closed curve then in order to glue a hyperbolic structure on S \ λ along λ ,the lengths of the boundary components must have equal length. Tracing through the combinatorializationby weighted arc systems, this implies that the A -length of the boundary components must be the same. Thefollowing lemma generalizes this observation to the case when S \ λ has crowned boundary (compare Lemma13.1 below for a similar discussion using hyperbolic geometry). Lemma 7.9.
Suppose that σ is a shear-shape cocycle for λ with underlying weighted arc system A , andlet µ be an orientable component of λ . Then the sum of the (signed) residues of the boundary componentsincident to µ is 0. Proof.
For any component µ of λ , let ∂ ( µ ) denote the boundary components (either closed or crowned)resulting from cutting along µ . For the purposes of this proof, let α ( µ ) denote the sub-arc system of α consisting of those arcs with endpoints on µ .Pick an orientation on µ ; this induces an orientation on each boundary component C ∈ ∂ ( µ ), and hencegives the metric residue of each such C a definite choice of sign. Since we are eventually going to prove thatthe sum of these residues is 0, it does not matter which orientation of µ we pick.As µ is orientable, picking an orientation on µ is also equivalent to picking one of the lifts ˆ µ of µ in theorientation cover (cid:98) λ ∪ ˆ α . Let (cid:91) α ( µ ) denote the set of all lifts of arcs of α ( µ ) which meet ˆ µ . Then since severing (cid:91) α ( µ ) disconnects ˆ µ from the rest of (cid:98) λ ∪ ˆ α , there is a relation (cid:88) ˆ α i ∈ (cid:91) α ( µ ) ε i ˆ t i = 0 in H ( (cid:98) N α , ∂ (cid:98) N α ; Z )where ε i is 1 if ˆ α i is on the left-hand side of ˆ µ and − t i is the (relative homologyclass of the) oriented standard transversal corresponding to ˆ α i . See Figure 7.Therefore, for any cohomology class σ ∈ H ( (cid:98) N α , ∂ (cid:98) N α ; Z ), and in particular any shear-shape cocycle,(13) (cid:88) ˆ α i ∈ (cid:91) α ( µ ) ε i σ (ˆ t i ) = 0 . Now ε i is positive when the arc is on the left–hand side of ˆ µ , equivalently (equipping µ ⊂ S with thecorresponding orientation) when S \ λ is on the left–hand side of µ . Similarly, ε i is negative when thecomplementary subsurface lies to the right of µ . Unraveling the definitions and partitioning the arcs of α ( µ )into their incident boundary components, we see that (13) is equivalent to the statement that (cid:88) C∈ ∂ ( µ ) res A ( C ) = (cid:88) α i ∈ α ( µ ) ε i c i = 0 , which is what we wanted to prove. (cid:3) Remark 7.10.
As with transverse cocycles, one can define shear-shape cocycles for any geodesic lamination,not just those which support transverse measures. The analogue of Lemma 7.9 is more complicated inthis case, as the corresponding homological relations may involve both the ˆ t i and other relative cycles (forexample, consider when λ contains a geodesic spiraling onto a closed leaf). We have omitted such a discussionas this level of generality will not be needed for our purposes. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 27 t t ˆ t ˆ t [ˆ t ] − [ˆ t ] = 0 in H µ (cid:98) µ Figure 7.
Severing ties with one of the lifts (cid:98) µ of an orientable component µ of λ . Thispartition induces a relation in homology, hence a restriction on shear-shape cocycles. Inthis figure the top surface contains (cid:98) λ while the bottom contains λ ; the shaded regions areneighborhoods of these laminations.7.3. Shear-shape cocycles as functions on arcs.
In analogy with Definition 7.4, we can also viewshear-shape cocycle as functions on transverse arcs which satisfy certain properties. While this definition ismore involved, it is more convenient for the calculations of Sections 13–15 and better reflects the process of“measuring” arcs by a shear-shape cocycle.As indicated by Lemma 7.9, we must first cut out the space of all possible weighted arc systems underlyinga shear-shape cocycle. Denote the complementary subsurfaces of λ ∈ ML ( S ) by Σ , . . . , Σ m , and set B ( S \ λ ) := (cid:110) A ∈ m (cid:89) j =1 | A fill (Σ j , ∂ Σ j ) | R (cid:12)(cid:12)(cid:12) (cid:88) C∈ ∂ ( µ ) res A ( C ) = 0 for all orientable components µ ⊂ λ (cid:111) where we recall that ∂ ( µ ) denotes the set of boundary components of S \ λ resulting from cutting along µ .By Theorem 6.4, we can reinterpret B ( S \ λ ) as the set of all hyperbolic structures on S \ λ so that themetric residues of the boundary components resulting from any orientable component µ of λ sum to zero.We note that when each component of λ is nonorientable, B ( S \ λ ) is just the product of the Teichm¨ullerspaces of the complementary subsurfaces. When λ is a simple closed curve, then B ( S \ λ ) consists of thosemetrics on S \ λ where the two boundary components have the same length.Using this reinterpretation together with Lemma 4.4, we see that B ( S \ λ ) is topologically just a cell: Lemma 7.11.
Let λ ∈ ML ( S ) with S \ λ = Σ ∪ . . . ∪ Σ m . Then B ( S \ λ ) ∼ = R d , where d = − n ( λ ) + m (cid:88) j =1 dim( T (Σ j ))where n ( λ ) is the number of orientable components of λ . Proof.
Let µ , . . . , µ n ( λ ) denote the orientable components of λ and fix an arbitrary orientation on each.Then the lemma follows from the observation that B ( S \ λ ) is a fiber bundle over n ( λ ) (cid:89) i =1 (cid:40) ( R ik ) ∈ R | ∂ ( µ i ) | (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) k R ik = 0 (cid:41) with fibers equal to [ Y, f ] ∈ m (cid:89) j =1 T (Σ j ) (cid:12)(cid:12)(cid:12)(cid:12) res( C k ) = R ik for each C k ∈ ∂ ( µ i ) . By Proposition 4.5, the fibers are each homeomorphic to R d , where d = m (cid:88) j =1 dim( T (Σ j )) − n ( λ ) (cid:88) i =1 | ∂ ( µ i ) | . Totalling the dimensions of base and fiber give the desired result. (cid:3)
We can now present our second definition of shear-shape cocycles.
Definition 7.12.
Let λ ∈ ML ( S ). A shear-shape cocycle for λ is a pair ( σ, A ) where A is a weighted fillingarc system A = n (cid:88) i =1 c i α i ∈ B ( S \ λ )and σ is a function which assigns to every arc k transverse to λ and disjoint from α := ∪ α i a real number σ ( k ), satisfying the following axioms:(SH0) (support): If k does not intersect λ then σ ( k ) = 0.(SH1) (transverse invariance): If k and k (cid:48) are isotopic through arcs transverse to λ and disjoint from α ,then σ ( k ) = σ ( k (cid:48) ).(SH2) (finite additivity): If k = k ∪ k where k i have disjoint interiors, then σ ( k ) = σ ( k ) + σ ( k ).(SH3) ( A -compatibility): Suppose that k is isotopic rel endpoints and transverse to λ to some arc whichmay be written as t i ∪ (cid:96) , where t i is a standard transversal and (cid:96) is disjoint from α . Then the loop k ∪ t i ∪ (cid:96) encircles a unique point p of λ ∩ α , and σ ( k ) = σ ( (cid:96) ) + εc i where ε denotes the winding number of k ∪ t i ∪ (cid:96) about p (where the loop is oriented so that theedges are traversed k then t i then (cid:96) ). See Figure 8.While axiom (SH3) may seem convoluted upon first inspection, its entire effect is to prescribe how thevalue σ ( k ) evolves as an endpoint of k passes through an arc of α . The sign change records whether the mapinduced by k = t i ∪ (cid:96) from the oriented simplex into S is orientation-preserving or -reversing. Remark 7.13.
In Section 9 (Proposition 9.5 in particular), we show that there exists a choice of “smoothing”for α which resolves condition (SH3) into an additivity condition. This is equivalent to prescribing that anarc k may only be dragged over a point of λ ∩ α in one direction.The equivalence between Definitions 7.5 and 7.12 is essentially the same as the equivalence of the coho-mological and axiomatic definitions of transverse cocycles [Bon96, pp. 248–9]. However, the A -compatibilitycondition (axiom (SH3)) contributes new technical difficulties, and so we have included a full proof forcompleteness. Proposition 7.14.
The cohomological and axiomatic definitions of shear-shape cocycles agree.
Proof.
Suppose first that σ is a cohomological shear-shape cocycle, that is, a cohomology class of the orien-tation cover (cid:98) N α of N α that is anti-invariant under the covering involution and that gives positive weight tothe canonical lifts of the standard transversals of each arc of a filling arc system α . We begin by buildingfrom σ a function f σ ; the basic idea is to restrict an arc to a neighborhood of λ , resulting in a relativehomology class, and to set f σ to be σ evaluated on this class. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 29
Suppose that k is any arc transverse to λ and disjoint from α . Choose a small neighborhood N α of λ ∪ α so that k meets ∂N α transversely and ∂k ∩ N α = ∅ ; then k | N α is a union of arcs with endpoints on ∂N α .Each arc k i of k | N α has two distinguished, oriented lifts k (1) i and k (2) i to (cid:98) N α that cross (cid:98) λ from right to left.As in Section 7.2, these distinguished lifts satisfy(14) ι ∗ ([ k (1) i ]) = − [ k (2) i ]in H ( (cid:98) N α , ∂ (cid:98) N α ; Z ), where ι is the covering involution of (cid:98) N α → N α . In particular σ ([ k (1) i ]) = σ ([ k (2) i ]) since σ is anti-invariant under ι . We therefore set f σ ( k ) := σ ([ k ])where [ k ] is the homology class of either lift of k | N α to (cid:98) N α .We now prove that f σ satisfies the axioms of Definition 7.12:(SH0) If k does not intersect λ then k | N α is empty and [ k ] = 0, implying f σ ( k ) = 0.(SH1) If k and k (cid:48) are isotopic through arcs transverse to λ and disjoint from α then k | N α and k (cid:48) | N α areproperly isotopic. One can lift this isotopy to the orientation cover to deduce that [ k ] = [ k (cid:48) ] for thecorrect choice of lifts, so f σ ( k ) = f σ ( k (cid:48) ).(SH2) Suppose that k = k ∪ k ; then so long as N α is small enough it is clear that k | N α = k | N α ∪ k | N α .Therefore, since a lift of k | N α consists of the union of lifts of k | N α and k | N α , we see that [ k ] =[ k ] + [ k ] and hence the corresponding equality of f σ values also holds.(SH3) Finally, suppose that k is isotopic (rel endpoints and transverse to λ ) to (cid:96) ∪ t i . Without loss ofgenerality, we assume that the restriction of each of k, (cid:96), t i to N is a single properly embedded arc(if not, simply break the arcs into smaller pieces and apply (SH1) and (SH2) repeatedly). We alsoassume the restrictions are all disjoint (even at their endpoints), appealing to (SH1) as necessary.The isotopy between k and (cid:96) ∪ t i induces a map from a disk ∆ to N α so that ∂ ∆ ⊂ ∂N α ∪ k ∪ (cid:96) ∪ t i .Refining N α , isotoping the arcs, and homotoping the map as necessary, we may assume that ∆embeds into N α , and therefore must occur in one of the configurations shown in Figure 8 below. α i α i (cid:98) λ (cid:98) N α (cid:98) ∆ (cid:98) ∆ k (cid:96)t i t i k (cid:96) [ k ] = [ t i ] + [ (cid:96) ] [ k ] + [ t i ] = [ (cid:96) ] Figure 8.
Possible configurations of the disk (cid:98) ∆ and the corresponding homological rela-tions.Now choose one of the lifts (cid:98) ∆ ⊂ (cid:98) N α of ∆; this choice specifies lifts of the arcs k , (cid:96) , and t i andtherefore (after equipping the lifts with their canonical orientations) relative homology classes [ k ],[ (cid:96) ], and [ t i ]. As these lifts together with ∂ ˆ N α bound the disk (cid:98) ∆, we therefore get the equality[ k ] = [ (cid:96) ] ± [ t i ]where the sign is determined by the relative configuration of the arcs. Inspection of Figure 8 revealsthat the sign coincides with the winding number of the loop k ∪ t i ∪ (cid:96) about p .Now suppose that ( σ, A ) is an axiomatic shear-shape cocycle in the sense of Definition 7.12. Pick a snugneighborhood N α of λ ∪ α ; our task to show that the function ( k (cid:55)→ σ ( k )) is indeed a cocycle (on theorientation cover, and is anti-invariant under the covering involution). We first show that σ naturally defines a cochain on (cid:98) N relative to ∂ (cid:98) N α which is anti-invariant by ι ∗ . Recallthat any arc in the orientation cover comes with a canonical orientation. We may then assign to any orientedarc ˆ k properly embedded in (cid:98) N α the value ± σ ( k ), where k is the image of ˆ k under the covering projectionand where the sign is positive if ˆ k is oriented according to the canonical orientation and negative otherwise.To the (canonically oriented lifts of the) standard transversals t i we assign the value c i . Anti-invariance thenfollows by construction (compare (14)).To see that this cochain is actually a cocycle, we show that it evaluates to 0 on every boundary. For thepurposes of this argument, it will be convenient to realize H ( (cid:98) N α , ∂ (cid:98) N α ; R ) in terms of simplicial (co)homology.The neighborhood N α may be triangulated as depicted in Figure 9 (compare [SB01, Figure 1]). In such atriangulation, each point of λ ∩ α and each switch of N α corresponds to a unique triangle, while the remainingbranches each contribute a rectangle which is in turn subdivided into two triangles. This triangulation clearlylifts to an ( ι -invariant) triangulation of (cid:98) N α . α i N α λ Figure 9.
A triangulation of a (snug) neighborhood of λ ∪ α . Axioms (SH0)–(SH3) implythat σ ( ∂ ∆) = 0 for each triangle ∆ in the triangulation, i.e., σ is a cocycle.It therefore suffices to prove that for each oriented triangle ∆ of (cid:98) N α we have that σ ( ∂ ∆) = 0 . There arethree types of triangles, each of which corresponds to a different axiom of Definition 7.12: • If ∆ is (the lift of) a triangle coming from a subdivision of a branch, then one if its sides does notintersect λ and is thus assigned the value 0 by (SH0). The other two sides are isotopic rel λ , cross λ with different orientations, and are assigned the same value by (SH1). Therefore σ ( ∂ ∆) = 0 . Similarly, if ∆ comes from a neighborhood of α , then the edges transverse to α are assigned the arcweight c i (with opposite signs) while the other edge gets zero weight, so σ ( ∂ ∆) = 0 . • Now suppose ∆ is (the lift of) a triangle corresponding to a switch of N α with ∂ ∆ = k + k − k .Then since the concatenation of k and k is isotopic transverse to λ to − k , axiom (SH2) implies σ ( k ) + σ ( k ) − σ ( k ) = 0and again σ ( ∂ ∆) = 0 . • Finally, suppose that ∆ is (the lift of) a triangle corresponding to a point of λ ∩ α , so ∂ ∆ is some signedcombination of the (canonically oriented) lifts of arcs k, (cid:96) and t where t is a standard transversal and k is isotopic rel endpoints and transverse to λ to (cid:96) ∪ t . Without loss of generality we assume that ∆is positively oriented; then depending on the configuration of k , t and (cid:96) we have either (cid:96) − k + t = 0 or (cid:96) − t − k = 0(as in Figure 8). In either case, axiom (SH3) implies that σ ( ∂ ∆) = 0.We have therefore shown that σ ( ∂ ∆) = 0 for every triangle of a triangulation and hence σ is indeed a1-cocycle on (cid:98) N α rel boundary, finishing the proof of the lemma. (cid:3) Measuring arcs along curves.
We will also want to associate a number σ ( k ) to certain arcs k thathave non-empty intersection with α ; this quantity should be invariant under suitable isotopy transverse to λ respecting the combinatorics of intersections with α . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 31
So suppose µ ⊂ λ is an isolated leaf, i.e. a simple closed curve. We say that an arc k transverse to λ ∪ α and contained in an annular neighborhood of µ is non-backtracking if any lift ˜ k of k to the universal coverintersects the entire preimage ˜ µ of µ exactly once and ˜ k crosses each lift of an arc of α at most once.If k is a non-backtracking arc, then one may orient k and give µ the orientation that makes k start tothe right of µ . Record the sequence of arcs β , ..., β m crossed by k , in order (note that arcs of α may repeatin this sequence). Then up to isotopy, we may assume that k is a concatenation of standard transversals t , ..., t m together with a small segment k disjoint from α crossing µ from right to left. Compare Figure 10.Since k is non-backtracking, the points β ∩ µ, ..., β m ∩ µ make progress around µ either in the positivedirection or the negative direction. Take ε = +1 in the former case and ε = − σ ( k ) := σ ( k ) + ε m (cid:88) j =1 c j where c j is the weight corresponding to the arc β j . Note that the value of ε only depends on k and not onits orientation, as reversing its orientation also reverses the orientation of µ . ε = +1 ˜ kβ β β (cid:101) µ ˜ k ˜ k (cid:48) (cid:101) N α Figure 10.
Since k makes progress around µ in the positive direction, ε = +1. Lemma 7.15.
Suppose that k and k (cid:48) are non-backtracking arcs transverse to λ ∪ α contained in an annularneighborhood of a simple closed curve component µ of λ . If there exist lifts ˜ k and ˜ k (cid:48) to (cid:101) S whose endpointslie in the same component of (cid:101) S \ ( (cid:101) λ ∪ ˜ α ) and k is isotopic to k (cid:48) transverse to λ , then σ ( k ) = σ ( k (cid:48) ). Proof.
Fix a snug neighborhood N α of λ ∪ α ; then we need only show that k | N α and k (cid:48) | N α define homologouscycles in the orientation cover.Now it is clear that the restrictions of k and k (cid:48) to the universal cover (cid:101) N α are homologous; indeed, onecan simply take an isotopy [0 , → (cid:101) S between lifts of the two (transverse to λ ) that leaves the endpoints inthe same component of (cid:101) S \ (cid:101) N α . Pushing the isotopy back down to S and restricting it to the neighborhood N α therefore gives a homology betweeen the restrictions of k and k (cid:48) .Since µ is orientable, an annular neighborhood of µ lifts homeomorphically to (cid:98) N α , as do k and k (cid:48) .Therefore, the isotopy between k and k (cid:48) (and the homology between their restrictions) also lifts to theorientation cover (cid:98) N α , showing that the (lifts of the) restrictions of k and k (cid:48) are homologous there as well.Compare Figure 10. (cid:3) The structure of shear-shape space
In this section, we investigate the global structure of the space of shear-shape cocycles. Whereas Bonahon’stransverse cocycles assemble into a vector space, the space SH ( λ ) of all shear-shape cocycles is more complexwhen λ is not maximal, forming an principal H ( λ )-bundle over B ( S \ λ ) (Theorem 8.1).After understanding the structure of shear-shape space, we define an intersection form on SH ( λ ) (Section8.2) and use it to specify the “positive locus” SH + ( λ ) (Definition 8.4) which we show in Sections 10 through15 serves as a global parametrization of both MF ( λ ) and T ( S ).8.1. Bundle structure.
Lemma 7.8 of the previous section parametrizes all shear-shape cocycles which arecompatible with a given weighted arc system. In this section, we analyze how these parameter spaces piecetogether to get a global description of the space of all shear-shape cocycles for a fixed lamination.
Let G be a topological group. A principal G -bundle is a fiber bundle whose fibers are equipped with atransitive, continuous G -action with trivial point stabilizers together with a bundle atlas whose transitionfunctions are continuous maps into G . We remind the reader that a principal G -bundle does not typicallyhave a natural “zero section,” but instead, any local section of the bundle defines an identification of thefibers with G via the G -action. Moreover, any two sections define local trivializations of the bundle thatdiffer by an element of G in each fiber. Theorem 8.1.
Let λ ∈ ML ( S ). The space SH ( λ ) forms a principal H ( λ )-bundle over B ( S \ λ ) whose fiberover A ∈ B ( S \ λ ) is SH ( λ ; A ). Proof.
There is an obvious map from SH ( λ ) to B ( S \ λ ) given by remembering only the values σ ( t i ) oftransversals to the arcs. For a given choice σ in the fiber SH ( λ ; A ) over A , Lemma 7.8 identifies SH ( λ ; A )with H ( λ ) via the assignment σ (cid:55)→ σ − σ .For any filling arc system α of S \ λ , the space SH ◦ ( λ ; α ) of shear-shape cocycles with underlying arcsystem α is naturally identified with the open orthant(16) (cid:110) σ ∈ H ( (cid:98) N α , ∂ (cid:98) N α ; R ) − : σ ( t ( j ) i ) > ∀ i, j = 1 , (cid:111) , where N α is a snug neighborhood of λ ∪ α on S .Consider the open cell B ◦ ( α ) ⊂ B ( S \ λ ) defined as all those weighted arc systems with support equalto a maximal arc system α . Using cohomological coordinates (16) for SH ◦ ( λ ; α ), we can find a continuoussection σ of SH ◦ ( λ ; α ) → B ◦ ( α ). Then φ σ : B ◦ ( α ) × H ( λ ) → SH ◦ ( λ ; α )( A, η ) (cid:55)→ σ ( A ) + η is a homeomorphism preserving fibers of the natural projections. For another choice of section σ (cid:48) , we have φ − σ ( φ σ (cid:48) ( A, η )) = (
A, η + σ (cid:48) ( A ) − σ ( A )) . Evidently, the map A (cid:55)→ σ (cid:48) ( A ) − σ ( A ) ∈ H ( λ ) is continuous.If N (cid:48) α is another snug neighborhood of λ ∪ α , then N α and N (cid:48) α share a common deformation retract.The composition of the linear isomorphisms induced on cohomology by inclusion of the deformation retractpreserves the orthants defined as in (16) as well as fibers of projection to B ( S \ λ ). This proves that theprincipal H ( λ )-structure of the bundle lying over B ◦ ( α ) does not depend on the snug neighborhood whosecohomology coordinatizes SH ◦ ( λ ; α ).To show that the principal H ( λ )-bundle structures over all cells of B ( S \ λ ) glue together nicely, we finda continuous section of SH ( λ ) → B ( S \ λ ) near any given weighted arc system A . Indeed, if α ⊂ β , theninclusion N α (cid:44) → N β of snug neighborhoods defines a map on cohomology. This map restricts to a linearisomorphism on the kernel of the evaluation map on the transversals to β \ α . Thus, the closure(17) SH ( λ ; β ) = (cid:91) β ⊇ αα fills S \ λ SH ◦ ( λ ; α )of SH ◦ ( λ ; β ) in SH ( λ ) may be realized as an orthant in H ( (cid:98) N β , ∂ (cid:98) N β ; R ) − with some open and closed faces;one of the closed faces corresponds to SH ◦ ( λ ; α ). Since the complex A fill ( S \ λ ) is locally finite, there are only finitely many arcs β , ..., β k disjoint from α .Let U ⊂ B ( S \ λ ) be a small neighborhood of A and σ be a continuous section of SH ( λ ; α ) → B ◦ ( α ) ∩ U . Foreach i , after including SH ◦ ( λ ; α ) as a face of SH ( λ ; α ∪ β i ), we may extend σ continuously on U ∩ B ◦ ( α ∪ β i ).Continuing this process, eventually extending σ to higher dimensional cells meeting U , we end up with acontinuous section U → SH ( λ ), as claimed. As before, trivializations defined by two different sections differby a continuous function U → H ( λ ); this completes the proof of the theorem. (cid:3) Since every bundle over a contractible base is trivial, this implies that
Corollary 8.2.
Shear-shape space SH ( λ ) is homeomorphic to R g − . When every component of S \ λ is simply connected, the empty set is a filling arc system. When this is the case, B ◦ ( ∅ )is identified with a point, while SH ( λ ; ∅ ) = H ( λ ). HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 33
Proof.
Let Σ , . . . , Σ m denote the complementary components of λ , where Σ j has genus g j with b j closedboundary components and k j crowns of types { c j , . . . , c jk j } . By Lemmas 7.11 and 4.4, we know that B ( S \ λ )is homeomorphic to a cell of dimension − n ( λ ) + m (cid:88) j =1 dim( T (Σ j )) = − n ( λ ) + m (cid:88) j =1 g j − b j + k j (cid:88) i =1 ( c ji + 3) Lemmas 7.8 and 4.6 together imply that SH ( λ ; A ) is an affine H ( λ )-space of dimension n ( λ ) − χ ( λ ) = n ( λ ) + 12 m (cid:88) j =1 k j (cid:88) i =1 c ji Putting these dimension counts together via Theorem 8.1, we see that SH ( λ ) is homeomorphic to a cell ofdimension m (cid:88) j =1 g j − b j + 32 k j (cid:88) i =1 ( c ji + 2) = 32 π m (cid:88) j =1 Area(Σ j ) = 32 π Area( S ) = 6 g − , where the first equality follows from (4). (cid:3) Intersection forms and positivity.
Now that we have a global description of shear-shape space, werestrict our attention to a certain positive locus SH + ( λ ) inside of SH ( λ ). The main result of this section isProposition 8.5, in which we identify SH + ( λ ) as an affine cone bundle over B ( S \ λ ). Positive transverse cocycles.
We begin by recalling the definition of positivity for transverse cocycles, asdeveloped in [Bon96, §
6] . Fixing some λ ∈ ML ( S ), we recall that a transverse cocycle for λ may be identifiedwith a relative cohomology class of the orientation cover (cid:98) N of a snug neighborhood N of λ (Definition 7.2).The intersection pairing of (cid:98) N therefore induces a anti-symmetric bilinear pairing ω H : H ( λ ) × H ( λ ) → R called the Thurston intersection/symplectic form . This form is nondegenerate when λ is maximal, and moregenerally, when λ cuts S into polygons each with an odd number of sides [PH92, § λ is in particular a transverse cocycle. Using the intersection form one cantherefore define a positive cone H + ( λ ) inside of H ( λ ) with respect to the (non-atomic) measures supportedon λ . Write λ = λ ∪ . . . ∪ λ L ∪ γ ∪ . . . ∪ γ M where the γ m are all weighted simple closed curves and the λ (cid:96) are minimal measured sub-laminations whosesupports are not simple closed curves. Then set(18) H + ( λ ) := { ρ ∈ H ( λ ) : ω H ( ρ, µ ) > µ ∈ L (cid:91) (cid:96) =1 ∆( λ (cid:96) ) } , where we recall that ∆( λ (cid:96) ) denotes the collection of measures supported on λ (cid:96) .The reason for this involved definition is that the Thurston form is identically 0 exactly when the under-lying lamination is a multicurve. Therefore, if the support of λ contains a simple closed curve γ , the pairingof γ with every transverse cocycle supported on λ is 0. On the other hand, so long as λ is not a multicurve then the Thurston form is not identically 0. In fact,the cone H + ( λ ) splits as a product H + ( λ ) = L (cid:77) (cid:96) =1 H + ( λ (cid:96) ) ⊕ M (cid:77) m =1 H ( γ m ) . As λ supports at most 3 g − H + ( λ (cid:96) ) is a cone with a side foreach (projective class of) ergodic measure supported on λ (cid:96) . This is because the components of the orientation cover are all annuli, whose first (co)homologies all have rank 1. Fornon-curve laminations, the homology has higher rank and so can support a nonzero intersection form.
When λ is a multicurve, then there are no λ (cid:96) ’s and so the condition of (18) is empty. As such, in thiscase we have that the space of positive transverse cocycles is the entire twist space: H + ( γ ∪ . . . ∪ γ M ) = H ( γ ∪ . . . ∪ γ M ) = M (cid:77) m =1 H ( γ m ) ∼ = R M . Therefore, we see that no matter whether γ is a multicurve or not, the space H + ( λ ) is a convex cone of fulldimension (where we expand our definition of “cone” to include the entire vector space). Positive shear-shape cocycles.
We now repeat the above discussion for shear-shape cocycles. By Defini-tion 7.5, any shear-shape cocycle ( σ, α ) may be identified with a relative cohomology class of the orientationcover (cid:98) N α of a neighborhood N α of λ ∪ α . As above, the intersection pairing of (cid:98) N then defines a pairingbetween any two shear-shape cocycles with underlying arc system contained inside of α . However, if theunderlying arc systems of σ, ρ ∈ SH ( λ ) are not nested then there is no obvious way to pair the two cocycles.While it does not make sense to pair two arbitrary shear-shape cocycles, we can always pair shear-shapecocycles with transverse cocycles. Recall from (the discussion before) Lemma 7.8 that H ( λ ) naturally embedsas a subspace of the cohomology of the neighborhood (cid:98) N α defining a shear-shape cocycle and may be identifiedwith the kernel of the evaluation map on transversals to α . Therefore, the intersection pairing on (cid:98) N α givesrise to a function ω SH : SH ( λ ) × H ( λ ) → R which we also refer to as the Thurston intersection form . Throughout the paper, we will differentiate betweenthe different intersection forms by indicating their domains in subscript.We record some of the relevant properties of ω SH below. Lemma 8.3.
The Thurston intersection form ω SH is a Mod( S )[ λ ]–invariant continuous pairing which ishomogeneous in the first factor and linear in the second. Moreover, for any A ∈ B ( S \ λ ) and ρ ∈ H ( λ ), thefunction ω SH ( · , ρ ) : SH ( λ ; A ) → R is an affine homomorphism inducing ω H ( · , ρ ) on the underlying vector space H ( λ ). Proof.
We begin by showing that the form is actually well-defined. Suppose first that α is maximal; thensince the (homological) intersection form is natural with respect to deformation retracts, and any two snugneighborhoods of λ ∪ α share a common deformation retract, we see that the form does not depend on thechoice of neighborhood.Now suppose that β is a filling arc system that is a subsystem of two different maximal arc systems α and α . Then one can take a snug neighborhood N β of λ ∪ β which includes into neighborhoods N i of λ ∪ α i for i = 1 ,
2. Now since the (homological) intersection form is also natural with respect to inclusions, we seethat the Thurston form must be as well. Therefore, for any σ ∈ SH ( λ ; β ) and ρ ∈ H ( λ ) it does not matterif we compute ω SH ( σ, ρ ) in N β , N , or N .Now that we have established that ω SH is well-defined, the other properties follow readily from propertiesof the (homological) intersection form. Since the homological intersection pairing is linear in each coordinate,we get that ω SH is in particular linear in the second coordinate. Similarly, for any A ∈ B ( S \ λ ) and any two σ , σ ∈ SH ( λ ; A ) we know that σ − σ is a transverse cocycle, and again by linearity of the homologicalintersection form we get that ω SH ( σ , ρ ) − ω SH ( σ , ρ ) = ω H ( σ − σ , ρ )for all ρ ∈ H ( λ ). Thus ω SH is affine on each SH ( λ ; A ).Finally, to see that the map ω SH ( · , ρ ) is continuous for a fixed ρ , we recall that for any maximal arcsystem α , the space SH ◦ ( λ ; α ) of shear-shape cocycles with underlying arc system α may be realized as anopen octant in cohomological coordinates (16), and this parametrization extends to its closure SH ( λ ; α ).Since the intersection pairing on cohomology is continuous, we see that for each maximal arc system α the function ω SH ( · , ρ ) is continuous on SH ( λ ; α ). But now since we have checked that the value of ω SH ( · , ρ )does not actually depend on the neighborhood, it agrees on the overlaps of closures SH ( λ ; α ) for maximal α .Therefore, since the cell structure of B ( S \ λ ) is locally finite we may glue together the functions ω SH ( · , ρ )(which are continuous on each SH ( λ ; α )) to get a globally continuous function on SH ( λ ). (cid:3) HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 35
With this intersection form in hand, we may now define a positive locus with respect to the set of measuressupported on λ . Definition 8.4.
The space of positive shear-shape cocycles SH + ( λ ) is the set SH + ( λ ) = { σ ∈ SH ( λ ) : ω SH ( σ, µ ) > µ ∈ ∆( λ ) } . Observe the difference between the definition above and the one appearing in (18): any positive shear-shape cocycle must also pair positively with all simple closed curves γ m appearing in the support of λ . Theessential difference between the two cases is that additional branches of τ α coming from the underlying arcsystem allows a shear-shape cocycle to meet each γ m without being completely supported on γ m . Indeed,one can check that the contribution to the Thurston form coming from the intersection of α with a simpleclosed curve component of λ is always positive (compare (20)). In particular, the positivity condition isautomatically fulfilled for any measure supported on a curve component of λ .On each cohomological chart (16) or (17) it is clear that SH + ( λ ) is an open cone cut out by finitely manylinear inequalities (one for each ergodic measure supported on λ , plus positivity of arcs weights). However,this does not yield a global description of SH + ( λ ). In order to get one, we must show that the linearsubspaces cut out by the positivity conditions intersect the H ( λ ) fibers transversely. Proposition 8.5.
The space SH + ( λ ) is an affine cone bundle over B ( S \ λ ) with fibers isomorphic to H + ( λ ).By an affine cone bundle, we mean that there is a (non-unique) section σ : B ( S \ λ ) → SH ( λ ) such that SH + ( λ ) ∩ SH ( λ ; A ) = σ ( A ) + H + ( λ )for every A ∈ B ( S \ λ ). Moreover, any two such sections differ by a continuous map B ( S \ λ ) → H ( λ ). Proof.
Choose mutually singular ergodic measures µ , . . . , µ N , γ , . . . , γ M on λ that span ∆( λ ), where thesupports of the µ n are non-curve laminations and the γ m are all simple closed curves. Pick an arbitrary σ ∈ SH ( λ ; A ), and define C ( σ ) := (cid:8) ρ ∈ H ( λ ) (cid:12)(cid:12) ω H ( ρ, µ n ) > − ω SH ( σ, µ n ) for all n = 1 , . . . , N (cid:9) . By linearity of ω H on H ( λ ), together with the fact that the pairing ω H ( · , µ n ) is not identically 0 since thesupport of µ n is not a simple closed curve, this is an intersection of N affine half-spaces which do not dependon our choice of ergodic measures µ i in their projective classes. Again by linearity, we see that this is just atranslate of H + ( λ ) and hence is a cone of full dimension.Now since ω SH ( · , µ j ) is an affine map on SH ( λ ; A ) for each j , we see that σ + C ( σ ) = (cid:8) η ∈ SH ( λ ; A ) (cid:12)(cid:12) ω SH ( η, µ n ) > n = 1 , . . . , N (cid:9) = SH + ( λ ) ∩ SH ( λ ; A )is an affine cone of full dimension (where the last equality holds because the positive discussion is automati-cally fulfilled for each γ m ). It is a further consequence of affinity that this identification does not depend onthe choice of σ . The bundle structure then follows from continuity of ω SH . (cid:3) Train track coordinates for shear-shape space
In this section, we introduced train track charts for shear-shape cocycles. In Section 9.1, we recall Bona-hon’s realization of transverse cocycles to a lamination in the weight space of a train track that snugly carriesit. In Section 9.2, we reinterpret the cohomological coordinate charts (16) for SH ◦ ( λ ; α ) by “smoothing” λ ∪ α onto a train track τ α (Construction 9.3) and realizing SH ◦ ( λ ; α ) as an orthant in the weight space of τ α (Proposition 9.5). This construction also has the added benefit of converting axiom (SH3) of Definition7.12 into a simpler additivity condition; this is convenient for computations and provides an explicit formula(20) for the Thurston intersection pairing. We rely on this formula in Section 10.2 to show that foliationstransverse to λ define positive shear-shape cocycles (Proposition 10.12).Later, in Section 9.3, we explain how the PIL structure of SH ( λ ) is manifest in train track coordinatesand provides a canonical measure in the class of Lebesgue. When λ is maximal, this measure is a constantmultiple of the symplectic volume element induced by ω H . Finally, in Section 9.4 we consider how traintrack charts facilitate an interpretation of SH ( λ ) as organizing the fragments of the cotangent space to ML at λ . Remark 9.1.
We advise the reader that two different types of train tracks appear below: those which carrytransverse cocycles for λ and give coordinates on the fiber SH ( λ ; A ), and those which carry shear-shapecocycles and give coordinates on the total space SH ( λ ).9.1. Train track coordinates for transverse cocycles.
We begin by recalling how transverse cocyclescan be parametrized by weight systems on (snug) train tracks. The advantage of these coordinates is thatthey determine the cocycle with only finitely many values (a main benefit of the cohomological Definition7.2), but do so using unoriented arcs on the surface, not the orientation cover (a main benefit of the axiomaticDefinition 7.4).Let τ be a train track snugly carrying a geodesic lamination λ and σ a transverse cocycle, thought of as afunction on transverse arcs. For each branch b of τ , pick a tie t b . Then one can assign to b the weight σ ( t b );by Axiom (H1) this value does not depend on the choice of tie, and by Axiom (H2) these weights necessarilysatisfy the switch conditions. Therefore, any transverse cocycle can be represented by a weight system on τ , and in fact this map is an isomorphism. Proposition 9.2 (Theorem 11 of [Bon97b]) . Let τ be a train track snugly carrying a geodesic lamination λ . Then the map σ (cid:55)→ { σ ( t b ) } b ∈ b ( τ ) is a linear isomorphism between H ( λ ) and W ( τ ), the space of all (real)weights on τ satisfying the switch conditions.On a given train track snugly carrying λ , the Thurston intersection form ω H is easily computable in termsof the weight systems. To wit, if σ, ρ ∈ H ( λ ) then their intersection is equal to(19) ω H ( σ, ρ ) = 12 (cid:88) s (cid:12)(cid:12)(cid:12)(cid:12) σ ( r s ) σ ( (cid:96) s ) ρ ( r s ) ρ ( (cid:96) s ) (cid:12)(cid:12)(cid:12)(cid:12) where the sum is over all switches s of τ and r s and (cid:96) s are the half-branches which leave s from the rightand the left, respectively. Compare [PH92, § Train track coordinates for shear-shape cocycles.
In order to imitate the above construction forshear-shape cocycles, we first must explain how to build a train track from λ and a filling arc system α onits complement.Suppose that τ carries λ snugly; then the complementary components of τ ∪ α correspond to those of λ ∪ α . A smoothing of τ ∪ α is a train track τ α which is obtained by choosing tangential data at each ofthe points of τ ∩ α and isotoping each arc of α to meet τ along the prescribed direction. Each componentof S \ τ inherits an orientation from S , which in turn gives an orientation to the boundary (of the metriccompletion) of each subsurface. A smoothing τ α is standard if for each switch of τ α with an incoming halfbranch corresponding to an arc α i ∈ α , the incoming tangent vector to α i is pointing in the positive directionwith respect to the boundary orientation of the component of S \ τ containing α i ; see Figure 11. Figure 11.
Left: a geometric train track neighborhood of (cid:101) λ together with an arc system (cid:101) α . Right: The (preimage of the) standard smoothing τ α .Recall (Construction 5.6) that a geometric train track τ constructed from a hyperbolic structure X ∈ T ( S ), λ ∈ ML ( S ), and (cid:15) > (cid:15) -neighborhood of λ in X (for small enough values of (cid:15) ). HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 37
Construction 9.3 (Geometric standard smoothings) . Let λ ∈ ML ( S ) and X be a hyperbolic metric on S .Let α be a filling arc system in S \ λ , realized orthogeodesically on X . For small enough (cid:15) > α ∩ N (cid:15) ( λ )lies in a finite collection of leaves of O λ ( X ) and so each end of each arc of α defines a point in the quotient τ = N (cid:15) ( λ ) / ∼ .The geometric standard smoothing τ α is then obtained by attaching α onto the geometric train track τ at these points and smoothing in the standard way.Since α is filling, the components of X \ ( λ ∪ α ) are topological disks. In a geometric standard smoothing τ α , each complementary disk incident to an arc α of α has at least one spike corresponding to an ends ofthat α . Since no arc of α joins asymptotic geodesics of λ , the complementary polygons all have at least 3spikes and so we see that τ α is indeed a train track. Remark 9.4.
A geometric standard smoothing keeps track of the intersection pattern of λ with α on “eitherside” of τ , and the endpoints of α on a geometric traintrack τ (cid:15) ⊂ X constructed from λ by a parameter (cid:15) > (cid:15) → τ α is reminiscent of the construction of completing λ to a maximal lamination λ (cid:48) by “spinning” the arcs of α around the boundary geodesics of complementary subsurfaces to λ in thepositive direction to obtain spiraling isolated leaves of λ (cid:48) in bijection with the arcs of α . In Proposition 9.5below, we observe that by smoothing α onto τ in a standard way, axiom (SH3) allows us to assign weightsto the branches of τ α in such a way that the switch conditions are satisfied. Thus, for a shear-shape cocyclecarried by τ α , the weights deposited on the branches α ⊂ τ α encode “shape” data, rather than “shear” data.As such, we do not think of a standard smoothing as corresponding to the completion of λ to a maximallamination λ (cid:48) . Proposition 9.5.
Every shear-shape cocycle ( σ, A ) ∈ SH ( λ ) may be represented by a weight system w α ( σ )on a standard smoothing τ α that also carries λ . Moreover, the map σ (cid:55)→ w α ( σ ) extends to a linear isomor-phism H ( (cid:98) N α , ∂ (cid:98) N α ; R ) − ∼ = W ( τ α )where N α is a neighborhood of λ ∪ α , (cid:98) N α is its orientation cover, and H ( (cid:98) N α , ∂ (cid:98) N α ; R ) − is the − ι ∗ .In particular, this isomorphism realizes SH ( λ ; α ) and SH + ( λ, α ) as convex cones (with some open andsome closed faces) inside of W ( τ α ). Proof.
Let τ α be a standard smoothing of τ ∪ α and for each branch b of τ α , let t b denote a tie transverse to b . Evaluating a shear-shape cocycle σ on t b yields an assignment of weights w α ( σ ) : b → σ ( t b ) . By axiom (SH1) of Definition 7.12, this weight system does not depend on the choice of tie.To check that w α ( σ ) satisfies the switch conditions, we observe that there are two types of switches of τ α :those that come from switches of τ and those that come from smoothings of points of λ ∩ α . Axiom (SH2)implies that the switch condition holds at each of the former, while axiom (SH3) together with our choiceof smoothing ensures that w α ( σ ) satisfies the switch conditions at each of the latter.We note that this discussion does not rely on the positivity of σ on standard transversals, and so can berepeated to realize an arbitrary element of H ( (cid:98) N α , ∂ (cid:98) N α ; R ) − as a weight system on τ α . (cid:3) Let A = (cid:80) c i α i ; then on any smoothing τ α the identification of Proposition 9.5 restricts to an isomorphism SH ( λ ; A ) ∼ = { w ∈ W ( τ α ) : w ( b i ) = c i } where b i is the branch of τ α corresponding to α i . Indeed, these coordinates together with the parametrizationof transverse cocycles by weight systems on τ ≺ τ α (Proposition 9.2) give another proof that the differenceof any two shear-shape cocyles compatible with a given A ∈ B ( S \ λ ) is a transverse cocycle (Lemma 7.8). Remark 9.6.
The metric residue condition (Lemma 7.9) is still visible in train track coordinates, thoughit is somewhat obscured. Indeed, suppose that λ contains an orientable component carried on a component ζ of the geometric train track τ ; fix an arbitrary orientation of ζ . Take a geometric standard smoothing τ α of τ ∪ α . Reversing the tangential information as necessary, wecan then construct a (non-standard) smoothing of τ ∪ α so that every arc of α is a small branch entering ζ according to the orientation. Moreover, by reversing the sign of the weight on each arc which has itssmoothing data modified, this non-standard smoothing still carries shear-shape cocycles as a weight systems.But then by conservation of mass the total sum of the weights on the branches entering ζ must be 0.Hence in this setting the metric residue condition manifests as a condition embedded in the recurrencestructure of smoothings.The extended intersection form on SH ( λ ) also has a nice formula in terms of train tracks. Let τ be a(trivalent) train track snugly carrying λ and let τ α be a standard smoothing of τ ∪ α ; then for σ ∈ SH ( λ )and ρ ∈ H ( λ ), we have(20) ω SH ( σ, ρ ) = 12 (cid:88) s (cid:12)(cid:12)(cid:12)(cid:12) σ ( r s ) σ ( (cid:96) s ) ρ ( r s ) ρ ( (cid:96) s ) (cid:12)(cid:12)(cid:12)(cid:12) where the sum is over all switches s of τ α and r s / (cid:96) s are the right/left small half-branches. The proof of thisformula is the same as that of (19) and is therefore omitted; the only thing to note in this case is that thevalue does not change if one completes α by adding in arcs of zero weight.9.3. Piecewise-integral-linear structure.
A piecewise linear manifold is said to be piecewise-integral-linear or PIL with respect to a choice of charts if the transition functions are invertible piecewise-linearmaps with integral coefficients. The track charts that we have constructed from standard smoothings in thissection endow each cell SH ( λ ; α ) with a PIL structure which clearly extends over all of SH ( λ ) (compare[PH92, § W ( τ α ) are invariant under coordinate transformation, thus the integerpoints SH Z ( λ ) ⊂ SH ( λ ) are well defined.The PIL structure defined by train track charts gives a canonical measure µ SH in the class of the (6 g − SH ( λ ). Namely, if B ⊂ SH ( λ ) is a Borel set, then(21) µ SH ( B ) := lim R →∞ R · B ∩ SH Z R g − . Since the symplectic intersection form ω SH is constant (19) in a train track chart, the volume element definedby the (3 g − ∧ ω SH is a constant multiple of µ SH on each chart.We note that B ( S \ λ ) is cut out of | A fill ( S \ λ ) | by linear equations with integer coefficients, as is each cellof | A fill ( S \ λ ) | . Therefore, the integer lattice SH Z ( λ ) restricts to a integer lattice in the bundle SH ( λ ; α ) overevery cell B ( α ). Thus we obtain a natural volume element on the bundle over the k -skeleton of B ( S \ λ ),whenever it is not empty.9.4. Duality in train track coordinates.
We now take a moment to discuss shear-shape coordinates fromthe point of view of train track weight spaces; this discussion is motivated by that in [Thu98], and is meantto clarify how shear-shape cocycles fit into the broader theory of train tracks.We begin by recalling the analogy between shear coordinates for Teichm¨uller space and the “horosphericalcoordinates” for hyperbolic space. As observed by Thurston [Thu98, p. 42], projecting the Lorentz model H n = { x + . . . + x n − x n +1 = − | x n +1 > } to (cid:104) x , . . . , x n (cid:105) along a family of parallel light rays gives a parametrization for H n in terms of a half-space.In these coordinates, horospheres based at the boundary point ξ ∈ ∂ ∞ H n corresponding to the choice oflight ray are mapped to affine hyperplanes and geodesics from ξ are mapped to rays from the origin. When λ is maximal and uniquely ergodic, Bonahon and Thurston’s shear coordinates similarly realize T ( S ) as the space of positive transverse cocycles H + ( λ ), in which planes parallel to the boundary are levelsets of the hyperbolic length of λ and rays through the origin are Thurston geodesics. Equivalently, if τ is atrain track carrying λ then shear coordinates identify T ( S ) as a half-space inside W ( τ ).However, shear coordinates are no longer induced by a global projection. Instead, as noted by Thurston,they can be thought of as a map that takes a hyperbolic structure X to (the 1-jet of) its length function We remark that this coordinate system is in some sense dual to the paraboloid model of [Thu97, Problem 2.3.13]. Horo-spherical coordinates place an observer looking out from the center of a family of expanding horospheres, whereas the paraboloidmodel places an observer at another boundary point looking in.
HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 39 with respect to a given lamination. Shear coordinates are then a map not into W ( τ ) but into its dual space W ( τ ) ∗ (which can be identified with W ( τ ) via the non-degenerate Thurston symplectic form). The imagecone is then the positive dual of the cone of measures on λ .This formalism then indicates how shear coordinates generalize to maximal but non-uniquely ergodiclaminations. The map is the same, but now the positive dual of ∆( λ ) has angles obtained from the intersectionof hyperplanes: one for each ergodic measure on λ . Rays in the cone still correspond to geodesics, and affineplanes parallel to the bounding planes correspond with the level sets of hyperbolic length of the ergodicmeasures on λ .Our shear-shape coordinates come into play when λ is not maximal. In this case, one can go through theabove steps for each maximal train track τ , obtained from a snug train tack carrying λ by adding finitelymany branches. Since λ is carried on a proper subtrack of τ its cone of measures lives in a proper subspace E ⊂ W ( τ ). Taking the positive dual of ∆( λ ) and applying the isomorphism W ( τ ) ∼ = W ( τ ) ∗ induced by theThurston form then realizes Teichm¨uller space as a cone C in W ( τ ). By definition C ∩ E is exactly H + ( λ ),and one can check this demonstrates C as an affine H + ( λ ) bundle.However, the base of this bundle structure is not canonically determined, in part because E < W ( τ )is generally not symplectic. Moreover, the same hyperbolic structure is parametrized by elements in manydifferent maximal completions, and to achieve Mod( S )-equivariance one needs to understand how to comparecoordinates for different completions. Shear-shape space is designed to solve both of these problems, pickingout geometrically meaningful completions and gluing together the corresponding cones all while preservingthe bundle structure.Indeed, the shear-shape coordinates defined in Section 13 below associate to each hyperbolic structure anatural finite set of completions (corresponding to standard smoothings of snug train tracks plus geometricarc systems) together with a weight system on each completion. The discussion of this section (Proposition9.5 especially) then implies that the associated shear-shape cocycle is independent of the choice of completion,and that the corresponding train track charts glue together according to the combinatorics of B ( S \ λ ). Inthis picture, level sets of the hyperbolic length now correspond to bundles over B ( S \ λ ) whose fibers areaffine subspaces parallel to the boundary of H + ( λ ), while rays in SH + ( λ ) correspond to scaling both thecoordinate in B ( S \ λ ) as well as the coordinate in H + ( λ ).10. Shear-shape cooordinates for transverse foliations
We now show how the familiar period coordinates for a stratum of quadratic differentials can be reinter-preted as shear-shape coordinates. The main construction of this section is that of the mapI λ : F uu ( λ ) → SH ( λ )which records the vertical foliation of a quadratic differential and should be thought of as a joint extensionof [Mir08, Theorem 6.3] and [MW14, Theorem 1.2].The idea is straightforward: given some quadratic differential q ∈ F uu ( λ ), the complement S \ Z ( q ) ofits zeros deformation retracts onto a neighborhood N α ( q ) of λ ∪ α ( q ) for some filling arc system α ( q ) (whosetopological type reflects the geometry of q ). We may therefore identify the period coordinates of q as arelative cohomology class in (the orientation cover of) N α ( q ) with complex coefficients. The imaginary partof this class corresponds to λ , while its real part is the desired shear-shape cocycle I λ ( q ).The only obstacle to this plan is in showing that S \ Z ( q ) can actually be identified with a neighborhoodof λ ∪ α ( q ). To overcome this, we recall first in Section 10.1 how to reconstruct the topology of S \ λ from thehorizontal separatrices of q ; this guarantees that all relevant objects have the correct topological types. Wethen describe in Section 10.2 how to build from S \ Z ( q ) a train track τ α snugly carrying λ ∪ α ( q ) (Lemma10.6); this in particular allows us to identify S \ Z ( q ) as a neighborhood of λ ∪ α ( q ). We may then defineI λ ( q ) using the strategy outlined above and identify it as a weight system on τ α (Lemma 10.10).Section 10.3 contains a discussion of the global properties of the map I λ : piecewise linearity, injectivity,and its behavior with respect to the intersection pairing. In this section, we also record Theorem 10.15, whichstates that I λ is a homeomorphism onto SH + ( λ ). For purposes of convenience, the proof of this theoremis deduced from our later (logically independent) work on shear-shape coordinates for hyperbolic structures(Sections 12–15). See Remark 10.16. i.e., those elements of W ( τ ) ∗ which pair positively with every element in ∆( λ ) via the intersection form. Separatrices and arc systems.
Given a quadratic differential with | Im( q ) | = λ , our first tasktowards realizing | Re( q ) | as a shear-shape cocycle is to build a filling arc system α ( q ) on S \ λ that encodesthe horizontal separatrices of q . We begin by recalling how to recover the topology of S \ λ from the realizationof λ as a measured foliation on q .Recall that a boundary leaf (cid:96) of a component of S \ λ is a complete geodesic contained in its boundary.Note that infinite boundary leaves of S \ λ are in 1-to-1 correspondence with leaves of λ which are isolatedon one side, while finite boundary leaves (i.e., closed boundary components) are in 2-to-1 correspondencewith closed leaves of λ . The corresponding notion for measured foliations is that of singular leaves . Let F be a measured foliationon S and (cid:101) F denote its full preimage to (cid:101) S under the covering projection; then a bi-infinite geodesic path ofhorizontal separatrices (cid:96) is a singular leaf of (cid:101) F if for every saddle connection s comprising (cid:96) , the separatricesadjacent to s leave from the same side of (cid:96) (i.e., always from the left or always from the right).There is a fundamental correspondence between boundary leaves of a lamination and singular leaves of afoliation which we record below. Heuristically, collapsing the complementary regions of a lamination yieldsa foliation; the deflation map of Section 5.3 is a geometric realization of this phenomenon. Compare [Lev83,Figure 2] as well as [Min92, Lemma 2.1]. Lemma 10.1.
Let λ be a measured lamination on S and let F be a measure-equivalent measured foliation.Then there is a one-to-one, π ( S )–equivariant correspondence between the boundary leaves of (cid:101) S \ ˜ λ andsingular leaves of (cid:101) F . Moreover, singular leaves of (cid:101) F that share a common separatrix correspond to boundaryleaves of the same component of (cid:101) S \ ˜ λ .This lemma in particular allows us to read off the topological type of S \ λ from the horizontal separatricesof q . Set Ξ( q ) to be the union of the horizontal separatrices of q , equipped with the path metric. This 1-complex also comes equipped with a ribbon structure (that is, a cyclic ordering of the edges incident to eachvertex) and by thickening each component of Ξ( q ) according to this ribbon structure we see that Ξ( q ) canbe regarded as a spine for the components of S \ λ .Our construction of α ( q ) then records the dual arc system to the spine Ξ( q ) of S \ λ . Construction 10.2.
Let q be a quadratic differential on S with | Im( q ) | = λ . By the correspondence ofLemma 10.1, each horizontal separatrix of q corresponds to a pair of boundary leaves of the same componentof S \ λ . Each infinite separatrix corresponds to a pair of asymptotic boundary leaves, while non-asymptoticboundary leaves are glued along horizontal saddle connections. Dual to each horizontal saddle connection ofΞ( q ) is a proper isotopy class of arcs on S \ λ , and we set α ( q ) to be the union of all of these arcs.Since Ξ( q ) is a spine for S \ λ and α ( q ) consists of arcs dual to its compact edges, we quickly see that Lemma 10.3.
The arcs of α ( q ) are disjoint and fill S \ λ . Proof.
Each component of (cid:101) S \ (cid:101) λ has a deformation retract onto the universal cover (cid:101) Ξ of a component of Ξ( q ).In particular, as the interiors of the edges of (cid:101) Ξ are disjoint, duality implies that the arcs of ˜ α ( q ) can all berealized disjointly. As this picture is invariant under the covering transformation, this implies that the arcsare disjoint downstairs in S \ λ .Similar considerations also imply that the arc system is filling: let Σ be a component of S \ λ with universalcover (cid:101) Σ with spine (cid:101)
Ξ. By construction, the edges of ˜ α ( q ) in (cid:101) Σ are dual to the edges of (cid:101)
Ξ. Since Ξ( q ) is aspine for S \ λ , any loop in Σ is homotopic to a union of saddle connections, implying that any nontrivialloop must pass through an edge of α ( q ). Hence α ( q ) fills S \ λ . (cid:3) Period coordinates as shear-shape cocycles.
Now that we understand the relationship between λ and the horizontal data of q , it is easy to build objects T ∗ \ H ∗ and T ∗ on q of the same topological typeas λ and λ ∪ α ( q ). However, it is not immediate to actually identify these objects as neighborhoods of λ and λ ∪ α ( q ). Below, we deduce this from the stronger statement that they admit smoothings onto train trackssnugly carrying λ and λ ∪ α ( q ); compare [Mir08, Sections 5.2 and 5.3]. Construction 10.4 (Train tracks from triangulations) . Let H denote the set of all horizontal saddle con-nections on q and let T be a triangulation of q containing H . Let T ∗ be the 1-skeleton of the dual complexto T and let H ∗ denote the edges of T ∗ dual to H . Note that T ∗ is trivalent by definition. This is true because we have insisted that λ support a measure, and so no non-closed leaf may be isolated from both sides. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 41
Let ∆ denote a triangle of T with dual vertex v ∆ in T ∗ . Using the | q | -geometry of ∆ we may assigntangential data to v ∆ as follows (compare Figures 12 and 13). • If no edge of ∆ is horizontal, then a unique edge e has largest (magnitude of) imaginary part. Assigntangential data to v ∆ so that the dual edge to e is a large half-branch. • Otherwise, some edge of ∆ is horizontal and the other two edges have the same imaginary parts. Inthis case, we choose tangential data so that the horizontal edge corresponds to a small half-branchand leaves the large half-branch from the right, as seen by the large half-branch.We denote the resulting train track by τ α . The subgraph T ∗ \ H ∗ can also be converted into a train track τ by deleting the branches of τ α dual to H . Remark 10.5.
We observe that the edges of H ∗ correspond to the arcs of α ( q ) and τ α is a standard smoothingof τ ∪ α ( q ). Our convention for “standard” ensures that additivity in period coordinates corresponds toadditivity in train track coordinates. Figure 12.
An example of the train track τ α around a saddle connection. The thick blacklines are stems of horizontal separatrices of q while the light black lines are non-horizontaledges of the triangulation T . The dashed line is a branch of τ α \ τ .By construction, the graph T ∗ (equivalently, the train track τ α ) is a deformation retract of S \ Z ( q ).Similarly, T ∗ \ H ∗ (and τ ) are deformation retracts of the complement of the horizontal saddle connections.Together with our discussion above, this implies that τ has the same topological type as λ and τ α has thesame topological type as λ ∪ α ( q ).In order to actually realize these objects as neighborhoods of λ , we observe that we can build an explicitcarrying map from (a foliation measure equivalent to) λ onto τ . Lemma 10.6.
The train track τ carries λ snugly. The weight system on τ that specifies λ is exactly the(magnitude of) the imaginary parts of the periods of the edges of T . Proof.
Let all notation be as above and let F denote the (singular) horizontal foliation of q .One can directly build a homotopy of the nonsingular leaves of F onto τ : in a neighborhood of each edge e of T \ H there is a homotopy of the leaves of F onto the branch of τ dual to e . Now any leaf of F which passesthrough a triangle ∆ of T does so (locally) only twice and must pass through the side of ∆ with the largestimaginary part, which corresponds to a large half-branch of τ . The complement of the separatrix meeting thevertex of ∆ opposite to the side with largest imaginary part separates the the (locally) non-singular leavesof F passing through ∆ into two packets that can be homotoped onto τ , respecting the smooth structure atthe switch dual to ∆; compare Figure 13. Now the horizontal foliation F of q is measure equivalent to λ , and so as τ carries F it carries λ (snugnessfollows as τ and λ have the same topological type). The statement about the weight system follows fromour description of the carrying map. (cid:3) Now that we have identified τ as a snug train track carrying λ , we may in turn identify a neighborhoodof λ ∪ α ( q ) with (a thickened neighborhood of) τ α . With this correspondence established, we may nowdefine I λ ( q ) as the image of the real part of the period coordinates of q under the natural isomorphism oncohomology. Construction 10.7 (Definition of I λ ( q )) . Let
S, λ, q, α ( q ), and τ α be as above, Set M α to be a thickenedneighborhood of τ α (in the flat metric defined by q ) and let N α be a snug neighborhood of λ ∪ α ( q ) (takenin some auxiliary hyperbolic metric). Perhaps by shrinking N α , we may assume it embeds into M α as adeformation retract (this follows by snugness).Now τ α is itself a deformation retract of S \ Z ( q ), so the inclusion M α (cid:44) → S \ Z ( q ) is a homotopy equivalence;composing inclusions N α (cid:44) → M α (cid:44) → S \ Z ( q ) and lifting to the orientation covers yields the isomorphism(22) H ( (cid:98) S, Z ( √ q ); C ) j ∗ −→ H ( (cid:98) N α , ∂ (cid:98) N α ; C )where the hats denote the corresponding orientation covers. As the composite retraction respects the coveringinvolution ι , this isomorphism also identifies − ι ∗ . We therefore defineI λ ( q ) = Re( j ∗ Per( q ))where Per( q ) are the period coordinates for q , and where the real part is taken relative to the natural splitting C = R ⊕ i R . Remark 10.8.
From the above construction, a basis consisting of branches for the weight space of τ α (equivalently, a basis for H ( (cid:98) N α , ∂ (cid:98) N α ; Z ) of dual arcs) picks out a basis for H ( (cid:98) S, Z ( √ q ); Z ). Moreover, eachrelative cycle is realized geometrically as a saddle connection (as opposed to concatenations, thereof).To see that I λ ( q ) is indeed a shear-shape cocycle, we need only observe that the values on standardtransversals to α ( q ) are all positive. This follows essentially by definition of the orientation cover andconstruction of α ( q ). To wit: if α is an arc of α ( q ) dual to a saddle connection s , and t is a standardtransversal to α , then the canonical lifts of t are mapped to those of s under the isomorphism (22). As theperiods of √ q increase as you move along the (oriented) horizontal foliation of ( (cid:98) S, √ q ), this implies that thevalue of I λ ( q ) on either of the lifts of t is exactly the length of the saddle connection s .Therefore, we see that the weighted arc system underlying I λ ( q ) is none other than A ( q ) := (cid:88) α ∈ α ( q ) c α α where c α is the | q | -length of the horizontal saddle connection dual to the arc α . Remark 10.9.
Naturality of all of the isomorphisms involved quickly implies that this construction doesnot depend on the choice of initial triangulation T . Indeed, suppose that T and T are two triangulationsgiving rise to train tracks τ and τ and hence shear-shape cocycles σ and σ . Since both τ i carry λ ∪ α ( q )snugly, Lemma 10.6 implies that they have a common refinement τ . Lifting the inclusions N ( τ ∪ α ( q )) (cid:44) → N ( τ i ∪ α ( q )) (cid:44) → S \ Z ( q )to their orientation covers and drawing the appropriate commutative diagram of cohomology groups, we seethat the shear-shape cocycles built from each T i coincide as weight systems on the common refinement τ .For use in the sequel, we record below the weight systems on τ α corresponding to λ and I λ ( q ). The prooffollows by combining the constructions above with the discussion in Section 9 and is therefore left to thefastidious reader. See also Figure 13.For a complex number z , define [ z ] + = (cid:26) z if arg( z ) ∈ [0 , π ) − z if arg( z ) ∈ [ π, π ) . Observe that [ z ] + = [ − z ] + for all z ∈ C . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 43
Lemma 10.10.
Let all notation be as above, and for each edge e of T let b e denote the branch of τ α dualto it. Then the assignment b e (cid:55)→ (cid:20)(cid:90) e √ q (cid:21) + defines a complex weight system w ( q ) on τ α satisfying the switch conditions. Moreover,Im( w ( q )) = λ and Re( w ( q )) = I λ ( q ) .b s (cid:96) s r s oo (cid:96) o r b s (cid:96) s r s o (cid:96) o r o Figure 13.
Local pictures of the different types of switches of τ α . Here we have illustratedthe images of each triangle under the holonomy map. The orientation of each edge shouldbe interpreted as indicating the value of [ · ] + so that the edge vector is exactly the complexweight assigned to the dual branch of τ α . The graphical conventions of this figure mirrorthose of Figure 12.10.3. Global properties of the coordinatization.
In this section, we show that the map I λ definedabove gives a global coordinatization of MF ( λ ) ∼ = F uu ( λ ). First, we record certain global properties of thismap; as it is defined by reinterpreting period coordinates as shear-shape cocycles, it preserves many of thestructures imposed by period coordinates.For example, it follows by construction that I λ respects the stratification of each space. That is, if q ∈ QT ( k , . . . , k n ) ∩F uu ( λ ), then the spine dual to α ( q ) has vertices of valence k +2 , . . . , k n +2. In a similarvein, since both F uu ( λ ) and SH ( λ ) have local cohomological coordinates (which induce PIL structures) wecan deduce the following: Lemma 10.11.
For any λ ∈ ML ( S ), the map I λ is Mod( S )[ λ ]–equivariant and PIL. Proof.
Equivariance follows from the naturality of our construction: all combinatorial data (arc systems,train tracks, etc.) can be pulled back to a reference surface equipped with λ , so changing the marking by anelement of Mod( S )[ λ ] acts by transforming the combinatorial data on the reference surface.The piecewise-linear structure on F uu ( λ ) (respectively, SH ( λ )) is given by period coordinates (respec-tively, cohomological coordinates in a neighborhood/train track coordinates) and so the map is by construc-tion piecewise-linear. Integrality comes from the fact that a homotopy equivalence induces an isomorphismon cohomology with Z -coefficients, hence takes integral points to integral points. (cid:3) The Thurston intersection pairing gives us a powerful tool to understand constraints on the image of I λ ;in particular, I λ ( q ) must be a positive shear-shape cocycle. Indeed, the tangential structure of the train track τ α at each switch provides us with an identification of each triangle ∆ of T with an oriented simplex. Withrespect to this orientation, we can compute the area of ∆ by taking (one half of) the cross product of twoof its sides. Comparing the formula for the cross product with the Thurston intersection pairing (20) thenallows us to see that the intersection of λ and I λ ( q ) is exactly the area of q ; compare [Mir08, Lemma 5.4]. We recall that a PL map between PIL manifolds is itself PIL if it sends integral points to integral points.
Proposition 10.12.
For all η ∈ MF ( λ ) and all µ ∈ ∆( λ ), ω SH (I λ ( η ) , µ ) = i ( η, µ ) . In particular, I λ ( MF ( λ )) ⊆ SH + ( λ ) . The proof of this lemma is made technical by the fact that if µ and µ (cid:48) ∈ ∆( λ ) are ergodic but notprojectively equivalent then they are mutually singular. To deal with this difficulty, we build a flat structureon the subsurface filled by µ by integrating against µ and I λ ( η ). The triangulation T then induces acombinatorially equivalent triangulation of this new flat structure by saddle connections, allowing us tocompare the area of this new flat metric (computed via cross products) with the Thurston form on ouroriginal train track τ α . This inverse construction will also be used in the proof of Proposition 10.14. Proof.
We begin by observing that since µ ∈ ∆( λ ), there is a union of minimal components of the horizontalfoliation of q ( η, λ ) that supports µ . Call this subfoliation F and let Y denote the subsurface filled by F on q ( η, λ ). Note that ∂Y must be a union of horizontal saddle connections, hence is contained in anytriangulation T used to define τ α . In particular, T | Y is a triangulation of Y .Since η and µ are realized transversely on q ( η, λ ) and this specific realization of η is non-atomic (as anyclosed leaves of η have become vertical cylinders), we can compute their intersection number as(23) i ( η, µ ) = (cid:90) S η × µ = (cid:90) Y η × µ. We now build a new flat structure on Y whose conical singularities coincide with those of Y ; the salientfeature is that T | Y can be straightened out to a triangulation by saddle connections on the new singular flatstructure that reflects the geometry of µ . To construct the new singular flat structure, we build charts froma neighborhood of each triangle ∆ ⊂ T | Y to C and describe the transitions.Each triangle ∆ of T is dual to a switch s with an edge that is dual to a large half-branch b incident to s . Orient τ α ∩ ∆ so that a train traveling along b toward s is moving in the positive direction. The otheredges r and (cid:96) of ∆ are dual to the half-branches of τ α to the right and left of s , respectively. The vertices o r , o (cid:96) are adjacent to r and (cid:96) , respectively, and the vertex o is opposite b ; see Figure 13. On the interior ofeach triangle ∆, we orient the leaves of F parallel to b . The leaves of η are given the orientation so that theordered basis of tangent vectors to λ and η at each point agree with the underlying orientation of S . Withthis orientation, the measures η and λ induce smooth real 1-forms dη and dλ that look locally like dx and dy , respectively (as opposed to | dx | and | dy | , respectively).Restricted to the interior of ∆, the local orientation of the leaves of η also gives the measure µ the structureof a measurable 1-form that we call dµ . Spreading out the measure on a closed leaf of µ over the horizontalcylinder of λ corresponding to its support as necessary, we get that the map F : p ∈ ∆ (cid:55)→ (cid:90) γ p dη + idµ ∈ C obtained by integrating along a path γ p from o r to p is continuous, isometric along leaves of F , and non-decreasing along leaves of η . We compute F ( o ) = I λ ( η )( r ) + iµ ( r ) and F ( o (cid:96) ) = I λ ( η )( b ) + iµ ( b ) . Transverse invariance and additivity of µ gives(24) F ( o (cid:96) ) − F ( o ) = I λ ( η )( (cid:96) ) + iµ ( (cid:96) ) . Since the real and imaginary parts of F are non-decreasing along the horizontal and vertical foliations, thepair ( F ( o ) , F ( o (cid:96) )) forms a positively ordered basis for C . Let ∆ (cid:48) be the convex hull of ( F ( o r ) , F ( o ) , F ( o (cid:96) )).The area of ∆ (cid:48) may be computed as half the cross product of F ( o ) and F ( o (cid:96) ). Using equation (24) andlinearity of the cross product, we have the formula(25) Area(∆ (cid:48) ) = 12 (cid:12)(cid:12)(cid:12)(cid:12) I λ ( η )( r ) I λ ( η )( (cid:96) ) µ ( r ) µ ( (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12) > . Now for each ∆ the map F may be extended to an open set U (∆) in Y \ Z ( q ) that contains ∆ (minus itsvertices) and so that for every p ∈ U (∆) there is a unique non-singular | q | -geodesic segment γ p joining o r to p . We claim that moreover, we may choose U (∆) so that ∆ (cid:48) ⊂ F ( U (∆)); see Figure 14. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 45 ∆ U (∆) oo r o (cid:96) F ∆ (cid:48) Figure 14.
Integrating against η and µ defines a new flat structure on triangles. Thesecharts piece together to give a new half-translation structure on the subsurface filled by µ .If not, there is some vertex v of T Y \ ∆ such that F ( v ) ∈ ∆ (cid:48) \ F (∆). Indeed, by construction, U (∆) isa star-shaped neighborhood about the vertex o r of ∆, so there is a saddle connection joining o r to v . Thissaddle connection passes through or shares a vertex of an edge e of ∆. Moreover, we may find v so that thetriangle ∆ formed by e and v is singularity free and contained in U (∆). But now, the straightening ∆ (cid:48) of F (∆ ) in C lies inside ∆ (cid:48) with the wrong orientation since F ( v ) lies between F ( e ) and the correspondingedge of ∆ (cid:48) . This is a contradiction to the fact that F is non-decreasing along leaves of η . So we may assumethat ∆ (cid:48) ⊂ F ( U (∆)).If ∆ ⊂ T Y shares an edge with ∆, then the construction of the map F associated to ∆ agrees with F on U (∆) ∩ U (∆ ) up to multiplication by ± ) and translation by the period of the arc connecting the basepoints o r of each triangle. Thus thesetriangles glue up to a half-translation structure on Y \ Z equipped with a triangulation by saddle connectionscorresponding to T | Y .In our new flat structure on Y , µ is measure equivalent to the horizontal foliation and (the restriction of) η is equivalent to the vertical foliation. Hence we obtain (cid:90) Y η × µ = (cid:88) ∆ i ∈ T | Y Area(∆ (cid:48) i ) = (cid:88) ∆ i ∈ T | Y (cid:12)(cid:12)(cid:12)(cid:12) I λ ( η )( r ) I λ ( η )( (cid:96) ) µ ( r ) µ ( (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12) = ω SH (I λ ( η ) , µ )where the second equality follows from (25) and the third from (20). Combining this with formula (23)completes the proof of the lemma. (cid:3) From the proof of Proposition 10.12 we extract the following, which allows us to reconstruct a (triangu-lated) quadratic differential from a sufficiently positive shear-shape cocycle, inverting Construction 10.4.
Lemma 10.13.
Let τ be a train track snugly carrying λ and let τ α be a standard smoothing of λ ∪ α .Suppose that σ ∈ SH ( λ ) is represented by a weight system on τ α so at every switch s of τ α , the contribution12 (cid:12)(cid:12)(cid:12)(cid:12) σ ( r s ) σ ( (cid:96) s ) λ ( r s ) λ ( (cid:96) s ) (cid:12)(cid:12)(cid:12)(cid:12) of s to ω SH ( σ, λ ) is positive. Then there exists a quadratic differential q ∈ F uu ( λ ) so that I λ ( q ) = σ and thedual triangulation to τ α is realized by saddle connections on q .This implies that we can locally invert I λ by building a quadratic differential out of triangles whose edgeshave specified periods. In particular, I λ is injective. Proposition 10.14.
For any λ ∈ ML ( S ), the map I λ is a homeomorphism onto its image. Proof.
To see that I λ is injective, we observe that Lemma 10.13 provides a (left) inverse map ∆ λ to I λ .Indeed, suppose that σ = I λ ( q ) for some q and pick a triangulation T as in Construction 10.4; let τ α denotethe dual train track. Applying Lemma 10.13 then constructs a quadratic differential q (cid:48) on which each edge of T is realized as a saddle connection. Since q and q (cid:48) have the same periods with respect to the same geometrictriangulation, they must be equal. To prove that I λ is continuous, we first observe that I λ is by definition continuous on the closure SH ( λ ; α ( q )) of any cell, as it is induced by a continuous mapping on the level of cohomology. In general,we need only exploit this fact together with a standard reformulation of sequential continuity: a function f : X → Y is continuous if and only if every convergent sequence x n → x has a subsequence x n k so that f ( x n k ) → f ( x ).So let q n → q ∈ F uu ( λ ). The polyhedral structure of SH ( λ ) is locally finite, so for n large enough, I λ ( q n )is contained in a finite union of cells. After passing to a subsequence q n k , we may assume that q n k all sharethe same underlying (maximal) arc system β completing α . In particular, I λ ( q n k ) ∈ SH ( λ ; β ) for all k andso I λ ( q n k ) → I λ ( q ) follows from continuity on cells. Therefore I λ is a continuous injective map betweenEuclidean spaces of the same dimension (Proposition 8.5 and Corollary 8.2) and so invariance of domainguarantees it is a homeomorphism onto its image. (cid:3) The image of I λ . In light of Lemma 10.13, to show that I λ surjects onto SH + ( λ ) it would suffice to showthat every positive shear-shape cocycle can be realized as a weight system on a train track where everyswitch contributes positively to the intersection form. However, it is rather complicated to show that everypositive shear-shape cocycle admits such a representation (see the discussion in Remark 10.16 just below).Instead, we deduce this fact using the commutativity of Diagram (2) and the results appearing in Sec-tions 12–15 coordinatizing hyperbolic structures by shear-shape cocycles. We emphasize, however, thatTheorem 10.15 is logically independent from the work done in Sections 12–15 that leads to its proof. Weinclude the statement here (as opposed to after Section 15) to provide some closure to our discussion of theparametrization of MF ( λ ) by shear-shape coycles. Theorem 10.15.
The map I λ : F uu ( λ ) → SH + ( λ ) is a homeomorphism. Proof.
In Section 13 we define the geometric shear-shape cocycle σ λ ( X ) ∈ SH ( λ ) associated to a hyperbolicmetric X ∈ T ( S ) and show (Theorem 13.13) that σ λ ( X ) = I λ ( O λ ( X )). In Section 15 we prove Theorem12.1, which states that the map σ λ : T ( S ) → SH + ( λ ) is a homeomorphism. In particular, σ λ is surjectiveand hence so is I λ . Together with Proposition 10.14 this implies the theorem. (cid:3) Remark 10.16. If λ is a maximal lamination, one can deduce surjectivity of I λ by appealing to the theoryof “tangential coordinates” for measured foliations transverse to λ . In general, given τ snugly carrying λ ,tangential coordinates can be constructed as a quotient of R b ( τ ) by a vector subspace spanned by vectorsthat model the change of length of branches of a train track on either side of a switch after a small “fold” or“unzip.” When λ is maximal, there is a linear isomorphism from shear coordinates to tangential coordinatesvia the symplectic pairing ω H ; we refer the interested reader to [Thu98, Section 9] or [PH92, § λ on τ together with positive tangential data give τ the structure of a bi-foliated Euclidean band complex. If the tangential data satisfy a collection of triangle-type inequalities, this band complex can be “zipped up” to obtain a bi-foliated flat surface with conicalsingularities. When defined, the linear transformation mapping tangential coordinates to shear coordinatespreserves the intersection number, hence positivity.A standard positivity argument (see [Thu82, Proposition 9.7.6] or [Thu98, Theorem 9.3]) shows thatany tangential data with positive intersection with λ has a positive representative, hence defines a foliationtransverse to λ . In particular, the map from MF ( λ ) to the space of tangential coordinates with positiveintersection with λ is surjective. As the space of tangential coordinates with positive intersection is isomorphicto the H + ( λ ), this completes the proof of surjectivity in the maximal case.This being considered, even in the case when λ is maximal “it is harder to see the [positivity] inequalitiessatisfied by the shear coordinates [than the tangential coordinates]” [Thu98, p. 45] and it is not clear how torun the “standard positivity argument” without passing through tangential coordinates. We have thereforechosen to prove Theorem 10.15 in a way that avoids developing a theory of tangential coordinates dual toshear-shape cocycles. Instead, we take advantage of the relationship between the Thurston intersection formon SH ( λ ) and the length of λ on a given hyperbolic surface, as exploited in the proof of Theorem 12.1 (seein particular Claim 15.8). Here, positive means that there is a representative of the tangential data that is positive on each branch of τ . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 47
Flat deformations in shear-shape coordinates
The identification of Section 10 between periods of saddle connections and the values of the shear-shapecocycle I λ ( q ) immediately allows us to transport certain flows on F uu ( λ ) to shear-shape space. Moreover,Theorem 10.15 affords a new perspective on the “tremor deformations” of [CSW20] (see Definition 11.3). The horizontal stretch.
We begin by observing that the space SH + ( λ ) carries a natural R > actiongiven by scaling both the underlying arc system A and the values assigned to test arcs (equivalently, thecorresponding cohomology class or the weights on a train track realization). Using our correspondencebetween period coordinates and shear-shape cocycles (Lemma 10.10), we see that this dilation expands thereal part of each period, so the corresponding flat deformation is just a horizontal stretch. Lemma 11.1.
Let q ∈ F uu ( λ ); then(26) I λ (cid:18)(cid:18) e t
00 1 (cid:19) q (cid:19) = e t I λ ( q )for all t ∈ R .In particular, we see that our coordinatization linearizes the expansion of the strong unstable foliationunder the Teichm¨uller geodesic flow. Horocycle flow and tremors.
We now consider the horocycle flow on F uu ( λ ), which is just the restrictionof the standard horocycle flow h s to the strong unstable leaf. An easy computation shows that for everysaddle connection e of q , one has(27) (cid:20)(cid:90) e (cid:112) h s q (cid:21) + = (cid:32) Re (cid:20)(cid:90) e √ q (cid:21) + + s Im (cid:20)(cid:90) e √ q (cid:21) + (cid:33) + i Im (cid:20)(cid:90) e √ q (cid:21) + (here we have invoked the [ · ] + function to avoid fussing over square roots and orientations).With the help of Lemma 10.10 we may translate this into the language of transverse and shear-shapecocycles to observe Lemma 11.2.
The map I λ takes horocycle flow to translation by λ in a time preserving way. In symbols,I λ ( h s q ) = I λ ( q ) + sλ. More generally, we can perform a similar deformation for any measure µ supported on λ , resulting inthe tremor flow along µ . First defined by Chaika, Smillie, and Weiss in the context of Abelian differentials,the tremor trem µ ( q ) of a quadratic differential q = q ( η, λ ) by a measure µ ∈ ∆( λ ) is the unique quadraticdifferential specified by shearing η by µ and leaving λ fixed. Why this makes sense (note that η and µ maynot fill S ) and why it can be continued for all time present significant technical challenges in [CSW20] (see §§ λ , let | ∆( λ ) | ± denote the vector space of all signed transverse measures on λ ; thisis naturally a vector subspace of H ( λ ) of dimension at most 3 g − λ . Definition 11.3.
Let q ∈ F uu ( λ ) and let µ ∈ | ∆( λ ) | ± . Then the tremor trem µ ( q ) of q along µ is the uniquequadratic differential specified by(28) I λ (trem µ ( q )) = I λ ( q ) + µ. Note that the fact that I λ ( q ) + µ ∈ SH + ( λ ) follows by affinity of the Thurston form (Lemma 8.3). Remark 11.4.
Technically, the deformation considered above is a “non-atomic tremor” in the language of[CSW20]. One can also consider “atomic tremors,” which transform q by twisting along certain admissibleloops of horizontal saddle connections.In shear-shape coordinates, these admissible loops correspond to certain simple closed curves in thecomplementary subsurfaces. Atomic tremors are then realized by appropriately shearing the underlying arcsystem A ( q ) along the curves and transporting the transverse cocycle using the affine connection coming This is just the Teichm¨uller geodesic flow normalized so that the horizontal foliation remains constant. Applying thestandard geodesic flow takes (I λ ( q ) , λ ) to ( e t/ I λ ( q ) , e − t/ λ ). from train-track coordinates. Of course, one can also define tremors along more complicated laminationscontained in S \ λ as well.For the convenience of the reader familiar with the terminology of [CSW20], we have included a dictionarywhich translates between our notation and theirs (at least when the horizontal lamination is filling — whenit is not, one must replace ∆( λ ) with a subset of the zero set of λ and take more care). See Figure 15.Shear-shape cocycles Foliation cocycles∆( λ ) C + q | ∆( λ ) | ± T q ω SH ( η, µ ) = i ( µ + , η ) − i ( µ − , η ) signed mass L q ( µ ) i ( µ + , η ) + i ( µ − , η ) total variation | L | q ( µ ) Figure 15.
Translating between our language of shear-shape cocycles and the “foliationcocycles” of [CSW20]. Throughout, we assume that q = q ( η, λ ) where λ is filling (equiva-lently, q has no loops of horizontal saddle connections). We have written a signed transversemeasure µ as µ = µ + − µ − ∈ | ∆( λ ) | ± , where µ ± ∈ ∆( λ ).We can now immediately deduce certain properties of the tremor map from the structure of SH + ( λ ) andthe intersection pairing. While we will not use these results in the sequel, we have chosen to include themin order to to demonstrate the utility of our new perspective on these deformations. For example, using ourcoordinates one can easily deduce that (non-atomic) tremors leave horizontal data invariant and hence canbe continued indefinitely while remaining in the same stratum. Lemma 11.5.
For any q ∈ F uu ( λ ) and µ ∈ | ∆( λ ) | ± , the tremor path trem tµ ( q ) is defined for all time andis completely contained in SH ( λ ; A ( q )). In particular, { trem tµ ( q ) } always remains in the same stratum. Remark 11.6.
The above Lemma is one specific instance of a much more general phenomenon. The globaldescription of F uu ( λ ) afforded by shear-shape coordinates allows one to formulate a general criterion forextending affine period geodesics, a topic which the authors hope to address in future work.Using our interpretation of tremors as translation, it is similarly easy to describe how tremors interactwith other flat deformations. Compare with Propositions 6.1 and 6.5 of [CSW20]. We leave proofs to thereader, as they follow immediately from (28) and (26). Lemma 11.7.
Let q ∈ F uu ( λ ). Then for any µ ∈ | ∆( λ ) | ± and for g t = (cid:18) e t/ e − t/ (cid:19) , we have that g t trem µ ( q ) = trem e t/ µ ( g t ( q )) . Additionally, for any µ , µ ∈ | ∆( λ ) | ± , we have thattrem µ ( q ) trem µ ( q ) = trem µ + µ ( q ) = trem µ ( q ) trem µ ( q ) . In particular, tremors commute with the horocycle flow.12.
Shear-shape coordinates for hyperbolic metrics
We now parametrize hyperbolic structures on S by shear-shape cocycles for a measured geodesic lamina-tion λ . With respect to the Lebesgue measure on ML ( S ), the generic lamination cuts a hyperbolic surfaceinto ideal triangles. As all ideal triangles are isometric, Bonahon and Thurston’s shearing coordinates needonly take into account the “shear” between pairs of complementary triangles to describe a hyperbolic struc-ture. As our objective is to generalize these coordinates to laminations with arbitrary topology, we musttherefore combine the data of the geometry of hyperbolic metrics in complementary subsurfaces with theshearing data between them. Shear-shape space SH ( λ ) is well suited to this task.In the following Sections 13–15, we explain how to associate a “geometric shear-shape cocycle” to ahyperbolic metric and prove that the space of positive shear-shape cocycles coordinatizes Teichm¨uller space: HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 49
Theorem 12.1.
The map σ λ : T ( S ) → SH + ( λ ) that associates to a hyperbolic metric its geometric shear-shape cocycle is a stratified real-analytic homeomorphism.As detailed in the Introduction, combining this theorem with Theorems 2.1 and 10.15 implies that theorthogeodesic foliation map O λ is a homeomorphism, and consideration of the earthquake/horocycle flowsin SH + ( λ ) coordinates then proves the conjugacy on slices (Theorem D). Fixed complementary subsurfaces.
By definition (see Section 13.2), the weighted arc system A ( X )underlying σ λ ( X ) exactly identifies the geometry of X \ λ via Theorem 6.4. Setting T ( S ; A ) := { X ∈ T ( S ) : A ( X ) = A } , Theorem 12.1 therefore implies that T ( S ; A ) is nonempty if and only if A ∈ B ( S \ λ ). Remark 12.2.
The authors do not know a proof of this fact that does not factor through Theorem 12.1except in some special cases (for example, when the complement of λ is polygonal, or when λ is a union ofsimple closed curves). In fact, since SH + ( λ ) is an affine cone bundle over B ( S \ λ ) (Proposition 8.5), we see that Corollary 12.3.
For each A ∈ B ( S \ λ ), the set T ( S ; A ) is a real-analytic submanifold of T ( S ) and therestriction of σ λ to T ( S ; A ) → SH + ( λ ; A ) ∼ = H + ( λ )is a real-analytic homeomorphism.In this setting, the correspondence between T ( S ; A ) and H + ( λ ) is a natural generalization of shearcoordinates, since the complementary subsurfaces to λ are always isometric. In fact, the shape-shiftingdeformations built to deform X by some s ∈ H ( λ ) (see the proof sketch of Theorem 12.1 just below) restrictto cataclysms/shear maps in the sense of [Bon96, Section 5]. In particular, if s represents a measure supportedon λ , then the shape-shifting deformation determined by s is part of an earthquake in s (Corollary 15.2);if s is a multiple of σ λ ( X ), the shape-shifting transformation can sometimes be identified with part of a(generalized) stretch ray (Propositions 15.12 and 15.17).In addition to being non-empty, T ( S ; A ) is structurally rich; the authors hope to explore this space furtherin future work. Of particular interest is the (degenerate) Weil–Petersson pairing on this locus and its relationwith the Thurston symplectic form and Masur–Veech measures. A sketch of the proof.
Since the proof of Theorem 12.1 spans several sections (two of which consist ofinvolved constructions of the relevant objects), we devote the remainder of this section to a broad-strokesoutline of the arguments involved. Our exposition throughout these sections is mostly self-contained, but wesometimes refer to [Bon96] for proofs and to [Thu98] for inspiration.We begin in Section 13 by defining the map σ λ . Under the correspondence established in Theorem 6.4, weassociate to X the weighted arc system A ( X ) recording the hyperbolic structure on X \ λ . We cut X alongthe (ortho)geodesic realization of λ ∪ α into a union of (degenerate) right-angled polygons, and measurethe shear between certain pairs of polygons. We then argue using train tracks that it suffices to record theshearing data comprising σ λ ( X ) on short enough arcs k transverse to λ and disjoint from α ( X ). The value of σ λ ( X ) on short k may then be defined by isotoping k to a path connecting vertices of the spine Sp and builtof segments alternating between leaves of λ and of O λ ( X ), then measuring the total (signed) length along λ . These measurements are equivalent to Bonahon and Thurston’s method of measuring shears (via thehorocyclic foliation) when k is short enough, but cannot be globally derived from theirs due to obstructionscoming from complementary subsurfaces.The proof that σ λ is a homeomorphism then follows the same general steps as appear in [Bon96]. Afterproving that σ λ is injective and lands inside of SH + ( λ ) (Proposition 13.12 and Corollary 13.14), we thenshow that it is open (Theorem 15.1) and proper. Since SH + ( λ ) is a cell (Proposition 8.5), invariance ofdomain then implies that σ λ must be a homeomorphism.Our proof of injectivity mirrors that of [Bon96, Theorem 12] with an additional invocation of Theorem 6.4.For properness we mostly appeal to [Bon96, Theorem 20] but need to discuss complications that arise from One can of course complete λ to a maximal lamination and then specify the shear coordinates on each of the added leaves,but then one must be very careful to ensure that these shears satisfy the relations coming from the metric residue condition.The argument then requires an involved computation with train tracks carrying the completed lamination. the piecewise-linear structure of shear-shape space. Similarly, our broad-strokes strategy to prove opennessparallels that of [Bon96, § ϕ s for all small enough deformations s of σ λ ( X ) (see Section 14). Deforming X by post-composing its charts to H with ϕ s then yields a surface X s with σ λ ( X s ) = σ λ ( X ) + s .It is in the construction of ϕ s , performed in Section 14, where our discussion truly diverges from [Bon96]and [Thu98]. When λ is maximal, one can specify ϕ s by shearing X along the leaves of λ (i.e., performinga cataclysm). Even in the maximal case this procedure is delicate, hinging on the convergence of infiniteproducts of small M¨obius transformations (compare Section 14.2). In the non-maximal case, we must alsosimultaneously account for the changing shapes of complementary subsurfaces (which also introduces extracomplications into the shearing deformations since the shapes of spikes are changing). See the introductionto Section 14 for a more granular description of the construction of ϕ s .13. Measuring hyperbolic shears and shapes
In this section, we take our first steps towards proving Theorem 12.1 by describing how to associate toany hyperbolic surface X a geometric shear-shape cocycle σ λ ( X ) in a natural way; this yields the map σ λ : T ( S ) → SH ( λ ) . After fixing some notational conventions that we will use throughout the sequel, we define σ λ ( X ) by firstspecifying its underling arc system A ( X ) in a variety of equivalent ways. After doing so, we define the shearbetween “nearby” hexagons analogously to Bonahon and Thurston; placing all of this data onto a standardsmoothing τ α of a geometric train track is therefore enough to specify σ λ ( X ) (Lemma 13.6).We then show that the data of shears between any two nearby hexagons can be recovered from the weightsystem on τ α , even if those hexagons are not “visible” to τ α (Lemma 13.9). This in particular implies thatour choice of τ α does not actually matter, and hence σ λ ( X ) is well-defined.We then conclude the section by proving some initial properties of σ λ . Proposition 13.12 shows that themap is injective following an argument of Bonahon, while in Theorem 13.13 we show that our map capturesthe geometry of the orthogeodesic foliation.13.1. Preliminaries and notation.
In this section, we discuss the geometry of a geodesic lamination on ahyperbolic surface and fix notation in preparation of our definition of the geometric shear-shape cocycle ofa hyperbolic structure.Throughout, we use the symbol λ to refer to both the measured lamination λ and its support, realizedgeodesically with respect to any number of hyperbolic metrics. We refer to Remark 7.10 for a discussionof how to relax the assumption that λ is measured. We reserve S to denote a topological surface and Σthe topological type of a component of S \ λ , while X and Y will denote their hyperbolic incarnations. Wealso adopt the following family of notational conventions: the expression g ⊂ λ means that g is a leaf of λ ,and Y ⊂ X \ λ means that Y is a component of (the metric completion of) X \ λ , etc. The notation of[Bon96] is used as inspiration, since we will make direct appeals to the results therein. However, our situationrequires more care, since we have more objects to keep track of. A key difference is that we will focus noton the relative shear between complementary subsurfaces of X \ λ , but on the relative positioning of pairsof boundary leaves of λ , equipped with a natural collection of basepoints determined by the orthogeodesicfoliation. Hexagons.
Given X ∈ T ( S ) and λ ∈ ML ( S ), realize λ geodesically on X . Construct the orthogeodesicfoliation O λ ( X ) on X with piecewise geodesic spine Sp and dual arc system α = α ( X ), realized orthogeodesi-cally with respect to X and λ . The union λ α = λ ∪ α is a geometric object on X that fills; that is, the metriccompletion of X \ λ α is a union of geometric pieces that are topological disks, possibly with some points onthe boundary removed corresponding to spikes. We lift the situation to universal covers (cid:102) λ α ⊂ (cid:101) X , where wehave also the full preimages (cid:102) Sp , (cid:101) λ , (cid:101) α , etc., of various objects.Let H be the vertex set of (cid:102) Sp ; we will sometimes refer to v ∈ H as a hexagon . Indeed, to v there isassociated a component H v of (cid:101) X \ (cid:102) λ α which is generically a degenerate right-angled hexagon, though H v may also be a regular ideal or right-angled polygon, for example. We reiterate that, by abuse of terminology, any complementary component H v of (cid:101) X \ (cid:102) λ α is called a hexagon , no matter its shape. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 51 If { H v : v ∈ H} contains components that are not degenerate right-angled hexagons in the usual sense,then α corresponds to a simplex of A fill ( S \ λ ) of non-maximal dimension (or the empty set, if λ is fillingand α is empty). One may always include α in a maximal arc system β , which necessarily defines a simplexof full dimension. The complementary components of (cid:101) X \ (cid:102) λ β are now degenerate right-angled hexagons inthe usual sense, and gluing them in pairs along β \ α gives the more general “hexagons” of (cid:101) X \ (cid:102) λ α . We willoften tacitly choose and work with a maximal arc system containing the original when convenient. Pointed geodesics.
We now define a natural family of basepoints associated to boundary leaves of (cid:101) λ . For v ∈ H and its associated hexagon H v , define the λ -boundary ∂ λ H v of H v to be the set of leaves of (cid:101) λ thatmeet ∂H v .For v ∈ H and g a leaf of ∂ λ H v , define p v to be the orthogonal projection of v to g . Observe that v and p v lie along the same (singular) leaf of O λ ( X ). The orientation of S gives H v an orientation and hence orients ∂H v ; this yields an orientation-preserving, isometric identification of ( g, p v ) with ( R , p v while ± x refer to thepoints at signed distance ± x from p v .For a pair v (cid:54) = w ∈ H not in the same component of (cid:102) Sp , there is a unique geodesic g wv ∈ ∂ λ H v thatseparates v from w . Symmetrically, there is such a pointed geodesic g vw ∈ ∂ λ H w separating w from v . Notethat g wv = g vw occurs if and only if this leaf is isolated, and by the assumption that λ is measured, projectsto a simple closed curve component of λ . Even in this case, the points p v and p w are in general different.13.2. The shear-shape cocycle of a hyperbolic structure.
Our first task towards defining the geometricshear-shape cocycle σ λ ( X ) of a hyperbolic structure X is to construct a weighted, filling arc system A ( X ) ∈ B ( S \ λ ) which records the shapes of the complementary subsurfaces.With the technology we have developed up to this point, we now have many ways of constructing A ( X ),all of which are easily seen to be equivalent: • To each α ∈ α ( X ), we associate the weight c α := i ( O λ ( X ) , e α ), where e α is the edge of Sp dual to α . Equivalently, c α is the length of the projection of e α to either of the two leaves of λ to which itis closest. Then set A ( X ) = (cid:80) c α α. • Each component Y ⊂ X \ λ is naturally endowed with a hyperbolic structure; by Theorem 6.4 thismetric corresponds to a weighted, filling arc system in | A fill ( Y, ∂Y ) | R , and we let A ( X ) denote theunion of these arc systems over all components of X \ λ . • Let q be the quadratic differential with | Re( q ) | = O λ ( X ) and | Im( q ) | = λ ; then set A ( X ) = A ( q ).The final definition together with the results of Section 10 implies that A ( X ) ∈ B ( S \ λ ) for every hyperbolicstructure X on S . In the interest of providing the reader with geometric intuition for this condition, we haveincluded an alternative, purely hyperbolic-geometric proof of this fact below. Lemma 13.1.
With notation as above, A ( X ) ∈ B ( S \ λ ). Proof.
By Theorem 6.4, it suffices to show that for each minimal, orientable component µ of λ , the sum ofthe metric residues of the crown ends of X \ λ incident to µ is 0. If µ is a simple closed curve, then themetric residue is just equal to the (signed) lengths of the boundary components resulting from cutting along µ , which clearly must match.So assume that µ is not a closed curve and pick an orientation. Construct a geometric train track τ snuglycarrying µ as in Construction 5.6; then τ inherits an orientation from the inclusion of µ and so has welldefined left- and right-hand sides. As in Section 5.2, every branch b of τ has a well-defined length along λ which we denote by (cid:96) τ ( b ) >
0. At each switch s of τ , let h s be the leaf of the horocyclic foliation of N (cid:15) ( µ )projecting to s . By assumption of snugness, the spikes of S \ τ correspond with the spikes of S \ µ , so theunion of the h s truncates each spike of each crown end incident to µ by h s .Each crown incident to µ inherits an orientation from the chosen orientation on µ , and we now computethe total metric residue with respect to these orientations and the truncations induced by the h s ’s. Recallthat the metric residue of an oriented crown C is the alternating sum of the lengths of the geodesic boundarysegments running between the truncation horospheres (Definition 4.3). Each such geodesic segment defines aco-oriented trainpath ( b · ... · b n , ± ) in τ (i.e., a trainpath and a distinguished side, left or right, correspondingto + and − , respectively) which runs along the entirety of a smooth component of the boundary of X \ τ . Using this identification, we may compute that the corresponding contribution to the total metric residue isgiven by ± (cid:80) i (cid:96) τ ( b i ).Finally, we observe that every branch of τ is a subpath of exactly two smooth boundary edges of X \ τ (corresponding to its left and right sides). Therefore, the sum of the metric residues of all of the crownends incident to µ is the sum of the contributions of the corresponding co-oriented trainpaths, which isnecessarily 0 since each branch is counted twice, once with positive and once with negative sign. Thus A ( X ) ∈ B ( S \ λ ). (cid:3) Shears between nearby hexagons.
Our second step towards defining σ λ ( X ) is to determine how torecord shearing data between two hexagons that lie in different components of (cid:101) X \ (cid:101) λ yet are close enoughtogether. Except for sign conventions (see Remark 13.3), our discussion is essentially identical to Bonahon’sdefinition of shearing between the plaques of a maximal lamination [Bon96, § (cid:101) X \ (cid:101) λ in a variety of ways.Given v, w ∈ H , consider the associated pointed geodesics ( g wv , p v ) ∈ ∂ λ H v closest to H w and ( g vw , p w ) ∈ ∂ λ H w closest to H v . We say that the geodesic segment k v,w ⊂ (cid:101) X joining p v to p w is a simple piece if k v,w projects to a simple geodesic segment in X and k v,w bounds a spike in every hexagon that it crosses. Thatis, if k v,w crosses H u for some u ∈ H , then k v,w ∩ H u bounds a triangle in H u , two sides of which lie onasymptotic leaves g vu and g wu defining a spike of (cid:101) λ . If k v,w is a simple piece, then we say that ( v, w ) is a simple pair .We observe that if v, w ∈ H are close enough together and lie in different components of (cid:102) Sp , then ( v, w ) isa simple pair. The exact value of “close enough” is unimportant, but we note that it suffices for d ( H u , H v )to be smaller than the length of the shortest arc of α ( X ).Now following Bonahon [Bon96, Section 2], let Λ v,w be the leaves of (cid:101) λ that separate g wv from g vw , equippedwith the linear order < induced by traversing k v,w from p v to p w . Since ( v, w ) is a simple pair, the subsetof those leaves that are also the boundary of a complementary component of (cid:101) X \ (cid:101) λ come in pairs that areasymptotic in one direction. The partial horocyclic foliations on the wedges bounded by pairs of asymptoticboundary leaves extend across the leaves of Λ v,w , foliating the region bounded by g wv and g vw . In particular,the leaf of the horocyclic foliation containing p v meets g vw (and the leaf containing p w meets g wv ).Since the orthogeodesic foliation is equivalent to the horocyclic foliation in spikes, we see that if ( v, w ) isa simple pair then the leaf of (cid:94) O λ ( X ) containing p v meets g vw (and the leaf containing p w meets g wv ). In fact,simplicity implies that the orthogeodesic foliation foliates the “quadrilateral” bounded by g vw , g wv , and thetwo leaves of (cid:94) O λ ( X ) containing p v and p w . Definition 13.2.
Suppose that ( v, w ) is a simple pair of hexagons. Using the orientation conventions ofSubsection 13.1, identify the corresponding pointed geodesics ( g wv , p v ) and ( g vw , p w ) with ( R , (cid:94) O λ ( X ) containing p v meets g vw in some point r ∈ R , andwe set σ λ ( X )( v, w ) = − r . See Figure 16.It is not hard to see that σ λ ( X )( v, w ) remains the same if we flip the roles of v and w . Indeed, followingalong the leaves of the orthogeodesic foliation defines an orientation reversing isometry from a subsegmentof g wv to a subsegment of g vw that takes t (cid:55)→ r − t . In particular, p v maps to a point on g vw that is positioned r signed units away from p w , and so we see that σ λ ( X )( v, w ) = σ λ ( X )( w, v ). Remark 13.3.
Our choice to set σ λ ( X )( v, w ) = − r instead of + r records “how far along g vw you must travelfrom r to get to p w .” Though this convention is the opposite of what appears in [Bon96], it allows us tocombine the data of σ λ ( X )( v, w ) and A ( X ) into a system of train track weights on a standard smoothing(Construction 13.5 just below). Our convention also parallels our choice of [ · ] + function when measuringperiods of a quadratic differential (Lemma 10.10), which makes the relationship between the hyperbolicgeometry of ( X, λ ) and the flat geometry of q ( O λ ( X ) , λ ) more transparent.Below, we give an elementary estimate that will be used in the proof of Proposition 13.12; compare with[Bon96, Lemma 8]. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 53 p v v p w wr = − σ λ ( X )( v, w ) Figure 16.
Computing the shears between two nearby hexagons v and w . In this example, r <
0, so σ λ ( X )( v, w ) > Lemma 13.4.
Suppose that ( v, w ) is a simple pair of hexagons. Let ( g wv , p v ) and ( g vw , p w ) be the associatedpointed geodesics. Then the geodesic segment k v,w joining p v to p w satisfies | σ λ ( X )( k v,w ) | ≤ (cid:96) ( k v,w ) . Proof.
As ( v, w ) is simple, the orthogeodesic foliation (cid:94) O λ ( X ) foliates the region U bounded by g vw , g wv , andthe two leaves of (cid:94) O λ ( X ) containing p v and p w . This foliation gives rise to a 1-Lipschitz retraction π from U to g vw defined by following the leaves of the orthogeodesic foliation to g vw . The image π ( k v,w ) is then equalto the segment of g vw joining p w to the point labeled by σ λ ( X )( v, w ), which has length | σ λ ( X )( v, w ) | . Thelemma follows. (cid:3) Hyperbolic shearing as train track weights.
Now that we have explained how to record the shapesof X \ λ (Lemma 13.1) and the shears between nearby hexagons (Definition 13.2), we can package thisinformation together to define the geometric shear-shape cocycle σ λ ( X ) of a hyperbolic structure X .Below, we realize the shape and shear information specified above as a weight system on a standardsmoothing of a geometric train track carrying λ ; this strategy allows us to specify σ λ ( X ) by a finite collectionof information. Once we show that the weights are well-defined and satisfy the switch conditions, we theninvoke Proposition 9.5 to interpret this weight system as an (axiomatic) shear-shape cocycle (see Definition13.8). This reinterpretation in turn makes it apparent that our initial choice of train track does not matter.Using Construction 5.6, choose a geometric train track τ ⊂ X that carries λ snugly and let τ α be astandard smoothing of τ ∪ α ( X ) (see Construction 9.3). Note that the components of (cid:101) X \ (cid:102) τ α are in bijectionwith H , and that the assumption that τ carries λ snugly ensures that if two hexagons correspond to adjacentcomponents of (cid:101) X \ (cid:102) τ α then they either share an edge of ˜ α or form a simple pair. Construction 13.5.
Fix τ α ⊂ X as above. We then associate a weight system w ( X ) ∈ R b ( τ α ) as follows: • To each branch corresponding to α ∈ α , assign the weight c α = i ( O λ ( X ) , e α ), where e α is the edgeof Sp dual to α . • For each branch b ⊂ τ α that dos not correspond to an arc of α , choose a lift ˜ b ∈ (cid:102) τ α . Let v, w ∈ H de-note the vertices of (cid:102) Sp corresponding to the hexagons adjacent to ˜ b , and set w ( X )( b ) = σ λ ( X )( v, w ). Lemma 13.6.
Let
X, λ, α and τ α be as above. Then the edge weights w ( X ) ∈ R b ( τ α ) given by Construction13.5 satisfy the switch conditions. Proof.
Reference to Figure 17 will provide clarity throughout. We note that τ α is generically trivalent, butmay be 4-valent if there are arcs α , α ∈ α whose endpoints on λ lie on a common leaf of the orthogeodesicfoliation. We give an argument only for the trivalent switches of τ α , and leave it to the reader to makethe necessary adjustments for 4-valent switches (the statement for 4-valent switches can also be deduced bycontinuity). Let s be a trivalent switch; then standing at s and looking into the spike, there are small half-branchesexiting s on our right and left; call these r and (cid:96) , respectively. By our convention on standard smoothings,every half-branch of τ α corresponding to an arc of α is a right small half-branch.If no branch of s corresponds to an arc of α , then the arguments appearing in [Bon96, Section 2] imply thatthe weights satisfy the switch conditions, because the orthogeodesic foliation is equivalent to the horocyclefoliation in near s . See also [Pap91, Section 6] for a discussion more similar in spirit to ours.Otherwise, the right small half-branch r is labeled by some α ∈ α . Let b be the large half-branch incidentto s . Give names also to the hexagons incident to s and their distinguished points on b or (cid:96) by projection;they are N, SE , and SW ∈ H , and p N , p SE , p SW respectively, where b and (cid:96) form part of the boundary of N , (cid:96) and r form part of the boundary of SE , and r and b form part of the boundary of SW . See Figure 17.Now take d = d ( p SW , p SE ), which is equal to w ( X )( r ) = c α > s := | w ( X )( b ) | = d τ ( p SW , p N ) and s := | w ( X )( (cid:96) ) | = d τ ( p N , p SE ) . Here, d τ is understood to mean the distance between leaves of the orthogeodesic foliation near τ , measuredalong any leaf of λ (see Section 5.2 for an explanation of why this value is well-defined). NSW SEp SE p N p SW b (cid:96)αs s = w ( X )( b ) d = w ( X )( α ) s = − w ( X )( (cid:96) ) Figure 17.
Left: A local picture of τ near s . Right: Case (3). The switch condition issatisfied because s = d − s .There are 3 kinds of configurations for the projection points p SW , p N and p SE that determine the signsof w ( X )( b ) and w ( X )( (cid:96) ):(1) The point p N precedes both p SW and p SE with respect to the orientation of τ on induced by H N ,so that w ( X )( b ) = − s and w ( X )( (cid:96) ) = − s with s > s . In this case, d = s − s and so d − s = − s , which is exactly the switch condition.(2) Both p SW and p SE precede p N , so that w ( X )( b ) = s and w ( X )( (cid:96) ) = s with s > s . This possibility gives that d = s − s and so d + s = s .(3) The point p SW precedes p N which in turn precedes p SE , so that w ( X )( b ) = s and w ( X )( (cid:96) ) = − s .In this case, d = s + s and so d − s = s , which is again the switch condition.Therefore, no matter the configuration of points p N , p SW , and p SE , we see that the switch conditions arefulfilled at s , completing the proof of the lemma. (cid:3) Remark 13.7.
It is important to note that w ( X ) is generally not the same as the weight system comingfrom the shear coordinates of a completion of λ (unless λ was maximal to begin with).Invoking Proposition 9.5 and Lemma 13.1, we see that the weight system w ( X ) defines a shear-shapecocycle with underlying arc system A ( X ). Definition 13.8.
The geometric shear-shape cocycle ( σ λ ( X ) , A ( X )) of a hyperbolic metric X is the uniqueshear-shape cocycle for λ corresponding to the weight system w ( X ) of Construction 13.5. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 55
The rule that assigns to a hyperbolic structure its geometric shear-shape cocyle therefore defines a map σ λ : T ( S ) → SH ( λ ) X (cid:55)→ σ λ ( X ) , which is the subject of the rest of this article. Train track independence.
We have employed the language of train tracks for convenience — the ties ofa train track are a useful class of measurable arcs in the sense that they can be made transverse to λ anddisjoint from α (or record the weight associated to an arc of α ). However, Construction 13.5 and Definition13.8 a priori depend on the choice of geometric train track τ α carrying λ .Now that we have identified the weight system w ( X ) with the shear-shape cocycle σ λ ( X ), however, wecan invoke both the axiomatic and cohomological interpretations (Definitions 7.5 and 7.12) to see that thevalue of σ λ ( X ) on any arc k transverse to λ but disjoint from α does not depend on the choice of geometrictrain track. Indeed, let k be any such arc; then by transverse invariance (axiom (SH1)) we may replace k with a concatenation of short geodesics, all of which are transverse to λ but disjoint from α . By additivity(axiom (SH2)), it therefore suffices to show that the value of σ λ ( X ) on any short geodesic disjoint from α does not depend on the train track. Lemma 13.9.
Let k be a short enough geodesic segment on X that is transverse to λ . Lift k to an arc ˜ k on (cid:101) X and let v and w be the hexagons containing the endpoints of ˜ k ; then σ λ ( X )( k ) = σ λ ( X )( v, w )where on the left σ λ ( X ) represents the axiomatic shear-shape cocycle and on the right σ λ ( X ) represents theshear between nearby hexagons (Definition 13.2). In particular, σ λ ( X )( k ) does not depend on the choice oftrain track employed in Definition 13.8.In fact, the conclusion of this lemma holds for all simple pairs. Proof.
So long as k is short enough (shorter than all arcs of α ( X )) we know that ( v, w ) is a simple pair.Using axiom (SH1), we may therefore isotope k through arcs transverse to λ but disjoint from α to an arc k (cid:48) , defined to be the concatenation of k v,w , the geodesic connecting the points p v and p w on the boundarygeodesics g wv and g vw , together with segments of the orthogeodesic foliation inside each hexagon H v and H w .Let τ be a geometric train track snugly carrying λ defined with parameter (cid:15) ; then the collapse map π : N (cid:15) ( λ ) → τ takes k (cid:48) to a train route on τ , hence on τ α . Orient k (cid:48) (and hence also the train route π ( k (cid:48) ))so that it travels from v to w . Let v = u , u , . . . , u N = w denote the sequence of hexagons correspondingto regions of (cid:101) X \ (cid:102) τ α bordering this train route, so that the regions corresponding to u i and u i +1 both meetthe same subsegment of π ( k (cid:48) ). Let p i denote their corresponding projections onto λ . Note that since π ( k (cid:48) )is carried on τ ≺ τ α , no pair of subsequent hexagons u i and u i +1 lies in the same component of (cid:102) Sp . Thisplus the construction of the train track implies that ( u i , u i +1 ) is a simple pair, and we can measure the shear σ λ ( X )( u i , u i +1 ) (up to sign) as the distance along the train track between π ( p i ) and π ( p i +1 ).Now given τ α carrying λ , we observe that k (cid:48) also determines a (pair of) relative cycle(s) in the corre-sponding (orientation cover of the) (cid:15) -neighborhood of λ α . The value σ λ ( X )( k ) = σ λ ( X )( k (cid:48) ) is then equalto the value of the cohomological shear-shape cocycle evaluated on either of the oriented lifts ˆ k (cid:48) of k (cid:48) whichcross the lift of λ with positive local orientation. We may therefore express[ˆ k (cid:48) ] = [ t ] − [ t ] + [ t ] − . . . ± [ t N − ]where t i is a (lift of a) tie corresponding to the branch of the train track connecting the regions correspondingto u i and u i +1 , lifted to the orientation cover to have positive intersection with λ . See Figure 18.But now by construction, we know that σ λ ( X ) evaluated on [ t i ] is just the shear σ λ ( X )( u i , u i +1 ). In turn,this shear is equal to the signed distance along the train track between π ( p i ) and π ( p i +1 ) (where the signis determined by the local orientation of λ ). Combining this with the expression for [ˆ k (cid:48) ] above, we see that σ λ ( X )( k ) is exactly equal to the signed distance along the train track between π ( p ) and π ( p N ), which isthe shear σ λ ( X )( v, w ). (cid:3) We note that in the proof above, the cohomological interpretation of shear-shape cocycles provides aconvenient workaround for the obstacle that the train route with dual transversals t , ..., t N − is not in p v = u p u p u p u p u p u = wα [ t ] − [ t ] [ t ] − [ t ] [ t ] k v,w Figure 18.
Measuring the shear of a small arc using a geometric train track. By isotoping k to a proper arc in the geometric train track neighborhood and then expressing its relativehomology class as a sum of the branches, we can compute its shear as the alternating sumof shears between adjacent hexagons.general isotopic to k through arcs transverse to λ . Regardless, the relative homology class defined by k (cid:48) ∩ N (cid:15) ( λ ) is homologous to a linear combination of { t i } in the orientation cover of N (cid:15) ( λ ). Remark 13.10.
The lemma above can also be proved by splitting any two geometric train tracks to a com-mon subtrack [PH92, Theorem 2.3.1]. Each splitting sequence can then be realized in the orthogeodesically-foliated neighborhood N (cid:15) ( λ ) ⊂ X by cutting along compact paths in the spine associated to a spike, as in[ZB04, Section 3]. Splits induce maps on weight spaces, and so Lemma 13.9 is essentially equivalent to thestatement that Construction 13.5 is compatible with splitting and collapsing. See also [Bon97b, Lemma 6]. The cocycle as a map on pairs.
It will be convenient to repackage the data provided by σ λ ( X ) in yetanother form, which also explains our choice of notation in Definition 13.2.If v, w ∈ H can be joined by a Lipschitz continuous segment k v,w which is transverse to λ , disjoint from α , and meets no leaf of (cid:101) λ twice, then we say that ( v, w ) is a transverse pair and that k v,w is a transversal .If ( v, w ) is a transverse pair, we say that r is between v and w if there is a transversal k v,w that decomposesas a concatenation of transversals k v,w = k v,r · k r,w . Finally, we define σ λ ( X )( v, w ) := σ λ ( X )( k v,w )and declare that σ λ ( X )( v, v ) = 0. Observe that if ( v, w ) are a simple pair then this agrees with our definitionof the shear between nearby hexagons (Definition 13.2). Lemma 13.11.
The shear-shape cocycle σ λ ( X ) defines a map on transverse pairs that satisfies(1) ( π -invariance) for each γ ∈ π ( X ), we have σ λ ( X )( γv, γw ) = σ λ ( X )( v, w ).(2) (finite additivity) If ( v, w ) is a transverse pair and r is between v and w , then σ λ ( X )( v, w ) = σ λ ( X )( v, r ) + σ λ ( X )( r, w ) . (3) (symmetry) σ λ ( X )( v, w ) = σ λ ( X )( w, v ).13.3. Injectivity and positivity.
We now record some initial structural properties of the map σ λ definedabove. In particular, we demonstrate that σ λ is injective and interacts coherently with the orthogeodesicfoliation map O λ and the shear-shape coordinatization I λ of transverse foliations.Observe that injectivity of σ λ is equivalent to the statement that if two hyperbolic structures have thesame complementary subsurfaces and same gluing data along λ , then they must be isometric. As thehorocyclic and orthogeodesic foliations are equivalent in spikes of complementary subsurfaces, the proofs of[Bon96, Lemma 11 and Theorem 12] may be invoked mutatis mutandis . We outline this argument below forthe convenience of the reader, and direct them to [Bon96] for a more thorough discussion of the estimatesinvolved. We remark that this strategy also appears in the proof of Proposition 15.12, where we use it topiece together Lipschitz-optimal homeomorphisms along λ . Proposition 13.12.
The map σ λ : T ( S ) → SH ( λ ) is injective. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 57
Sketch of proof.
Fix homeomorphisms ( (cid:101) S, (cid:101) λ ) with ( (cid:101) X i , (cid:101) λ ) that lift the markings S → X i and so that eachcomponent (cid:101) Σ ⊂ (cid:101) S \ (cid:101) λ maps homeomorphically to a component (cid:101) Y i ⊂ (cid:101) X i \ (cid:101) λ for i = 1 , σ λ ( X ) = σ λ ( X ); then in particular A ( X ) = A ( X ) and so by Theorem 6.4, the comple-mentary subsurfaces X \ λ and X \ λ are isometric. Therefore, for a given component Σ ⊂ S \ λ , we canfind an π (Σ) equivariant isometry ϕ Σ : (cid:101) Y → (cid:101) Y . Define ϕ : (cid:101) X \ λ → (cid:101) X \ λ to be the union of these mapson each complementary component; by construction, ϕ is an isometry.We need to show that ϕ extends to a π ( S )-equivariant isometry ϕ : (cid:101) X → (cid:101) X . To prove this, we applythe arguments of [Bon96, Lemma 11], which we summarize presently. The first step is to construct a locallyLipschitz continuous extension of ϕ ; this step employs the length bound of Lemma 13.4 and some elementaryhyperbolic geometry, and the arguments of the first ten paragraphs of [Bon96, Lemma 11] may be appliedverbatim.As in Bonahon’s original proof, we now show that ϕ is actually 1-Lipschitz, given that it is locally Lipschitz.We first show that ϕ does not increase the length of leaves of the orthogeodesic foliation.Given any segment (cid:96) of a leaf of the orthogeodesic foliation (cid:94) O λ ( X ), the length of (cid:96) restricted to anyhexagon H u where u ∈ H is completely determined by the isometry type of H u and the distance along (cid:101) λ from p u ∈ ∂ λ H u . As σ λ ( X ) determines the shape of X \ λ , we can recover this information and hencedetermine the length of (cid:96) ∩ H u just from the data of σ λ ( X ).From σ λ ( X ) = σ λ ( X ), we deduce that the length of (cid:96) in any hexagon of (cid:101) X is equal to the length of ϕ ( (cid:96) ) in the corresponding hexagon of (cid:101) X . Moreover, since ϕ is locally Lipschitz, the 1-dimensional Lebesguemeasure of ϕ ( (cid:96) ) ∩ ϕ ( (cid:101) λ ) at most the 1-dimensional Lebesgue measure of (cid:96) ∩ (cid:101) λ . By a now classical fact thelatter is zero [BS85], hence so is the former. Therefore, the length of (cid:96) in X is equal to the length of (cid:96) in X .Now there is a path joining any two points in (cid:101) X built from geodesic segments and segments of leaves ofthe orthogeodesic foliation. The argument above shows that ϕ preserves the lengths of such paths, so ϕ isglobally 1-Lipschitz. The construction is completely symmetric, so ϕ − is 1-Lipschitz as well. Now every1-Lipschitz homeomorphism between metric spaces with 1-Lipschitz inverse is necessarily an isometry, andequivariance of ϕ is immediate from the construction. Therefore X and X must be isometric. (cid:3) The diagram commutes.
We have now developed sufficient technology to prove that the geometric shear-shape cocycle of a hyperbolic metric is the same as the shear-shape cocycle associated to its orthogeodesicfoliation. In other words, Diagram (2) commutes. Compare with [Mir08, Proposition 6.1].
Theorem 13.13.
For all λ ∈ ML and all X ∈ T ( S ) we have σ λ ( X ) = I λ ◦ O λ ( X ). Proof.
Fix a standard smoothing τ α of a geometric train track τ for λ on X . Our approach is to computeboth σ λ ( X )( b ) and I λ ◦ O λ ( X )( b ) for each branch b of τ α . These numbers will coincide, so by Proposition9.5, σ λ ( X ) = I λ ◦ O λ ( X ).Let T X ⊂ X be the piecewise geodesic triangulation of X whose vertices are the vertices of Sp , so thatthere is an edge between v, w ∈ Sp if the corresponding regions of X \ τ α share a branch. This recipegenerically yields a triangulation, but may have quadrilaterals in the case that two points of α ( X ) ∩ λ lieon the same leaf of O λ ( X ) ∩ N (cid:15) ( λ ). In this case, we may either choose a smaller initial neighborhood todefine our geometric train track so that this does not occur, or these points correspond to arcs that meet anisolated leaf of λ on either side; in the latter case, choose either diagonal that crosses the quadrilateral toinclude into T X .Let q = q ( O λ ( X ) , λ ), and recall that Proposition 5.9 provides a homotopy equivalence D λ : X → q inthe correct homotopy class satisfying D λ ∗ O λ ( X ) = V ( q ) and D λ ∗ λ = H ( q ) both leafwise and measurably.Furthermore, D λ maps Sp ⊂ T X to the horizontal saddle connections H of q , so D λ ( T X ) is homotopic relvertices to a triangulation T q ⊃ H of q by saddle connections. Construction 10.4 then provides us with atrain track τ (cid:48) dual to T q \ H that carries λ , as well as a standard smoothing τ (cid:48) α of τ (cid:48) with respect to α ( q ).Evidently, D λ ( τ ) is isotopic to τ (cid:48) and D λ ( τ α ( X ) ) is isotopic to τ (cid:48) α ( q ) , so we may speak of τ ⊂ τ α withoutambiguity. It remains to show that the shear-shape cocycles σ λ ( X ) and I λ ( q ) deposit the same weights oneach branch of τ α . By definition, A ( X ) = A ( q ), so let b be a branch of τ α not corresponding to an arc of thearc system. Dual to b there is a non-horizontal saddle connection e on q , and we recall that up to sign, the weight I λ ( q )( b ) is exactly the real part of the period of e , which is just the geometric intersection number i ( O λ ( X ) , e ).On the other hand, we have that σ λ ( X )( b ) is equal to the shear between the two hexagons on either sideof b . This in turn is equal to the geometric intersection number i ( O λ ( X ) , k v,w ) up to sign, where k v,w is thegeodesic connecting the vertices p v and p w of λ ∩ α ( X ). Since D λ takes k v,w to an arc transversely isotopicto e , we have that | σ λ ( X )( b ) | = | I λ ( q )( b ) | .Finally, to show that the signs are equal, fix matching orientations on k v,w and e . These induce a localorientations on the leaves of λ so that the algebraic intersection of λ with k v,w , respectively e , is positive.In turn, this induces a local orientation on the leaves of O λ ( X ) near k v,w , respectively e , and our signconventions are equivalent to stipulating that the sign is positive if k v,w , respectively e , crosses O λ ( X ) fromleft to right and negative if it crosses from right to left (compare [Mir08, § σ λ ( X )( b ) = I λ ( q )( b ) for all branches b , completing the proof of the theorem. (cid:3) Corollary 13.14.
For all µ ∈ ∆( λ ), we have an equality ω SH ( σ λ ( X ) , µ ) = i ( O λ ( X ) , µ ) = (cid:96) X ( µ ) > . In particular, σ λ ( T ( S )) ⊆ SH + ( λ ). Proof.
The first equality is a direct consequence of Theorem 13.13 and Proposition 10.12. The secondequality was proved in Lemma 5.7. (cid:3)
Shape-shifting cocycles
In the previous section, we explained how to associate to each hyperbolic structure X a shear-shape cocycle σ λ ( X ). In this one, we explain how to upgrade a small deformation s of the cocycle into a deformation ofthe hyperbolic structure; this is eventually used to prove that σ λ : T ( S ) → SH + ( λ ) is open (Theorem 15.1below). The main issue that we need to overcome is that we must simultaneously change the geometry ofthe non-rigid components of X \ λ while shearing these subsurfaces along one another.The goal of this section is therefore to build, for every small enough deformation s of σ λ ( X ), a π ( S )-equivariant shape-shifting cocycle that records how to adjust the relative position of geodesics of λ : ϕ s : ∂ λ H × ∂ λ H →
Isom + (cid:101) X where ∂ λ H := { ( h v , p v ) ⊂ ∂ λ H v : v ∈ H} is the set of boundary geodesics of (cid:101) λ equipped with basepointsobtained from projections of the vertices of (cid:102) Sp . See Proposition 14.26.In Section 15.1 below, we explain how to modify the developing map (cid:101) X → H according to ϕ s , resultingin a new (equivariant) hyperbolic structure X s with geometric shear-shape cocycle σ λ ( X ) + s (Lemma 15.6).By fixing a pointed geodesic ( h v , p v ) ∈ ∂ λ H we identify Isom + ( (cid:101) X ) with T (cid:101) X , so that the projection of { ϕ s (( h v , p v ) , ( h w , p w )) | ( h w , p w ) ∈ ∂ λ H} to (cid:101) X is then be the geodesic realization of (cid:101) λ in the new metric (cid:101) X s .When the deformation s preserves A ( X ), the cocycle ϕ s corresponds to a cataclysm map: the complemen-tary components of (cid:101) X \ (cid:101) λ are sheared along the leaves of (cid:101) λ and map isometrically into the deformed surface X s . When s alters A ( X ), we must shear the complementary subsurfaces while also simultaneously changingtheir shape, introducing complications not present in Bonahon and Thurston’s original considerations. Deforming the cocycle.
We first make explicit what we mean by a deformation of a shear-shape cocycle;we quantify what we mean by “small” in Section 14.2.Observe that if σ and σ (cid:48) in SH + ( λ ) are close, then by Proposition 8.5 we know that their underlyingweighted arc systems A and A (cid:48) are close in B ( S \ λ ). In particular, the corresponding unweighted arc systems α and α (cid:48) must both live in some common top-dimensional cell of B ( S \ λ ), i.e., must both be containedin some common maximal arc system β . Let τ be some snug train track for λ and let τ β be a standardsmoothing of τ ∪ β . By Proposition 9.5, we may then identify σ and σ (cid:48) as weight systems on τ β ; the difference σ − σ (cid:48) ∈ W ( τ β ) is then a deformation of σ .In general, if ( σ, A ) ∈ SH + ( λ ) and β is any maximal arc system containing the support of A , then thedeformations we consider in this section are those s ∈ W ( τ β ) such that σ + s ∈ W ( τ β ) corresponds to apositive shear-shape cocycle. Passing between equivalent definitions of shear-shape cocycles, we see thatwe may also think of s as a “shear-shape cocycle with negative arc weights.” The underlying weighted arc HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 59 system of any deformation s will be denoted by a ; while its coefficients are not necessarily positive, they willsatisfy the zero total residue condition of (13) by construction.By Theorem 6.4, the arc system A + a gives each component of S \ λ a new complete hyperbolic metric Y with (non-compact) totally geodesic boundary. Since the supports of A and A + a are both contained insideof some common maximal β , one may set up a correspondence between the complementary components of X \ λ α with the components of Y \ supp( A + a ) (adding in weight 0 edges as necessary). A blueprint.
To help guide the reader through this rather intricate construction, we include here a top-leveloverview of the necessary steps, together with an outline of the section. Briefly, our strategy is to explicitlydefine ϕ s on two types of pairs of pointed geodesics: the “simple pairs” between which the orthogeodesicfoliation is comparable to the horocyclic, and the pairs which live in the boundary of a common subsurface.Piecing together these basic deformations then allows us to define ϕ s on arbitrary pairs of pointed geodesics.Our construction of ϕ s for simple pairs parallels Bonahon’s construction of shear maps [Bon96, Section 5],and as such requires a detailed analysis of the geometry of the spikes of (cid:101) X \ (cid:101) λ . We therefore devote Section14.1 to recording a number of useful notions and estimates from [Bon96]. In this section, we also introducethe “injectivity radius of X along λ ,” which measures the length of the shortest curve carried on a maximalsnug train track for λ and plays a crucial role in our convergence estimates.After these preliminary considerations, we turn in Section 14.2 to the actual construction of ϕ s on simplepairs. As in [Bon96], the map is defined by adjusting the lengths of countably many horocyclic arcs in anappropriate neighborhood of λ , compensating for changing shears between hexagons. Unlike in [Bon96], wemust also adjust the arcs to account for the changing shapes of each of the spikes (as we are deformingthe complementary subsurfaces). Convergence of the resulting infinite product of parabolic transformationsis delicate; our approach follows [Bon96, Section 5] with influence from the more geometric approach of[Thu98].We then turn in Sections 14.3 and 14.4 to defining ϕ s on pairs of geodesics in the boundary of the samehexagon or the same complementary subsurface, respectively. It is here that our work significantly differsfrom that of Bonahon and Thurston. In these sections we also develop the idea of “sliding” a deformedcomplementary subsurface along the original; this viewpoint allows us to easily demonstrate a number ofotherwise nontrivial relations between M¨obius transformations (see Propositions 14.18, 14.19, and 14.24).Finally, in Section 14.5 we build the shape-shifting cocycle ϕ s from these pieces; the cocycle relation(Proposition 14.26) then follows from the cocycle relations for pieces and the separation properties of (cid:101) λ . Note.
We remark that throughout this section and the next, we consider isometries via their action on apointed geodesic, and compositions should be read from right to left.14.1.
Geometric control in the spikes.
We first record some useful definitions and associated geometricestimates. These estimates play a crucial role in establishing convergence of the infinite products appearingin Section 14.2 below. Many of our definitions follow Bonahon’s, but in order to contend with the factthat the complementary subsurfaces of λ are not always isometric, we must relate certain constants to thegeometry of λ on X (see Lemma 14.5, in particular).Our discussion will take place with certain data fixed. Choose a hyperbolic surface X ∈ T ( S ) and ameasured lamination λ ∈ ML ( S ). Let (cid:15) > (cid:15) -geometric train track τ on X carries λ snugly. The standard smoothing τ α for the arc system α = α ( X ) provides us with a vector space W ( τ α ) that models SH ( λ ; α ). With τ α fixed, we endow the vector space of weights on branches of τ α withthe sup norm (cid:107) · (cid:107) τ α , and restrict this norm to the weight space W ( τ α ).Let k b be an oriented geodesic transverse to a branch b ∈ τ that also avoids α . Following Bonahon, wedefine the divergence radius or depth r b ( d ) ∈ Z > of a component d of k b \ λ to be “how long the leaves of λ incident to d track each other,” as viewed by τ .More precisely, lift everything to the universal cover (cid:101) X . By convention, set r b ( d ) = 1 if d contains oneof the endpoints of k b . Otherwise, d is contained in a spike of H v for some v ∈ H , i.e., d connects a pairof asymptotic geodesics g − d and g + d . The divergence radius r b ( d ) is then the largest integer r ≥ π ( g + d ) and π ( g − d ) successively cross the same sequence of branches b − r +1 , b − r +2 , ..., b , ...b r − , b r − of (cid:101) τ , where b is the lift of b meeting ˜ k b and π : N (cid:15) ( (cid:101) λ ) → (cid:101) τ is the collapse map. By equivariance, r b ( d ) isclearly independent of the choice of lift ˜ k b of k b . Remark 14.1.
After projecting back down to τ ⊂ X , either b − r +1 · ... · b or b · ... · b r − defines a trainroute γ d in τ that starts at b and terminates by “opening up” into the projection of H v in X . That is, thegeodesics g + d and g − d diverge from each other (at scale (cid:15) ) at the terminus of γ d .Now there are boundedly many spikes of X \ λ , and for each r ≥ d ⊂ k b \ λ with depth exactly r . This gives us the following bound: Lemma 14.2 (Lemma 4 of [Bon96] and Lemma 5 of [SB01]) . For any branch b of τ and any transversal k b ,the number of components d of k b \ λ with r b ( d ) = r is at most 6 | χ ( S ) | .The train track interpretation of the depth of a segment also allows us to bound the value of a shear-shapecocycle s in terms of its weights on a snug train track and the depth of its endpoints.More specifically, for each component d of k b \ λ , let k db be the subarc of k b joining the initial point of k b to any point of d . Then for any combinatorial deformation s and b a branch of τ α , there is an explicitformula for s ( k db ) as a linear function of the weights of s on τ α with at most r b ( d ) terms [Bon97b, Lemma6]. Conceptually, this formula arises by splitting τ α open along the spike s containing d , until d is “visible”in some new track τ (cid:48) α carried by τ α (see also the proof of Lemma 13.9).The exact expression for s ( k db ) will not be important for us; instead, we record the following estimate,which follows by considering the growth of edge weights upon splitting. Lemma 14.3 (Lemma 6 of [Bon96] and Lemma 6 of [SB01]) . Let k b be a transversal of a branch b . Thereis a bound | s ( k db ) | ≤ (cid:107) σ (cid:107) τ α r b ( d )for every σ ∈ SH ( λ ; α ) and every component d of k b \ λ . Geometric estimates on depth.
The depth of a component d of k b \ λ is proportional to the distancefrom a lift ˜ d to the vertex u ∈ H inside of the corresponding spike. The constant of proportionality in turndepends on how quickly the spike of H u containing ˜ d returns to k b on X ; we now identify a quantity thatwill allow us to estimate this constant.Let k be any geodesic arc transverse to λ such that each lift ˜ k to (cid:101) X bounds a spike in every hexagon thatit crosses; equivalently, the endpoints of ˜ k lie in a simple pair of hexagons. As in Section 13.2, it suffices for k to be shorter than the shortest arc of α ( X ). Now for each leaf g of (cid:101) λ , there is a bound R k ( g ) > g between intersections of g with different lifts ˜ k and ˜ k of k . Indeed, any two lifts of k meeting g differ by a deck transformation γ ∈ π ( X ) determined by a path in X that traces along the projection ofa segment in g and then closes up along k .We then define the injectivity radius of X along λ to beinj λ ( X ) := inf k (cid:116) λ inf g ⊂ (cid:101) λ R k ( g )where the infimum is taken over all transverse arcs k whose endpoints lie in a simple pair of hexagons.Equivalently, the injectivity radius of λ may also be computed by taking a (cid:15) so that the geometric traintrack τ max built from N (cid:15) ( λ ) is snug and so that for all (cid:15) (cid:48) > (cid:15) , the train track built from N (cid:15) ( λ ) is the same(not just equivalent) to τ max , as follows. For each branch of τ max , choose a tie t b (that is, a leaf of the orthogeodesic (or horocyclic) foliationrestricted to N (cid:15) ( λ ) that is transverse to b ). The injectivity radius along λ is then equal to the infimum ofthe recurrence times of λ to any t b . Using the “length along a geometric train track” function (cid:96) τ max definedin Section 5.2, we may therefore write(29) inj λ ( X ) = inf γ ≺ τ max (cid:96) τ max ( γ )where the infimum is taken over all simple closed curves γ carried on the train track τ max . Remark 14.4.
The length of the systole of X is clearly a lower bound for inj λ ( X ), which is thereforepositive. However, inj λ ( X ) can be much larger than the length of the systole.For example, if λ does not fill the surface then there can be a disjoint curve of arbitrarily small length. Inaddition, X may have a very short curve γ transverse to λ , and if λ does not twist around γ , then inj λ ( X )is necessarily very large. Observe that any (cid:15) sufficiently close to the supremum of (cid:15) for which N (cid:15) ( λ ) is snug satisfies these conditions. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 61
We can now relate the geometry of small arcs to their depth and injectivity radius along λ . Lemma 14.5 (Lemmas 3 and 5 of [Bon96] and Lemma 4 of [SB01]) . Given a branch b of a geometric traintrack τ constructed from λ on X and a short transversal k b , there exists B > d of k b \ λ with depth r b ( d ), (cid:96) X ( d ) ≤ Be − D λ ( X ) r b ( d ) , where D λ ( X ) = inj λ ( X ) / | χ ( S ) | . Proof.
The idea is the same as in the reference, but our constants are different. Small geodesic arcs meetinga spike s of a hexagon H v transversely and far away from the vertex v look like horocycles, which have lengththat decays exponentially in distance from v . Therefore, we just need to give a lower bound for the distancebetween d and v ∈ H v along the spike s in terms of inj λ ( X ) and the topological complexity of S .Consider the train path γ d starting at b that defines r b ( d ). By definition, γ d traverses exactly r b ( d )branches of τ (counted with multiplicity). Now γ d decomposes as a concatenation of maximal sub–trainpaths with embedded interiors, each forming a simple loop in τ . The depth r b ( d ) is thus bounded above by the number of consecutive simple loops in γ d times the sizeof the longest simple loop in τ . The size of a simple loop in τ is in turn bounded above by the number ofbranches of τ , which is at most 9 | χ ( S ) | . Finally, since each simple loop in γ d is carried on τ ≺ τ max it musthave length at least inj λ ( X ) by (29).Putting the above estimates together, we see that the distance between v and d in H v is at leastinj λ ( X ) · { simple loops in γ d } ≥ inj λ ( X ) r b ( d )size of the longest simple loop in τ ≥ inj λ ( X )9 | χ ( S ) | r b ( d ) , and the lemma follows. (cid:3) Shape-shifting in the spikes.
Our discussion now begins to diverge from [Bon96]. While pairs ofasymptotic geodesics are all isometric, the spikes of X \ λ come with extra decoration, namely, a choice ofhorocycle at each cusp (equivalently, basepoints which lie on a common leaf of the orthogeodesic foliation).In this section, we explain how to use these decorations to define the shape-shifting cocycle ϕ s on pairs ofbasepointed geodesics coming from simple pairs of hexagons.We remind the reader that X , λ , and τ α are fixed so that geometric objects like geodesic segments,hexagons, arcs of α ( X ), etc. are understood to live in and be realized (ortho)geodesically on X . Throughoutthis section we will fix A = A ( X ) and use it to denote both a weighted arc system as well as the inducedmetric on S \ λ. Finally, we recall that s is a combinatorial deformation of σ λ ( X ) which changes A by a ; wewill refer to the deformed hyperbolic structure on S \ λ by A + a and its hexagonal pieces by G u for u ∈ H . Shapes of spikes.
The group
PSL ( R ) acts transitively on pairs of asymptotic geodesics but, having doneso, cannot further act on the family of horocycles based at the spike. To measure this failure, we associatebelow a geometric parameter which records the placement of basepoints in each spike.Suppose that u ∈ H is a hexagon of (cid:101) X \ (cid:102) λ α and s is a spike of H u , that is, a pair of asymptotic geodesics g and g (cid:48) . Both g and g (cid:48) come with basepoints p and p (cid:48) obtained by projecting u to these geodesics. We thenassociate to s the number h A ( s ) which measures the length of either of the orthogeodesic leaves connecting u to p or p (cid:48) : h A ( s ) := d ( p, u ) = d ( p (cid:48) , u ) . Our notation reflects the fact that this function clearly depends only on the geometry of X \ λ and not theshearing along λ . The reader familiar with the literature will observe that this parameter is essentially anorthogeodesic version of the “sharpness functions” appearing in [Thu98].In order to measure the difference in sharpness functions between the realizations of s in A and in thedeformed metric A + a , we superimpose the hexagons H u and G u and measure the distance between theirboundary basepoints.More concretely, choose an arbitrary orientation (cid:126)s of the spike s and fix realizations of both H u and G u inside of H . As PSL ( R ) acts simply transitively on triples in ∂ H , there is a unique isometry that takesthe realization of s in G u to its realization in H u . The vertex u of Sp is realized in both H u and G u ; let p and q denote the projections of these points to one of the boundary geodesics g of s . See Figure 19. A simple loop on a train track is a carried curve which traverses each branch at most once.
Lemma 14.6.
With all notation as above, the signed distance from q to p along g is(30) f X, s ( (cid:126)s ) := ε log (cid:18) tanh h A ( s )tanh h A + a ( s ) (cid:19) ∈ R , where ε = +1 if (cid:126)s is oriented towards the shared ideal endpoint, and ε = − q p (cid:126)sf X, s ( (cid:126)s ) G u H u Figure 19.
Superimposing hexagons to measure the difference in the shapes of their spikes.The parameter f X, s ( (cid:126)s ) plays a crucial role below in our definition of the shape-shifting map on spikes. Inour convergence estimates, we will also need to consider the parameter(31) (cid:107) s (cid:107) ∞ := max s | f X, s ( (cid:126)s ) | < ∞ which quantifies the maximum distance that the deformation s moves a basepoint in a spike. Proof.
We compute in the upper half plane; up to isometry, we may assume that s is bound by the imaginaryaxis V = i R > and its translate 2 + V ; the spine of the orthogeodesic foliation in this spike is a subsegmentof the vertical line 1 + V . With this choice fixed, the projections p and q of u to V may be identifiedwith ie a and ie b for some a and b , respectively. Without loss of generality, we may also assume that V isoriented upwards (towards ∞ ); the opposite choice of orientations simply reverses all signs at the end of thecomputation.Now for t ≥
0, the path t (cid:55)→ tanh t + i sech t is the unit speed parametrization of the orthogeodesicemanating from V at i . Observe that the isometry z (cid:55)→ e a z stabilizes V and takes this segment to anorthogeodesic segment emanating from ie a = p which is distance a from i . Since the orthogeodesic segmentthrough p meets the spine 1 + V after traveling distance h A ( s ) (by definition), this implies that e a = tanh h A ( s ) . Similarly, we have that e b = tanh h A + a ( s ). Together, these imply thattanh h A ( s u )tanh h A + a ( s u ) = e a − b . Taking logarithms, we see that a − b is the signed distance from q to p along V , as claimed. (cid:3) Remark 14.7.
Note that by Theorem 6.4, the parameter f X, s ( (cid:126)s ) varies analytically in a (hence s ). Orientation conventions.
We now specialize to the case where ( v, w ) is a simple pair of hexagons withassociated oriented geodesic k v,w running between p wv on g wv (the projection of v to the boundary leaf of ∂ λ H v closest to w ) and p vw on g vw .Each leaf g ⊂ (cid:101) λ crossed by k v,w inherits an orientation by declaring that turning right onto g whiletraveling from v to w along k v,w is the positive direction. We remark that if k v,w crosses a hexagon H u ,then the induced orientation of g wu , the geodesic in ∂ λ H u closest to w , is the opposite of the orientation of g wu induced as a part of the boundary of H u . On the other hand, the two orientations on g vu induced by k v,w and coming from H u agree. This is an artifact of our sign convention for measuring shears; see Remark 13.3.If g is a complete oriented geodesic in the hyperbolic plane and t ∈ R , we let T tg be the hyperbolic isometrystabilizing g and acting by oriented translation distance t along g . The opposite orientation of g will bedenoted ¯ g , so that T t ¯ g = T − tg . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 63
For an oriented spike (cid:126)s = ( g vu , g wu ), its opposite orientation is (cid:126)s = (¯ g wu , ¯ g vu ). In particular, we note that if (cid:126)s is an oriented spike of H u crossed by k v,w , then (cid:126)s is an oriented spike crossed by k w,v = ¯ k v,w . Shape-shifting in spikes.
Suppose ( v, w ) is a simple pair and suppose u is between v and w . Let (cid:126)s = ( g vu , g wu ) be the spike of u crossed by k v,w with basepoints p v and p w . We define the elementary shapingtransformation A ( (cid:126)s ) ∈ Isom + ( (cid:101) X ) = PSL R determined by X , s , and s to be(32) A ( (cid:126)s ) := T f X, s ( (cid:126)s ) g vu ◦ T − f X, s ( (cid:126)s ) g wu . Ultimately, the element A ( (cid:126)s ) will be the value of the shear-shape cocycle ϕ s on the pair ( g vu , g wu ); see justbelow for an explanation of how we think of A ( (cid:126)s ) as “changing the shape” of s .Observe that A ( (cid:126)s ) is a parabolic transformation preserving the common ideal endpoint of s . A familiarcomputation shows that in the spike determined by g vu and A ( (cid:126)s ) g wu , the orthogeodesics emanating from p v and A ( (cid:126)s ) p w meet at a point distance h A + a ( s ) from each (supposing that the deformation is small enough).To the oriented spike (cid:126)s of u , we also associate the elementary shape-shift (33) ϕ ( (cid:126)s ) := T s ( v,u ) g vu ◦ A ( (cid:126)s ) ◦ T − s ( v,u ) g wu = T s ( v,u )+ f X, s ( (cid:126)s ) g vu ◦ T − ( s ( v,u )+ f X, s ( (cid:126)s )) g wu where we recall that the value s ( v, u ) is obtained by thinking of s as a function on transverse pairs (`a laLemma 13.11). Note that ϕ ( s ) depends on our reference point v : whereas A ( (cid:126)s ) is eventually identified as avalue of the shape-shifting cocycle ϕ s , the elementary shape-shifts ϕ ( (cid:126)s ) are only building blocks for valuesof ϕ s .For the opposite orientation (cid:126)s = (¯ g wu , ¯ g vu ), we check(34) A ( (cid:126)s ) = T f X, s ( (cid:126)s )¯ g wu T − f X, s ( (cid:126)s )¯ g vu = T f X, s ( (cid:126)s ) g wu T − f X, s ( (cid:126)s ) g vu = A ( (cid:126)s ) − . Since s ( v, u ) = s ( u, v ), we may similarly observe that ϕ ( (cid:126)s ) = ϕ ( (cid:126)s ) − .Take H v,w to be the set of hexagons between v and w equipped with the linearing order u < u inducedby the orientation of k v,w . Let H ⊂ H v,w be any finite subset and order its elements H = { u , ...u n } . Forshort, we denote hexagons H i := H u i , spikes s i := (cid:126)s u i , geodesics g vi := g vu i , etc.To the finite, ordered set H we associate the product(35) ϕ H := ϕ ( s ) ◦ ... ◦ ϕ ( s n ) ◦ T s ( v,w ) g vw ∈ Isom + ( (cid:101) X ) . The goal of the rest of the section is then to extract a meaningful limit from ϕ H as H increases to H v,w .Ultimately, this limit is how we will define the shape-shifting cocycle ϕ s on the boundary geodesics g wv and g vw corresponding to the simple pair ( v, w ). Remark 14.8.
In the case that λ is maximal, each H i is an ideal triangle and so A = A + a = ∅ . In thiscase, each spike parameter f X, s ( s i ) is 0 and we recover the formula from [Bon96, p. 255]. Geometric explanation of (33) . Before proving convergence, however, let us explain the intuition behindthe formulas above. In order to interpret A ( (cid:126)s ) and ϕ ( (cid:126)s ) as deformations of the hyperbolic structure X ,we will switch our viewpoint to think of them as values of a deformation cocycle, and so as affecting theplacement of pointed geodesics relative to each other. For brevity, let f X, s ( (cid:126)s ) = t .Let us focus first on the shaping transformation A ( (cid:126)s ). The oriented spike (cid:126)s in the hexagon H u is formedby two pointed geodesics ( g vu , p vu ) and ( g wu , p wu ). Fixing our viewpoint at ( g vu , p vu ), we may think of A ( (cid:126)s ) asdeforming (cid:101) X by holding ( g vu , p vu ) fixed and identifying ( g wu , p wu ) with A ( (cid:126)s ) · ( g vu , p vu ). This has the overall effectof “widening” the spike s so that its sharpness parameter increases from h A to h A + a .If instead we fix our basepoint to be outside of H u , say at the basepoint p wv on g wv ⊂ ∂ λ H v , then thistransformation can viewed as a composition of left and right earthquakes. Let Q w and Q v denote the half-spaces to the left of the oriented geodesics g wu and g vu , respectively. Note that Q w ⊂ Q v . The deformation A ( (cid:126)s ) may then be thought of as first transforming all geodesics of (cid:101) λ that lie in Q w by T − tg wu ; this has theeffect of breaking (cid:101) X open along g wu and sliding Q w to the left by distance t along g wu while keeping (cid:101) X \ Q w fixed. The deformation then further transforms all geodesics in Q v by T tg vu ; this is equivalent to the rightearthquake with fault locus g vu that slides Q v to the right while keeping (cid:101) X \ Q v fixed. The cumulative effectis then that the spike s has been “pushed” in the direction of (cid:126)s by distance t . See Figure 20. Remark 14.9.
We give one final interpretation of A ( (cid:126)s ) as “sliding G u along H u ” in the proof of Proposition14.19 below (see also Figure 24), once we have set up the framework to understand the utility of thisviewpoint. g wu g vu vwu T − tg wu | Q w T tg vu | Q v Figure 20.
The effect of A ( (cid:126)s ) when considered as a composition of left and right earth-quakes.In particular, note that the shear from H v to H u measured from p wv to the image of p vu under thiscomposition of earthquakes has increased by t = f X, s ( (cid:126)s ). Therefore, if we let q vu denote the basepoint on g vu corresponding to the hexagon G u , then the shear from H v to H u measured from p wv to the image of q vu under the deformation is exactly the original shear σ λ ( X )( v, u ) between v and u .The elementary shape-shift ϕ ( (cid:126)s ) can be interpreted in much the same way, but now the spike should bepushed distance f X, s ( (cid:126)s ) + s ( v, u ) so that the resulting shear (measured between p wv and the image of q vu ) isexactly σ λ ( X )( v, u ) + s ( v, u ).Finally, the composition (35) can be thought of as a composition of the operations described above (readfrom right to left). Therefore, ϕ H first performs a right earthquake along g wv by s ( v, w ), then performs anelementary shape-shift to pushing the spike s n by s ( v, u n ) + f X, s ( s n ), then performs a shape-shift for s n − ,etc. Observe that if q vi denotes the basepoint in g vi corresponding to G u i , then by construction the shearbetween v and each u i measured from p wv to the image of q vi under the composite deformation is exactly thedesired shear σ λ ( X )( v, u i ) + s ( v, u i ).Assuming the convergence of ϕ H to a limit ϕ v,w (a step performed just below), we see that the placementof ϕ H ( g vw , p vw ) limits to that of ϕ v,w ( g vw , p vw ). This in turn will be the placement of the geodesic ( g vw , p vw )relative to ( g wv , p wv ) straightened in the deformed surface (cid:101) X s ; see Lemma 15.6. Convergence.
We now consider the limiting behavior of ϕ H as H → H v,w ; that a limit exists is almostexactly the content of [Bon96, Lemma 14]. We give a proof here for convenience of the reader and to makesure that we are extracting the correct radius of convergence, i.e., that the modifications in the cusps actuallydo not affect the radius of convergence (even though there are countably many contributions from changingthe shape of each spike).Recall from Lemma 14.5 that the function D λ ( X ) = inj λ ( Y ) / | χ ( S ) | gives a bound for the rate of decayof the length of a piece of a leaf of O λ ( X ) in terms of its divergence radius. Lemma 14.10 (compare Lemma 14 of [Bon96]) . If (cid:107) s (cid:107) τ α < D λ ( X ), then ϕ H converges to a well-definedisometry ϕ v,w as H tends to H v,w . Definition 14.11.
The limiting isometry ϕ v,w is called the shape-shifting map for the simple pair ( v, w ). Remark 14.12.
After combining all of our deformations in Section 14.5, the shape-shifting map ϕ v,w will beidentified as the value of the shape-shifting cocycle ϕ s on the pair ( g wv , g vw ). However, due to the asymmetryof our current definition, it is not clear that ϕ − v,w = ϕ w,v . See Lemma 14.14. Proof of Lemma 14.10.
For brevity, we set D = D λ ( X ) for the remainder of the proof.Identify (cid:101) X with H and Isom + ( (cid:101) X ) with the unit tangent bundle T H so that the identity I is the vectorover p v ∈ (cid:101) X that is tangent to g wv and pointed in the positive direction with the orientation on g wv induced HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 65 by k v,w . Equip T H with a left-invariant metric d that is right-invariant with respect to the stabilizer of p v .Finally, for A ∈ Isom + ( (cid:101) X ), let (cid:107) A (cid:107) := d ( I, A ), so that (cid:107) AB (cid:107) ≤ (cid:107) A (cid:107) + (cid:107) B (cid:107) holds by the triangle inequality.We first show that ϕ H stays in a compact set in Isom + ( (cid:101) X ). Using boundedness, we then show that anysequence H → H is in fact Cauchy with respect to d , hence converges.We start by bounding the lengths of segments of the form k v,w ∩ H u , where u ∈ H v,w . To this end,construct a geometric train track τ from λ on X , and assume that the projection of k v,w meets τ transversely.Subdivide k v,w into arcs k , ..., k m whose projections meet τ once in branches b , ..., b m . For all but finitelymany u ∈ H v,w , we have k v,w ∩ H u ⊂ k j \ (cid:101) λ for some j = 1 , ..., m .If d ⊂ k j \ (cid:101) λ , we set r ( d ) to be the depth r b j ( d ) of d with respect to b j and r ( d ) = 1, otherwise. By Lemma14.5, there is B > u ∈ H v,w , (cid:96) ( k v,w ∩ H u ) ≤ Be − Dr ( k v,w ∩ H u ) . With this estimate in mind, our next task is to give a uniform bound on (cid:107) ϕ H ◦ T − s ( v,w ) g vw (cid:107) for all finite H ⊂ H v,w . For each i , let γ i ∈ Isom + ( (cid:101) X ) be the isometry corresponding to the tangent vector over k v,w ∩ g vi pointing toward the positive endpoint of g vi . Unpacking definitions, we may therefore write the shape-shift ϕ ( s i ) as ϕ ( s i ) = γ i T s ( v,u i )+ f X, s ( s i ) g wv T − ( s ( v,u i )+ f X, s ( s i )) h i γ − i , where h i := γ − i g wi .An explicit computation (in the upper half plane model, say) shows that (cid:13)(cid:13)(cid:13) T s ( v,u i )+ f X, s ( s i ) g wv T − ( s ( v,u i )+ f X, s ( s i )) h i (cid:13)(cid:13)(cid:13) ≤ e | s ( v,u i )+ f X, s ( s i ) | (cid:96) ( k v,w ∩ H i ) ≤ Be | s ( v,u i )+ f X, s ( s i ) |− Dr ( k v,w ∩ H i ) . By Lemma 14.3 and the triangle inequality, we have that | s ( v, u i ) + f X, s ( s i ) | ≤ (cid:107) s (cid:107) τ α r ( k v,w ∩ H i ) + (cid:107) s (cid:107) ∞ and so we conclude that (cid:13)(cid:13) γ − i ϕ ( s i ) γ i (cid:13)(cid:13) ≤ B (cid:48) e r ( k v,w ∩ H i )( (cid:107) s (cid:107) τα − D ) for B (cid:48) = Be (cid:107) s (cid:107) ∞ .Notice now that conjugation by γ i changes the reference point of our calculation at a distance in the planeat most (cid:96) ( k v,w ), so the effect of γ − i ϕ ( s i ) γ i on g vi ∩ k v,w is a displacement by e (cid:96) ( k v,w ) times the quantityindicated above. Since this is independent of H , we have that(36) (cid:107) ϕ ( s i ) (cid:107) = O (cid:16) e r ( k v,w ∩ H i )( (cid:107) s (cid:107) τα − D ) (cid:17) for any spike s i corresponding to any hexagon u between v and w .Expanding out ϕ H in terms of the ϕ ( s i ) (see (35)), we have that (cid:13)(cid:13)(cid:13) ϕ H ◦ T − s ( v,w ) g vw (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:89) i =1 ϕ ( s i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n (cid:88) i =1 (cid:107) ϕ ( s i ) (cid:107) = O (cid:32) n (cid:88) i =1 e r ( k v,w ∩ H i )( (cid:107) s (cid:107) τα − D ) (cid:33) . Since there are a uniformly bounded number of gaps with given depth (Lemma 14.2), the last expression isbounded by the sum of at most 6 | χ ( S ) | many geometric series which are convergent so long as (cid:107) s (cid:107) τ α < D .Therefore, we see there is a compact set K in Isom + ( (cid:101) X ) so that ϕ H ∈ K for any finite subset H ⊂ H v,w .Now that we have shown the family of isometries { ϕ H } to be uniformly bounded, we can show thatany sequence of refinements is in fact Cauchy. So suppose that H n increases to H v,w and |H n | = n . Byconstruction, we may therefore write ϕ n = ψψ (cid:48) and ϕ n +1 = ψϕ ( s u ) ψ (cid:48) , where H n +1 = H n ∪ { u } and ψ, ψ (cid:48) ∈ K . Writing ϕ n +1 = ψψ (cid:48) ϕ ( s u )[ ϕ ( s u ) − , ψ (cid:48)− ] , we have that d ( ϕ n , ϕ n +1 ) = (cid:13)(cid:13)(cid:13) ϕ ( s u ) (cid:104) ϕ ( s u ) − , ψ (cid:48)− (cid:105)(cid:13)(cid:13)(cid:13) . The zeroth order term in the Taylor expansion near the identity for the function X (cid:55)→ (cid:107) [ X, ψ (cid:48)− ] (cid:107) is 0,because [ I, ψ (cid:48)− ] = I . Since ψ (cid:48)− stays in a compact set, (cid:13)(cid:13)(cid:13)(cid:104) ϕ ( s u ) − , ψ (cid:48)− (cid:105)(cid:13)(cid:13)(cid:13) = O ( (cid:107) ϕ ( s u ) (cid:107) )(see [Thu97, Theorem 4.1.6] or [Gel14, Lemma 1.1 of Lecture 2]).Combining this estimate with the triangle inequality and (36), we get that d ( ϕ n , ϕ n +1 ) = O ( (cid:107) ϕ ( s u ) (cid:107) ) = O (cid:16) e r ( k v,w ∩ H u )( (cid:107) s (cid:107) τα − D ) (cid:17) . Now there are at most 6 | χ ( S ) | many u ∈ H v,w with r ( k v,w ∩ H u ) = r (Lemma 14.2), so as n → ∞ we musthave that r → ∞ , and hence d ( ϕ n , ϕ n +1 ) →
0. Moreover, since this goes to 0 exponentially quickly, thesequence is in fact Cauchy. This completes the proof of the Lemma. (cid:3)
Shape-shifting as a limit of signed earthquakes.
Here we give a different description of the shape-shifting map which forgoes approximations by “pushing spikes” in favor of approximations by left and rightsimple earthquakes (compare [EM06, Section III]). While this reformulation is symmetric and geometri-cally meaningful, it comes at the cost of restricting which approximating sequences
H → H actually yieldconvergent sequences of deformations ϕ H . See also the remark at the top of page 261 in [Bon96].Let ( v, w ) be a simple pair and fix a geometric train track τ snugly carrying λ . So long as τ is built from asmall enough neighborhood, we may assume that the geodesic k v,w is transverse to the branches of τ . Thenfor each integer r ≥
0, let H rv,w denote the set of hexagons such that k v,w ∩ H u has depth at most r withrespect to the branches of τ . Order H rv,w = ( u = v, u , ..., u n , u n +1 = w ) , and for each i = 0 , . . . , n , choose a geodesic h i that separates the interior of H i from the interior of H i +1 .Orient each h i so that it crosses k v,w from left to right and set(37) ϕ rv,w = T s ( u ,u ) h ◦ A ( s ) ◦ T s ( u ,u ) h ◦ A ( s ) ◦ ... ◦ A ( s n ) ◦ T s ( u n ,u n +1 ) h n . We now wish to show that ϕ rv,w → ϕ v,w as r → ∞ . As in the case of ϕ H → ϕ v,w , this argument will parallelthat of [Bon96], with the extra complicating factor of the adjustments A ( s i ) to the shape of cusps.The interpretation of (37) as a deformation cocycle is now similar to that of (35), but is now a combinationof spike-shaping transformations together with simple earthquakes.Let us give a description of the action of this deformation on the pointed geodesic ( g vw , p vw ) in ∂ λ H w closestto v . Reading the formula from right to left, we can obtain ϕ rv,w ( g vw , p vw ) by first breaking (cid:101) X along h n = g vw and sliding the closed half-space containing H w signed distance s ( u n , u n +1 ), keeping the open half-spacecontaining H v fixed. Applying the spike shaping transformation A ( s n ) then preserves the natural basepoints p vn and p wn but increases the sharpness parameter h A ( s n ), making it match that of the spike in the hexagon G u n in the deformed metric A + a . We then simply continue moving from w to v (i.e., backwards along k v,w ), alternating between signed earthquakes in the h i and shaping the next spike until we reach g wv . Notethat unlike the deformations associated to ϕ H , each step of the process requires only local information aboutthe spike s i and the shear between u i and u i +1 . Lemma 14.13 (Lemma 16 of [Bon96]) . So long as (cid:107) s (cid:107) τ α < D λ ( X ) /
2, we have that lim r →∞ ϕ rv,w = ϕ v,w . Proof.
Using additivity, s ( u i , u i +1 ) = s ( v, u i +1 ) − s ( v, u i ), we observe that(38) ϕ rv,w = (cid:16) T s ( v,u ) h A ( s ) T − s ( v,u ) h (cid:17) (cid:16) T s ( v,u ) h A ( s ) T − s ( v,u ) h (cid:17) . . . (cid:16) T s ( v,u n ) h n − A ( s n ) T − s ( v,u n ) h n (cid:17) T s ( v,w ) h n . So ϕ rv,w is obtained from ϕ H rv,w by replacing each term of the form ϕ ( s i ) = T s ( v,u i ) g vi A ( s i ) T − s ( v,u i ) g wi with φ ( s i ) := T s ( v,u i ) h i − A ( s i ) T − s ( v,u i ) h i and T s ( v,w ) g vw with T s ( v,w ) h n . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 67
The basic estimate we need is approximately how close ϕ ( s i ) is to φ ( s i ) in Isom + ( ˜ X ) as r tends to infinity.For this we will want to understand how closely h i − approximates g vi near its intersection with k v,w , as wellas for g wi and h i .By construction, h i must be between g wi and g vi +1 for each i = 1 , ..., n and h is between g wv and g v . But g wi and g vi +1 follow the same edge path of length 2 r in τ ⊂ τ α , for otherwise, there would be a another u ∈ H rv,w such that H u separates H i from H i +1 . Thus h i follows the same edge path and fellow travels g wi and g vi +1 for length at least O (2 rD λ ( X )); using negative curvature, we have that h i is O ( e − D λ ( X ) r ) close toboth g wi and g vi +1 near k v,w .From closeness of these geodesics from the previous paragraph (and our estimates for (cid:107) ϕ ( s i ) (cid:107) from Lemma14.10) it is possible to obtain the basic estimate (cid:107) φ ( s i ) − ϕ ( s i ) (cid:107) = O (cid:0) exp (cid:0) (cid:107) s (cid:107) τ α r ( k v,w ∩ H i ) − rD λ ( X ) (cid:1)(cid:1) , which is small when (cid:107) s (cid:107) τ α < D λ ( X ). Notice that we have also used the fact that the adjustment parameterassociated to each spike s ( s i ) is uniformly bounded; that said, even if it grew linearly in r we would obtainthe same estimate (up to a multiplicative factor).The rest of the argument ensuring that ϕ rv,w and ϕ H rv,w have the same limit as long as (cid:107) s (cid:107) τ α < D λ ( X ) / / (cid:3) The following simple fact was not apparent from the definition of ϕ v,w due to its lack of symmetry.Fortunately, the approximation of ϕ v,w by ϕ rv,w gives us a symmetric description of ϕ v,w . Corollary 14.14.
If ( v, w ) is simple and (cid:107) s (cid:107) τ α < D λ ( X ) /
2, then ϕ w,v = ϕ − v,w . Proof.
We observe first that H rv,w = H rw,v , so the each term of ϕ rv,w appears in ϕ rw,v with the oppositeorientation. Now by (34), the inverse of the shaping transformation of an oriented spike is equal to theshaping transformation of the same spike with opposite orientation. Therefore we have that ϕ rv,w = ( ϕ rw,v ) − for all r , and the equality holds as we take r → ∞ . (cid:3) Shape-shifting in hexagons.
In this section, we explain how to define the shape-shifting cocycle ϕ s on pairs of basepointed geodesics that lie in the boundary of a common hexagon; this will encode the changein hyperbolic structure on X \ λ. While in this setting we do not have to worry about delicate convergence results, we must be more diligentabout recording the placement of basepoints on each geodesic of ∂ λ H u . Moreover, the cocycle condition(Propositions 14.18 and 14.19) only becomes apparent once we reinterpret the shaping deformations definedbelow as “sliding the deformed hexagon along the original.”Throughout this section, we have extended both A and A + a to some common maximal arc system α byadding in arcs of weight 0 as necessary. We remind the reader that s ( α ) denotes the coefficient of α in a . Notations and orientations.
Let H u ⊂ (cid:101) X \ (cid:102) λ α be a nondegenerate right-angled hexagon and enumeratethe λ -boundary components of H u as ∂ λ H u = { ( h , p ) , ( h , p ) , ( h , p ) } , cyclically ordered about u . Let α i ∈ (cid:101) α be the orthogeodesic arc opposite to h i , and denote by p ij the vertex of H u meeting both h i and α j .See Figure 21. If H u is a degenerate hexagon (i.e., a pentagon with one ideal vertex or a quadrilateral withtwo) then we label only those points and geodesics which appear in its boundary.Each choice of orientation (cid:126)α of α induces orientations of h and h so that α leaves from the left-handside of h j and arrives on the right-hand side of h k for { j, k } = { , } ; an example is pictured in Figure 21.Observe that the opposite orientation (cid:126)α induces the opposite orientations on h and h . Throughout thissection, we also adopt similar conventions for each orientation of α and α .Recall that (by Theorem 6.4) the deformation s induces a new metric on (cid:101) X \ (cid:101) λ denoted by A + a andwhich contains a hexagon G u corresponding to H u . The corresponding basepointed λ -boundary geodesicsand vertices of G u will be denoted by ( g i , q i ) and q ij , respectively. We adopt similar orientation conventionsas above for the realizations of α j in G u . Shapes of Hexagons.
Paralleling our discussion for spikes, we first need to define geometric parametersthat measure the shape of the hexagon as well as the difference of the placements of the basepoints p i and up p (cid:126)α p p p α p p p α p h h h Figure 21.
Distinguished points on a hexagon H u and induced orientations on h , h ∈ ∂ λ H u . q i on the geodesics h i and g i . For concreteness, we only consider α below; the parameters for α and α are defined symmetrically.We begin by associating to α the parameter (cid:96) s ( α ) := (cid:96) A + a ( α ) − (cid:96) A ( α ) ∈ R . which measures the difference in the hyperbolic length of α in the metric determined by A + a versus in theoriginal metric A induced by X .Now fix an orientation (cid:126)α of α ; as above, this induces orientations of the geodesics h , h , g , and g . Let d A ( (cid:126)α , u ) be the signed distance from p to p on h ; the local symmetry of the orthogeodesic foliationimplies that d A ( (cid:126)α , u ) can also be computed as the signed distance from p to p on h . Define similarly d A + a ( (cid:126)α , u ) as the distance from q to q on g (equivalently, the signed distance from q to q on g ).To all of this information, we associate the parameter f X, s ( (cid:126)α , u ) := d A + a ( (cid:126)α , u ) − d A ( (cid:126)α , u ) ∈ R which measures the difference in how far u is from α in G u versus in H u . More precisely, considering H u and G u in the hyperbolic plane, we can use an element of PSL ( R ) to line up ( h , p ) with ( g , q ) so thatthe basepoints and orientations agree. The parameter f X, s ( (cid:126)α , u ) then measures the distance from q to p along h = g . See Figure 22. Of course, symmetry shows that it is equivalent to align ( h , p ) with ( g , q )and measure the signed distance from q to p along h = g .Note that reversing orientations reverses signs, so that d A ( (cid:126)α , u ) = − d A ( (cid:126)α , u ) and hence f X, s ( (cid:126)α , u ) = − f X, s ( (cid:126)α , u ) . The parameters associated to the hexagons which border a given arc are related in the following way:
Lemma 14.15.
Let α be any edge of (cid:101) α and let H u and H v be its adjoining hexagons. Then f X, s ( (cid:126)α, u ) + f X, s ( (cid:126)α, v ) = s ( α )where the orientation (cid:126)α is chosen so that u is on its left (equivalently, (cid:126)α is oriented so that v is on its left). Proof.
The proof is an exercise in unpacking the definitions and being careful with orientations; compareFigure 22. Let p u and p v denote the projections of u and v to either of geodesics common to ∂ λ H u and ∂ λ H v , and let q u and q v play similar roles for G u and G v . The parameter d A ( (cid:126)α , u ) is called the “ t -coordinate” of the arc α in the hexagon H u in [Luo07]. See also [Mon09b,Proposition 2.10], where a formula is given in terms of the lengths of { α i : i = 1 , , } . HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 69 G u q u H u p u f X, s ( (cid:126)α , u ) (cid:126)α G v q v H v p v − f X, s ( (cid:126)α , v ) h g h = g Figure 22.
The parameter f X, s ( (cid:126)α , u ) for two adjacent hexagons. We have decorated thebasepoints on h = g with a superscript to emphasize their dependence on the hexagon.We can then write f X, s ( (cid:126)α, u ) + f X, s ( (cid:126)α, v ) = f X, s ( (cid:126)α, u ) − f X, s ( (cid:126)α, v )= d A + a ( (cid:126)α, u ) − d A ( (cid:126)α, u ) − d A + a ( (cid:126)α, v ) + d A ( (cid:126)α, v )= d h ( p u , p v ) − d g ( q u , q v ) = s ( α )where we recall that s ( α ) denotes the coefficient of α in a and where d h and d g represent the signed distancemeasured along h and g , equipped with the orientation induced by (cid:126)α . (cid:3) Remark 14.16.
Using Theorem 6.4 and some hyperbolic trigonometry, one may show that f X, s ( (cid:126)α , u )depends analytically on both A and a for fixed α and u . Shaping Hexagons.
To the hexagon H u and oriented arc (cid:126)α in its boundary, we associate the shapingtransformation A ( (cid:126)α , u ) by(39) A ( (cid:126)α , u ) := T − f X, s ( (cid:126)α ,u ) h ◦ T (cid:96) s ( α ) (cid:126)α ◦ T f X, s ( (cid:126)α ,u ) h ∈ Isom + ( (cid:101) X ) , where T (cid:126)α denotes translation along the complete oriented geodesic extending (cid:126)α . The shaping transforma-tion is explicitly constructed so that if H u and G u are superimposed with ( h , p ) = ( g , q ), then A ( (cid:126)α , u )( h , p ) = ( g , q ) . This claim is not immediately apparent from the expression of (39), but is easy to verify once we reinterpret A ( (cid:126)α , u ) as “sliding G u along H u .”To wit, suppose that we superimpose H u and G u so that ( h , p ) = ( g , q ). Now consider what happensas we apply A ( (cid:126)α , u ) to G u while holding H u fixed; the first term T f X, s ( (cid:126)α ,u ) h translates G u along h sothat q = p , and the right angle formed by α and g in G u lines up with the same angle in H u . Thetransformation T (cid:96) s ( α ) (cid:126)α then slides T f X, s ( (cid:126)α ,u ) h G u along α so that ( h , q ) = ( g , p ). Finally, T − f X, s ( (cid:126)α ,u ) h slides T (cid:96) s ( α ) (cid:126)α T f X, s ( (cid:126)α ,u ) h G u along h = g so that q lines up with p . See Figure 23.Summarizing, we have shown that A ( (cid:126)α , u ) takes a superimposition of G u on H u with ( h , p ) = ( g , q )to another superimposition with ( h , p ) = ( g , q ). In particular, this implies that applying A ( (cid:126)α , u ) to( h , p ) takes it to the position of ( g , q ) in the latter placement of G u , which is what we claimed. Remark 14.17.
An elementary hyperbolic geometry argument similar to that in the proof of Lemma14.6 shows that if α in X degenerates to an oriented spike (cid:126)s then the corresponding geometric parameter f X, s ( (cid:126)α , u ) limits to the parameter f X, s ( (cid:126)s ). In particular, along this degeneration the corresponding hexagon-shaping transformation A ( (cid:126)α , u ) converges to the spike-shaping transformation A ( (cid:126)s ). A cocycle condition for hexagons.
A number of relations hold between the shaping transformations fordifferent arcs and different orientations; eventually, these relations are what ensure that the deformations weare currently building package together into an honest cocycle (see Proposition 14.26). p = q G u H u (cid:126)α h h p p = q p = q p = q T f X, s ( (cid:126)α ,u ) h T (cid:96) s ( α ) (cid:126)α T − f X, s ( (cid:126)α ,u ) h Figure 23.
How the shaping transformation A ( (cid:126)α , u ) slides G u along H u .First, we observe that reversing the orientation of α inverts the shaping transformation:(40) A ( (cid:126)α , u ) = T − f X, s ( (cid:126)α ,u )¯ h ◦ T (cid:96) s ( α ) (cid:126)α ◦ T f X, s ( (cid:126)α ,u )¯ h = T − f X, s ( (cid:126)α ,u ) h ◦ T − (cid:96) s ( α ) (cid:126)α ◦ T f X, s ( (cid:126)α ,u ) h = A ( (cid:126)α , u ) − . Now suppose that H u is on the left and H v is on the right of the oriented arc (cid:126)α . Combining the relationof Lemma 14.15 with the definition of the shaping transformation, we have that(41) A ( (cid:126)α , u ) = T s ( α ) h ◦ A ( (cid:126)α , v ) ◦ T − s ( α ) h . This equation is used frequently in Section 14.4 just below.Finally, a beautiful and important relationship holds among the three shaping transformations in a singleright-angled hexagon. Our proof utilizes the “sliding” viewpoint explained above; the statement seemsdifficult to prove just by writing down a string of M¨obius transformations.
Proposition 14.18.
Let u ∈ H be a nondegenerate right-angled hexagon with boundary arcs (cid:126)α , (cid:126)α , (cid:126)α ,oriented so that H u lies to the left of each (cid:126)α i . Then A ( (cid:126)α , u ) ◦ A ( (cid:126)α , u ) ◦ A ( (cid:126)α , u ) = 1 . A similar statement clearly holds for any cyclic permutation of (3 , , Proof.
In order to prove the lemma, we superimpose G u on top of H u so that ( g , q ) = ( h , p ). Holding H u fixed, the first shaping transformation A ( (cid:126)α , u ) slides G u along h , then along α , then along h sothat ( g , q ) lines up with ( h , p ). The second shaping transformation A ( (cid:126)α , u ) then acts on this translatedcopy of G u by sliding it along h , then α , then h so that ( g , q ) = ( h , p ). Finally, the last term slides A ( (cid:126)α , u ) A ( (cid:126)α , u ) G u along the edges of H u so that ( g , q ) returns to ( h , p ) (with the same orientation).Therefore, since A ( (cid:126)α , u ) ◦ A ( (cid:126)α , u ) ◦ A ( (cid:126)α , u ) preserves a unit tangent vector and Isom + ( (cid:101) X ) acts simplytransitively on T (cid:101) X , the composition of the three shaping transformations must be trivial. (cid:3) A similar result holds for degenerate right-angled hexagons, with the hexagon-shaping transformationreplaced with the corresponding spike-shaping transformation.
Proposition 14.19.
Suppose that u ∈ H is a pentagon with two orthogeodesic arcs α , α and one spike s , labeled so that ( α , α , s ) runs counterclockwise around u . Orient each α j so that H u is on its left andorient s so that it is pointing towards the ideal vertex of H u . Then A ( (cid:126)s ) ◦ A ( (cid:126)α , u ) ◦ A ( (cid:126)α , u ) = 1 . Similarly, if u ∈ H is a quadrilateral with one orthogeodesic edge α and two spikes s and s (labeled sothat ( α, s , s ) is read counterclockwise), then A ( (cid:126)s ) ◦ A ( (cid:126)s ) ◦ A ( (cid:126)α, u ) = 1where all orientation conventions are as above. Proof.
We only explain how to interpret the spike-shaping transformation A ( (cid:126)s ) in our “sliding” framework;once we have done so, the rest of the proof is completely analogous to that of Proposition 14.18.So let (cid:126)s be a spike of H u , oriented as described; suppose that its left and right boundary geodesicsare h and h . Recall that A ( (cid:126)s ) is constructed so that if we superimpose G u and H u with ( g , q ) =( h , p ) then A ( (cid:126)s )( h , p ) = ( g , q ). This can equivalently be interpreted by superimposing G u on H u with HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 71 ( g , q ) = ( h , p ); then applying the shaping transformation to G u while leaving H u fixed takes G u toanother superimposition where ( g , q ) = ( h , p ). See Figure 24. (cid:3) p = q G u H u h h p h = g p = q T − f X, s ( (cid:126)s ) h T f X, s ( (cid:126)s ) h Figure 24.
Interpreting the spike-shaping transformation A ( (cid:126)s ) as sliding G u along H u .Note that in this picture, we have f X, s ( (cid:126)s ) < Shape-shifting along the spine.
In this section we package together the hexagon-shaping deforma-tions defined in (39) into deformations of entire complementary subsurfaces of (cid:101) X \ (cid:101) λ . As always, we willexhibit this deformation by explaining how to adjust the positions of the pointed geodesics in the boundaryof each component of (cid:101) X \ (cid:101) λ relative to one another. This in turn requires some book-keeping of orientationsand a liberal application of the cocycle relation (Propositions 14.18 and 14.19).Throughout this section, we fix some component Y of (cid:101) X \ (cid:101) λ . We remind the reader that the deformation s induces a new hyperbolic structure A + a on Y whose hexagons and basepointed geodesics correspond tothose of Y . Hexagonal hulls and induced orientations.
Suppose that v, w ∈ H are distinct hexagons of Y . Sincethe corresponding component of (cid:102) Sp is a tree it contains a unique oriented non-backtracking edge path [ v, w ]joining v to w . We then define the hexagonal hull H ( v, w ) of ( v, w ) to be the union of all of the hexagonscorresponding to the vertices of [ v, w ]. Define also the truncated hexagonal hull (cid:98) H ( v, w ) by truncating eachspike of H ( v, w ) by the horocycle through the basepoints that are closest to the ideal vertex. Note that both H ( v, w ) and (cid:98) H ( v, w ) come with ( π ( Y )-equivariant) collections of basepoints on their boundaries obtainedby projecting each of the vertices of [ v, w ] onto the associated boundary geodesics.Now for any ( h v , p v ) ∈ ∂ λ H v and ( h w , p w ) ∈ ∂ λ H w , we have that ∂ (cid:98) H ( v, w ) \ { p v , p w } consists of two paths δ ± . We orient each of δ ± so that they both travel from p v to p w . See Figure 25. p v p w v wδ − δ + p p p p p Figure 25.
The truncated hexagonal hull (shaded) of the path [ v, w ] and the inducedorientations on the paths δ ± from p v to p w in its boundary.With this induced orientation, the path δ + passes through a sequence of basepoints p v = p , p , . . . , p n +1 = p w . We then associate a shaping transformation A i to each subsequent pair of basepoints as follows: • If p i , p i +1 are in different hexagons, then they must lie on the same geodesic h i of ∂Y and correspondto two hexagons H i and H i +1 both adjacent to an arc α i . In this case, define A i = T s ( α i ) h i where h i is given the orientation induced by δ + and where we recall that s ( α i ) is the coefficient of α i in a . • If p i , p i +1 are in the same hexagon H u i but do not lie on a common spike, then necessarily they lieon geodesics connected by some arc α i . In this case, define A i = A ( (cid:126)α i , u i ) where the orientation on α i is induced from δ + . • If p i and p i +1 lie on a common spike s i , then we define A i = A ( (cid:126)s i ), where the orientation on s i is such that the horocyclic segment of δ + cutting off s i runs from the left of one of the orientedgeodesics to the right of the other.Finally, we then combine all of this information to define the shape-shifting transformation(42) A ( δ + ) := A ◦ A ◦ . . . ◦ A n , where we recall that we are multiplying from right to left. Define A ( δ − ) analogously; the point is, however,that the choice of ± does not matter. Lemma 14.20.
We have A ( δ − ) = A ( δ + ). Definition 14.21.
We call ϕ p v ,p w := A ( δ + ) = A ( δ − ) the shape-shifting map for the pair (( h v , p v ) , ( h w , p w )). Proof.
The proof follows by induction on the length of [ v, w ]. If [ v, w ] has length 0, i.e., v = w , then thisstatement is exactly the content of the cocycle relation for hexagons (Propositions 14.18 and 14.19).Now suppose that [ v, w ] has length n and let u be the penultimate vertex in [ v, w ]. Let α denote thearc separating u from w , and choose the orientation (cid:126)α so that u lies on its left. Up to relabeling, we mayassume that the orientation of δ + agrees with the orientation of ∂ (cid:98) H ( v, w ). Denote by ( h ± u , p ± u ) ∈ ∂Y thelast basepoints of H u visited by δ ± and let γ ± denote the subpaths of δ ± from p v to p ± u in the boundary ofthe truncated hexagonal hull (cid:98) H ( v, u ) . Define A ( γ ± ) analogously to A ( δ ± ). Then we may write A ( δ + ) A ( δ − ) − = A ( γ + ) T s ( α ) h + u B B T − s ( α ) h − u A ( γ − ) − = (cid:0) A ( γ + ) A ( (cid:126)α, u ) A ( γ − ) − (cid:1) · A ( γ − ) (cid:0) A ( (cid:126)α, u ) − T s ( α ) h + u B B T − s ( α ) h − u (cid:1) A ( γ − ) − where B and B are the shaping transformations corresponding to arcs and spikes of w that are differentfrom α (labeled counterclockwise from α ), oriented either so that w lies on the left of the arc or so that thespike points into the common ideal endpoint.Now observe that A ( γ + ) A ( (cid:126)α, u ) A ( γ − ) − is trivial by the inductive hypothesis, as it corresponds to thecomparison between the two possible definitions of ϕ p v ,p − u . We also note that A ( (cid:126)α, u ) − T s ( α ) h + u B B T − s ( α ) h − u is conjugate to T − s ( α ) h − u A ( (cid:126)α, u ) − T s ( α ) h + u B B = A ( (cid:126)α, w ) B B = 1where the first equality follows from (41) (note the reversals in orientations of h ± u ) and the second follows fromthe cocycle relation (Propositions 14.18 and 14.19). Therefore, we see that the entire term A ( δ + ) A ( δ − ) − istrivial, which is what we wanted to show. (cid:3) Remark 14.22.
The above statement can also be proven by interpreting A ( δ ± ) in terms of sliding. Inparticular, let Z denote the π ( Y )-equivariant hyperbolic structure on Y corresponding to the weighted arcsystem A + a . Then superimposing Z on Y so that ( g w , q w ) = ( h w , p w ), one can consecutively apply theshaping transformations A i to Z while keeping Y fixed.Doing so, we see that A n moves Z so that ( g n , q n ) = ( h n , p n ), then A n − ◦ A n moves Z so that( g n − , q n − ) = ( h n − , p n − ), etc. At the end of this process, we have applied A ( δ + ) to Z and by con-struction, the pointed geodesic ( g v , q v ) matches up with ( h v , p v ). Since the final positioning of Z is the samerelative to Y whether we used A ( δ + ) or A ( δ − ), this allows us to conclude that the two compositions definethe same element. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 73
Remark 14.23.
While we used the distinguished boundary paths δ ± to define the shape-shifting map, onecould in fact use any path from p v to p w in Y ∪ (cid:101) α . In this case, one must take more care to enumeratebasepoints so that p i , p i +1 always either lie on the same geodesic or in the same hexagon.Observe that reversing the orientation [ v, w ] = [ w, v ] also reverses the sequence p n +1 , . . . , p of base-points that the boundary paths δ ± meet. Since flipping the order of p i and p i +1 inverts each of the A i transformations defined above, we therefore discover that ϕ p w ,p v = ϕ − p v ,p w .In a similar vein, it is not hard to see that the shape-shifting maps satisfy a cocycle relation. Proposition 14.24.
For any triple of pointed geodesics ( h u , p u ), ( h v , p v ), and ( h w , p w ) of ∂Y , we have that ϕ p u ,p v ◦ ϕ p v ,p w ◦ ϕ p w ,p u = 1 . Proof.
This follows immediately from the definitions when v lies on either of the paths δ ± from u to w .Otherwise, note that the intersection of paths [ u, v ] ∩ [ v, w ] ∩ [ w, u ] is a point x ∈ H . Choosing a basepoint p x ∈ ∂ λ H x , compute the shape-shifting transformations using the boundary arcs that pass through x . Thenwe may express ϕ p u ,p v = ϕ p u ,p x ◦ ϕ p x ,p v and using the observation about inverses above, we realize that ϕ p u ,p v ◦ ϕ p v ,p w ◦ ϕ p w ,p u = ϕ p u ,p x ◦ ( ϕ p x ,p v ◦ ϕ p v ,p x ) ◦ ( ϕ p x ,p w ◦ ϕ p w ,p x ) ◦ ϕ p x ,p u = 1 . This completes the proof of the proposition. (cid:3)
The shape-shifting cocycle.
We now combine the shape-shifting maps for simple pairs (Definition14.11) with those for complementary subsurfaces (Definition 14.21) into the promised shape-shifting cocycle(Proposition 14.26), which is well-defined as long as the combinatorial deformation s is small enough. Asusual, we construct a geometric train track τ from λ on X so that the weight space of τ α provides a notionof size for s . Admissible routes.
For v, w ∈ H and Y a component of (cid:101) X \ (cid:101) λ , we say that Y is thick with respect to v and w if one of the two possibilities occur:(1) either Y contains v and/or w , or(2) v and w lie in different components of (cid:101) X \ Y and the boundary leaves of Y closest to v and w arenot asymptotic.Observe that in the first case, there is either no or one boundary geodesic of Y separating v from w (dependingif v and w are both in Y or not), while in the second, there are exactly two boundary components of Y separating v from w .Now let v, w ∈ H be any pair of distinct hexagons that do not lie in the same component of (cid:101) X \ (cid:101) λ andlet ( h v , p v ) and ( h w , p w ) be a pointed geodesic in ∂ λ H v and ∂ λ H w . Then there is a unique (possibly empty)sequence h , . . . , h n of boundary geodesics of thick subsurfaces separating p v from p w , ordered by proximityto v (with h closest). If one of the h i lies in the boundary of two complementary subsurfaces (socorresponds to a lift of a curve component of λ ) then we record it twice, one time for each of the adjoiningsubsurfaces. Additionally, if either h v or h w is a boundary geodesic separating v from w , then we do notrecord it as one of the h i . See Figure 26.We now define an admissible route from p v to p w to be any sequence of basepoints p v = p , p ∈ h , . . . , p n ∈ h n , p n +1 = p w coming from the projections of the central vertices u i of hexagons H u i to h i ∈ ∂ λ H u i . If any geodesic h i = h i +1 is repeated then we require that v and u i lie on one side of h i and that w and u i +1 lie on theother. Observe that the sequence of pairs ( u i , u i +1 ) necessarily alternates between simple pairs/pairs sharinga boundary geodesic and pairs which lie in the same (thick) subsurface. Shape-shifting along admissible routes.
To any admissible route we can then define a shape-shiftingtransformation by concatenating the shape-shifting transformations for subsequent pairs:(43) ϕ p v ,p w := ϕ p ,p ◦ . . . ◦ ϕ p n ,p n +1 Note that this sequence is necessarily finite, as the distance that any geodesic travels in a thick subsurface is boundedbelow by the shortest arc of α (compare the discussion of “close enough” pairs of hexagons in Section 13.2). p v p p p p p p p p w Figure 26.
Thick subsurfaces between v and w and an admissible route from p v to p w . Inthe figure we have highlighted a path from p v to p w through the p i ; each subpath from p i to p i +1 specifies a factor in the shape-shifting transformation.where ϕ p i ,p i +1 is as in Definition 14.11 if ( u i , u i +1 ) is simple and as in Definition 14.21 if u i and u i +1 lie inthe same subsurface. If h i = h i +1 , then we orient h i to the right as seen from u i and set ϕ p i ,p i +1 = T s ( u i ,u i +1 ) h i (recall that we can associate a shear value to the pair ( u i , u i +1 ) by (15)). Lemma 14.25.
The shape-shifting map ϕ p v ,p w is independent of the choice of admissible route (as long asit is defined). Proof.
Since the h i are uniquely determined, it suffices to change one point at a time.So suppose that p i and p (cid:48) i are both basepoints on the geodesic h i ; we then demonstrate the equality ϕ p i − ,p i ◦ ϕ p i ,p i +1 = ϕ p i − ,p (cid:48) i ◦ ϕ p (cid:48) i ,p i +1 from which the lemma follows. Orient h i so that it runs to the right as seen from v or u i − .Without loss of generality, we may assume that the hexagons u i and u i +1 lie in the same subsurface.Otherwise, the hexagons u i − and u i lie in the same subsurface and so ( p i , p i +1 ) is either simple or thepoints lie on the same isolated leaf. If this happens we prove that ϕ p i +1 ,p i ◦ ϕ p i ,p i − = ϕ p i +1 ,p (cid:48) i ◦ ϕ p (cid:48) i ,p i − , which is equivalent to the equation above since each of the shape-shifting factors inverts when one flips theorder of the points.We first consider the shape-shifting transformations coming from comparing p i or p (cid:48) i with p i +1 . By ourreduction above, u i and u (cid:48) i lie in the same thick subsurface Y . Let α , . . . , α m denote the arcs of (cid:101) α ∩ Y encountered when traveling from p (cid:48) i to p i along h i ; then our definition of shape-shifting in subsurfacesassociates the transformation ϕ p (cid:48) i ,p i = T ε (cid:80) mj =1 s ( α j ) h i where ε = 1 if h i is oriented from p (cid:48) i to p i and − ϕ p (cid:48) i ,p i +1 = ϕ p (cid:48) i ,p i ◦ ϕ p i ,p i +1 = T ε (cid:80) mj =1 s ( α j ) h i ◦ ϕ p i ,p i +1 . We now turn our attention to the transformation ϕ p i − ,p (cid:48) i . Consider first the case when ( p i − , p i ) is simple;since p i and p (cid:48) i both lie on h i this implies that ( p i − , p (cid:48) i ) is also simple. Moreover, since the geodesics H i − ,i that separate p i − from p i are the same that separate p i − from p (cid:48) i , we may write ϕ p i − ,p i = lim H→H v,w ϕ ( s ) ◦ . . . ◦ ϕ ( s n ) ◦ T s ( u i − ,u i ) h i HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 75 and similarly for ϕ p i − ,p (cid:48) i . In particular, each approximation for ϕ p i − ,p i differs from the approximation for ϕ p i − ,p (cid:48) i by translation along h i , and so the same is true in the limit:(45) ϕ p i − ,p (cid:48) i = ϕ p i − ,p i ◦ T s ( u i − ,u (cid:48) i ) − s ( u i − ,u i ) h i . Applying axiom (SH3) for shear-shape cocycles (Definition 7.12) multiple times, we compute that(46) s ( u i − , u (cid:48) i ) − s ( u i − , u i ) = ε m (cid:88) j =1 s ( α j )where ε = +1 if p i precedes p (cid:48) i along h i and − p (cid:48) i precedes p i . Combining (44), (45), and (46), ϕ p i − ,p (cid:48) i ◦ ϕ p (cid:48) i ,p i +1 = ϕ p i − ,p i ◦ T ε (cid:80) mj =1 s ( α j ) h i ◦ T ε (cid:80) mj =1 s ( α j ) h i ◦ ϕ p i ,p i +1 = ϕ p i − ,p i ◦ ϕ p i ,p i +1 since ε = − ε . This completes the proof of the lemma in the case when ( u i − , u i ) is simple.Similarly, if p i − and p i lie on the same isolated leaf of λ then so must p (cid:48) i . Unpacking the definitions showsthat (45) holds in this case, and Lemma 7.15 implies that (46) does as well. Therefore, in this case we alsosee that the desired equality holds. (cid:3) Finally, now that we have constructed shape-shifting maps for arbitrary pairs of pointed geodesics in ∂ λ H we can prove that they piece together into an Isom + ( (cid:101) X )-valued cocycle. Proposition 14.26.
The map constructed from X and s ϕ s : ∂ λ H × ∂ λ H →
Isom + ( (cid:101) X )(( h v , p v ) , ( h w , p w )) (cid:55)→ ϕ p v ,p w is a π ( X )-equivariant 1-cocycle, as long as (cid:107) s (cid:107) τ α < D λ ( X ) / Proof.
That ϕ is π ( X )-equivariant means that ϕ γp v ,γp w = γ ◦ ϕ p v ,p w ◦ γ − for γ ∈ π ( X ); this followsdirectly from the construction.That ϕ is a 1-cocycle means it satisfies the familiar cocycle condition on triples, i.e., ϕ p u ,p v ◦ ϕ p v ,p w = ϕ p u ,p w . Note that if p v lies on some admissible route from p u to p w then this is fulfilled automatically by unpackingthe definitions and invoking Lemma 14.25.One special case of the cocycle condition is when p u = p w ; in this case we must show that ϕ p v ,p w = ϕ − p w ,p v .To demonstrate this, observe that reversing an admissible route from v to w produces an admissible routefrom w to v . Moreover, by Corollary 14.14 in the simple case and by definition in the other cases, each ϕ p i ,p i +1 also inverts when we flip i and i + 1, proving that reversing v and w inverts ϕ p v ,p w .Now suppose that u, v, and w are all distinct; then there exists a unique subsurface Y of (cid:101) X \ (cid:101) λ such thateach component of (cid:101) X \ Y contains at most one of u , v , or w (note that some of u, v, w may be inside of Y ).Choose basepoints r u , r v , and r w on the boundary components of Y that are closest to u , v , and w (if any • ∈ { u, v, w } is in Y then set r • = p • ). See Figure 27.Choose an admissible route from p u to p v containing r u and r v , and similarly for the other two pairs.Then by Lemma 14.25 and the observation that the cocycle condition holds along admissible routes, we maywrite ϕ p u ,p v = ϕ p u ,r u ◦ ϕ r u ,r v ◦ ϕ r v ,p v and similarly for the other two pairs. Combining all three equations and applying the cocycle relation for Y (Proposition 14.24), we see that ϕ p u ,p v ◦ ϕ p v ,p w = ϕ p u ,r u ◦ ϕ r u ,r v ◦ ϕ r v ,p v ◦ ϕ p v ,r v ◦ ϕ r v ,r w ◦ ϕ r w ,p w = ϕ p u ,r u ◦ ϕ r u ,r w ◦ ϕ r w ,p w = ϕ p u ,p w , finishing the proof. See Figure 27 for a graphical depiction of this argument. (cid:3) p v = r v r w p w r u p u ϕ p u ,p v ϕ p v ,p w ϕ p w ,p u Figure 27.
The cocycle relation for admissible routes.15.
Shear-shape coordinates are a homeomorphism
We now finish the proof of Theorem 12.1 by proving that the map σ λ : T ( S ) → SH + ( λ ) is open (Theorem15.1) and proper and thus, by invariance of domain, a homeomorphism.In Section 15.1, we use the shape-shifting cocycle ϕ s , built in the previous section, to deform the represen-tation ρ : π ( S ) → PSL R that induces the hyperbolic structure X . The deformed representation ρ s is thendiscrete and faithful (Lemma 15.3) and the quotient surface X s has the desired shear-shape cocycle (Lemma15.6). In particular, this gives us a continuous local inverse to σ λ , proving openness. These statementsare similar in spirit to those in [Bon96], but the specifics of our proofs are different. In particular, insteadof adjusting the relative placements of ideal triangles of (cid:101) X \ (cid:101) λ we adjust the relative position of pointedgeodesics comprising (cid:101) λ .We then prove properness of σ λ in Section 15.2, concluding the proof of Theorem 12.1. Here we returnto Bonahon’s argument [Bon96, Theorem 20], but applying this strategy in our setting still requires a bit ofextra care due to the polyhedral structure of SH + ( λ ).Finally, in Section 15.3 we show that the action of R > on SH + ( λ ) by dilation produces lines in T ( S )that can sometimes be identified with directed Thurston geodesics.15.1. Deforming by shape-shifting.
In this section, we show that any positive shear-shape cocycle closeenough to σ λ ( X ) is actually the geometric shear-shape cocycle of a hyperbolic structure. Compare with[Bon96, Proposition 13]. Theorem 15.1.
Let β be a maximal arc system containing α ( X ) and let τ β be a standard smoothing. Thenfor any s ∈ W ( τ β ) such that (cid:107) s (cid:107) τ β < D λ ( X ) / σ λ ( X ) + s represents a positive shear-shapecocycle, there exists X s ∈ T ( S ) close to X with σ λ ( X s ) = σ λ ( X ) + s . In particular, the image of σ λ ( X ) is open in SH + ( λ ).The proof of this theorem appears at the end of this subsection as the culmination of a series of structurallemmas. Our strategy is to explicitly define X s by using the shape-shifting cocycle constructed in Section14 to deform the hyperbolic structure on X . Before proceeding we note the following Corollary 15.2.
For all t ∈ R , and for all µ ∈ ∆( λ ), we have the following identity σ λ (Eq tµ ( X )) = σ λ ( X ) + tµ. Proof.
That the earthquake Eq tµ ( X ) is defined for all time is a consequence of countable additivity (equiv-alently, positivity) of µ ; a complete proof can be found in [EM06, Section III]. Viewing the set of mea-sures supported on λ as a subset of H ( λ ), the formula is immediate from Theorem 15.1 once we note thatEq tµ ( X ) = X tµ , which follows from the description of ϕ tµ as a limit of simple left (or right) earthquakes;see (37) and Lemma 14.13. (cid:3) HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 77
Fix s as in the statement of the theorem and pick an arbitrary v ∈ H and ( h v , p v ) ∈ ∂ λ H v . Identifying (cid:101) X isometrically with H and ( h v , p v ) with a pointed line picks out a representation ρ : π ( S ) → PSL R thatinduces X . Since (cid:107) s (cid:107) τ α < D λ ( X ) /
2, Proposition 14.26 allows us to construct the shape-shifting cocycle ϕ s .We may now deform the representation ρ by ϕ s by defining ρ s : π ( S ) → PSL R γ (cid:55)→ ϕ p v ,γp v ◦ ρ ( γ ) . The equivariance and cocycle properties of Proposition 14.26 ensure that ρ s is itself a representation. Indeed: ρ s ( γ γ ) = ϕ p v ,γ γ p v ◦ ρ ( γ γ )= ϕ p v ,γ p v ◦ ϕ γ p v ,γ γ p v ◦ ρ ( γ ) ◦ ρ ( γ )= ϕ p v ,γ p v ◦ ρ ( γ ) ◦ ϕ p v ,γ p v ◦ ρ ( γ ) − ◦ ρ ( γ ) ◦ ρ ( γ )= ρ s ( γ ) ◦ ρ s ( γ )for all γ , γ ∈ π ( S ). Our goal in the rest of the section is then to show that ρ s is discrete and faithful, andthat the quotient surface has the correct geometric shear-shape cocycle. Adjusting geodesics.
To show that ρ s has the desired properties, we use ϕ s to adjust the position of (cid:101) λ in (cid:101) X . Ultimately, these adjusted geodesics correspond to the realization of λ on the quotient surface (cid:101) X/ im ρ s .Let G ( (cid:101) X ) be the space of geodesics in (cid:101) X , and let ∂ (cid:101) λ ⊂ G ( (cid:101) X ) denote the set of boundary leaves of (cid:101) λ .Define a map Φ p v : ∂ (cid:101) λ → G ( (cid:101) X ) , as follows: if h is a leaf of ∂ (cid:101) λ , then h = h u for some ( h u , p u ) in ∂ λ H . The map Φ p v then takes ( h u , p u )isometrically to the pointed geodesic ϕ p v ,p u ( h u , p u ) ⊂ (cid:101) X . Note that if h u = h w for some other ( h w , p w ) in ∂ λ H , then ϕ − p v ,p u ◦ ϕ p v ,p w = ϕ p u ,p w by the cocycle relation (Proposition 14.26) and ϕ p u ,p w is by definition a translation along h . ThereforeΦ p v h w = Φ p v h u , so Φ p v is indeed well defined. Lemma 15.3.
The representation ρ s constructed above is discrete and faithful. Proof.
For distinct leaves h u and h w ∈ ∂ (cid:101) λ , we claim that Φ p v ( h u ) is disjoint from Φ p v ( h w ). Indeed, by thecocycle relation, the position of Φ p v ( h w ) relative to Φ p v ( h u ) is the same as the position of ϕ p u ,p w h w relativeto h u . Every finite approximation of ϕ p u ,p w by compositions of elementary shape-shifting transformationspreserves the property that the image of h w is disjoint from h u , so the same is true in the limit.So, as long as ρ ( γ ) does not stabilize h v then Φ p v ( ρ ( γ ) h v ) = ρ s ( γ ) h v is different from Φ p v ( h v ) = h v . If ρ ( γ ) is a translation along h v , we can find γ such that ρ ( γ γγ − ) h v (cid:54) = h v . Together with the previous line,this implies that ρ s is faithful.Since π ( S ) is a non-elementary group and ρ s is faithful, im ρ s is a non-elementary subgroup of isometries.So assume towards contradiction that ρ s is indiscrete; then im ρ s must be dense in PSL R . In particular,there is an element γ ∈ π ( S ) such that ρ s ( γ ) is arbitrarily close to a rotation of angle π/ p v . Then ρ s ( γ ) h v = Φ p v ρ ( γ ) h v meets h v in a point, which is impossible, because Φ p v ( h v ) is either equal to or disjointfrom Φ p v ( ρ ( γ ) h v ). We conclude that ρ s is discrete, completing the proof of the lemma. (cid:3) By Lemma 15.3, the quotient X s = (cid:101) X/ im ρ s is a hyperbolic surface equipped with a homeomorphism S → X s in the homotopy class determined by ρ s . As such, ρ s induces a ( ρ, ρ s )-equivariant homeomorphism ∂ (cid:101) X → ∂ (cid:101) X , hence a continuous, equivariant map on the space of geodesics. Lemma 15.4.
The map Φ p v extends continuously to (cid:101) λ , and Φ p v ( (cid:101) λ ) descends to the geodesic realization of λ on X s . Proof.
By equivariance, the induced map on geodesics agrees with Φ p v on ∂ (cid:101) λ . The leaves of ∂ (cid:101) λ are densein (cid:101) λ , so the closure of the image of Φ p v is the geodesic realization of (cid:101) λ on (cid:101) X s , which is invariant under theaction of ρ s . (cid:3) Since Φ p v ( (cid:101) λ ) is the lift of the realization of λ on X s , we may now leverage our understanding of theshape-shifting cocycle to show that the complementary subsurfaces of X s \ λ have the desired shapes. Lemma 15.5.
We have A ( X s ) = A ( X ) + a . Proof.
Recall that by construction the unweighted arc systems of X \ λ and X s \ λ are both contained insome joint maximal arc system β , leading to an identification of hexagons of (cid:101) X \ λ with those of (cid:101) X \ λ s .So let β be an arc of β , realized orthogeodesically in X s . Let β also denote a choice of lift, orthogonal to λ s in (cid:101) X , and let u and w denote the hexagons adjacent to β . Choose either of the geodesics g of λ meeting β and let q u and q w be the basepoints of u and w on g . Then by the cocycle relation (Proposition 14.26),we know that ϕ p v ,p w = ϕ p v ,p u ◦ ϕ p u ,p w and so applying equivariance we see that the placement of q w relative to q u differs from the placement of p w relative to p u only by ϕ p u ,p w .But now since u and w are in the same subsurface, we see by definition that ϕ p u ,p w is translation along g by exactly s ( β ). Therefore, the distance along ϕ p u ,p w g between q u and q w is exactly the distance along g between p u and p w plus s ( β ). Translated into arc weights, σ λ ( X s )( β ) = σ λ ( X )( β ) + s ( β ) , completing the proof of the lemma. (cid:3) Now that we know that the “shape” part of the data of σ λ ( X s ) is what it is supposed to be, we need onlycheck that the “shearing” data is as specified. Compare [Bon96, Lemma 19]. Lemma 15.6.
The surface X s has geometric shear-shape coycle σ λ ( X s ) = σ λ ( X ) + s . Proof.
Observe that by the cocycle relation (Proposition 14.26) and the discussion in Section 13.2, it sufficesto compute the change in shearing data between simple pairs.So suppose that ( v, w ) is simple. For each integer r , recall that H rv,w = ( u i ) ni =1 denotes the set of hexagonssuch that the intersection of the geodesic from p v to p w with u i has depth at most r with respect to a fixedgeometric train track. Set v = u and w = u n +1 , and let h i = g vu i , the pointed boundary geodesic of u i closest to v . Then by Lemma 14.13, we know that ϕ rp v ,p w = T s ( u ,u ) h ◦ A ( s ) ◦ T s ( u ,u ) h ◦ A ( s ) ◦ ... ◦ A ( s n ) ◦ T s ( u n ,u n +1 ) h n is a good approximation of ϕ p v ,p w for large enough r .Now for each r we can deform the hyperbolic structure on (cid:101) X by ϕ rp v ,p w (sacrificing equivariance) andmeasure the shear σ r ( v, w ) between v and w in that deformed structure. More precisely, we recall that if h (cid:48) i denotes the other geodesic in u i that separates v from w , the spike-shaping transformation is equal to atranslation along h (cid:48) i then along h i . We may then deform (cid:101) X by replacing each translation in the factorizationof ϕ rp v ,p w with a (right) earthquake along the same geodesic; compare with our “geometric explanation” ofspike-shaping in Section 14.2.Since each translation T s ( u i ,u i +1 ) h i appearing in ϕ rp v ,p w shears (cid:101) X along a leaf of λ , it preserves the ortho-geodesic foliation in complementary components. Therefore, each such term in the deformation thus changesthe shear between v and w by exactly s ( u i , u i +1 ).On the other hand, each spike-shaping transformation A ( s i ) is a parabolic transformation fixing the vertexof the spike and thus preserves horocycles based at that point. In particular, the distinguished basepointsof each h i and h (cid:48) i remain on the same horocycle and hence deforming by A ( s i ) does not affect σ r ( v, w ).In summary, deforming (cid:101) X by the approximation ϕ rp v ,p w changes the shear between v and w by σ r ( v, w ) − σ ( v, w ) = n (cid:88) i =0 s ( u i , u i +1 ) = s ( v, w )where the last equality follows from finite additivity (axiom (SH2)).Since this equality holds in each approximation and ϕ rp v ,p w → ϕ p v ,p w as r → ∞ , the equality holds inthe limit as well. Therefore, deforming (cid:101) X by ϕ p v ,p w changes the shear between v and w by exactly s ( v, w ),which is what we needed to show. (cid:3) HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 79
Proof of Theorem 15.1. As (cid:107) s (cid:107) τ α < D λ ( X ) /
2, Lemmas 14.10/14.13 ensure that the limits in the definitionof ϕ p v ,p w make sense for all simple pairs ( v, w ). Proposition 14.26 then allows us to construct ϕ s . By Lemma15.3, the deformed representation ρ s = ϕ s · ρ is discrete and faithful, and by Lemma 15.6 the quotient surfacehas the correct geometric shear-shape cocycle.Finally, we observe that the values of shape-shifting cocycle ϕ s all converge to the identity as (cid:107) s (cid:107) τ α → X s → X. This completes the proof of the theorem. (cid:3)
The global structure of the shear-shape map.
We have already proven in Proposition 13.14 thatthe image of σ λ lies in SH + ( λ ). We now show that this containment is in fact an equality, completing theproof of Theorem 12.1.We proceed in two steps; the first is to show that Proposition 15.7.
The shear-shape map σ λ is a homeomorphism onto its image. Proof.
Proposition 13.12 (injectivity of σ λ ) allows us to invert σ λ on its image and for each X ∈ T ( S ),Theorem 15 . σ λ ( X ) ∈ SH + ( λ ) on which σ − λ is defined andcontinuous. By Proposition 8.5, we have that SH + ( λ ) ⊂ SH ( λ ) is an open cell of dimension 6 g − σ − λ and hence σ λ are local homeomorphisms. An additionalapplication of Proposition 13.12 implies that σ λ is globally injective, so σ λ is a homeomorphism onto itsimage as claimed. (cid:3) The second step is to prove that σ λ : T ( S ) → SH + ( λ ) is a proper map. That is, we must show that when X k escapes to infinity in T ( S ), the corresponding shear-shape cocycles σ λ ( X k ) must diverge in SH + ( λ ). Sinceproper local homeomorphisms are coverings and SH + ( λ ) is a cell, the map σ λ must be a homeomorphism.The proof we present below is essentially just that of [Bon96, Theorem 20], but we have to address theadditional complications introduced by the PL structure of SH + ( λ ); this manifests itself in the stratified real-analytic structure of the map. Proof of Theorem 12.1.
We begin by recording an estimate for the geometry of surfaces near the boundaryof the image of σ λ (where “near” is measured in a train track chart).So suppose that X ∈ T ( S ), set α = α ( X ), and build a standard smoothing τ α carrying λ geometrically on X . Fix (cid:15) > s ∈ W ( τ α ) with (cid:107) s (cid:107) τ α < (cid:15) such that σ λ ( X ) + s ∈ SH + ( λ )is not in the image of σ λ ; then Theorem 15.1 implies that D λ ( X ) / ≤ (cid:107) s (cid:107) τ α < (cid:15). The following claim can be extracted from the proof of [Bon96, Theorem 20]; we outline a proof for theconvenience of the reader.
Claim 15.8.
There is a transverse measure µ ∈ ∆( λ ) with χ ( S ) ≤ (cid:107) µ (cid:107) τ α ≤ (cid:96) X ( µ ) = ω SH ( σ λ ( X ) , µ ) < (cid:15). Proof of Claim 15.8.
If there is a simple closed curve component of λ with length at most (cid:15) , then we aredone. Otherwise, even though A + a defines a hyperbolic structure on each piece of S \ λ , the overall shear-shape cocycle σ λ ( X ) + s does not define a hyperbolic structure on S because the proof of Lemma 14.10 orLemma 14.13 fails. Therefore, there is a simple pair ( v, w ) for which the finite products ϕ H (or ϕ rv,w ) fail toconverge as H tends to H v,w (or r → ∞ ).We claim that there exists u between v and w and a spike s = ( g, h ) of H u such that the following holds:for any geodesic transversal k ⊂ X to λ meeting the spike s , the countably many points of ˜ k ∩ g ⊂ g (labeledby r ∈ N ) exiting one end of g escape at a rate strictly slower than (cid:15) ( r − d r ⊂ g such that (cid:96) X ( d r ) ≤ (cid:15) ( r −
1) and d r meets k exactly r times.If this were not the case, then as in the proof of Lemma 14.5, the “gaps” c r ⊂ k v,w \ λ have length (cid:96) X ( c r ) = O ( e − (cid:15)r ), where c r ∩ g is labelled by r ∈ N . This estimate on the decay of gaps implies that ϕ H converges as H → H v,w and that ϕ rv,w → ϕ v,w as r → ∞ (see the proof of Lemma 14.10), contradicting ourassumption. Recall that H + ( λ ) is an open cone with finitely many faces in a vector space, while SH + ( λ ) is an affine cone bundle overa piecewise linear space with no obvious way of extending the smooth structure over faces of B ( S \ λ ). Now consider the weight system w r on τ α (not satisfying the switch conditions) defined by counting thenumber of times d r travels along each branch of τ α , and dividing by the total number of branches n r that d r traverses, with multiplicity. Observe that n r ≥ r by definition. Then (cid:107) w r (cid:107) τ α ≤ R b ( τ α ) and w r takes value zero on branches corresponding to arcs of α . Moreover, w r is non-negative on eachbranch and approaches the weight space W ( τ ) ⊂ R b ( τ ) as r → ∞ . Since w r are built from leaves of λ , anylimit point µ defines a transverse measure supported on λ (compare also [PH92, Proposition 3.3.2]).There are at most 9 χ ( S ) branches of τ α , so by the pigeonhole principal there is a branch such that each w r has mass at least 1 / χ ( S ), and therefore so must µ . But now by construction, (cid:96) X ( µ ) = lim r →∞ (cid:96) X ( d r ) n r < ( r − (cid:15)n r < (cid:15), providing the desired measure. (cid:3) Now suppose towards contradiction that α is maximal and σ λ ( X k ) ∈ SH + ( λ ; α ) is a sequence approachingsome σ ∈ SH + ( λ ; α ) that is not in the image of σ λ . We may then apply the above construction to σ − σ λ ( X k )to extract a family of measures µ k on λ satisfying 1 / χ ( S ) ≤ (cid:107) µ k (cid:107) τ α ≤ ω SH ( σ λ ( X ) , µ k ) →
0. Bycompactness of the set measures on λ with norm bounded away from zero and infinity, there is some non-zeroaccumulation point µ of µ k . Continuity of ω SH (Lemma 8.3) then gives ω SH ( σ λ ( X k ) , µ k ) → ω SH ( σ, µ ) = 0 , and so we see that σ (cid:54)∈ SH + ( λ ; α ), a contradiction. Hence im( σ λ ) ∩ SH + ( λ ; α ) is relatively closed. On theother hand, σ λ is a local homeomorphism by Proposition 15.7, hence im( σ λ ) ∩ SH + ( λ ; α ) is relatively open.If we knew that the projection of im( σ λ ) surjects onto B ( S \ λ ) (or at least meets each top-dimensionalface) we would be done. Since we do not a priori have this information, we instead work our way out in B ( S \ λ ) cell by cell.To wit, we may invoke Theorem 15.1 once more to deduce that im( σ λ ) ∩ SH + ( λ ; α (cid:48) ) is relatively openfor every filling arc system α (cid:48) that shares a common filling arc subsystem with α (hence SH + ( λ ; α ) and SH + ( λ ; α (cid:48) ) intersect). Repeating the argument above for these cells, we have that im( σ λ ) ⊃ SH + ( λ ; α (cid:48) ) aswell. Since B ( S \ λ ) is connected, iterating this procedure allows us to deduce that im( σ λ ) ⊃ SH + ( λ ). Thereverse inclusion follows from Corollary 13.14, so σ λ is a homeomorphism onto SH + ( λ ).To address the regularity of σ λ , we note that while T ( S ) has a natural R -analytic structure, SH ( λ )does not. However, for each arc system α , filling or not, the open cell B ◦ ( α ) has a well defined analyticstructure compatible with that of the analytic submanifold of T ( S \ λ ) that it parametrizes. The totalspace of the bundle SH ◦ ( λ ; α ) → B ◦ ( α ) also carries an analytic structure, invariant under train trackcoordinate–transformations (Proposition 8.5); thus SH ( λ ) has a stratified R -analytic structure.The shape-shifting cocycle ϕ s , hence the surface X s , then depends real-analytically on s ∈ W ( τ α ) (where α here is equal to the support of A ( X ), not a maximal completion). The reason for this is clear: all elementaryshape-shifting transformations are products of small parabolic transformations (see [Thu98, Section 9] or[Bon96, Theorem A]) or translations with translation distance that are (restrictions of) real-analytic functionson (an analytic submanifold of) T ( S \ λ ). These products converge absolutely to the shape-shifting cocycle,hence uniformly on compact sets to an analytic deformation. (cid:3) Dilation rays and Thurston geodesics.
Using our coordinatization, we can define an extension ofthe earthquake flow to an action by the upper-triangular subgroup.
Definition 15.9.
Given a measured geodesic lamination λ , a hyperbolic surface X ∈ T ( S ), and t ∈ R ,define an analytic path of surfaces { X tλ } t ∈ R by X tλ := σ − λ ( e t σ λ ( X )) , called the dilation ray based at X directed by λ . We are abusing terminology here by declaring that the image of R under an analytic mapping is a ray. Our aim is toemphasize that the dilation ray should be thought of as directed toward the future, even though it can be defined for all time. HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 81
As the earthquake flow acts by translation in coordinates (Corollary 15.2), we see that dilation andearthquake along λ (together with scaling the measure on λ ) fit together into an action by the upper-triangular subgroup B < GL +2 R on PT g . More explicitly, we can specify an action of B on T ( S ) × R > λ (byhomeomorphisms) by setting(47) (cid:18) a b c (cid:19) · ( X, λ ) := ( σ − λ ( aσ λ ( X ) + bλ ) , cλ ) . These B -actions assemble into a Mod( S )-equivariant B -action on PT g (observe that σ λ depends only on thesupport of λ and not the actual measure). Quotienting by the mapping class group and restricting to theunit length locus then gives a P -action on P M g , and since dilation preserves the property of being regular,a P -action on each stratum P M reg g ( κ ). We call any such action an action by stretchquakes .Using the commutativity of Diagram (2) (Theorem 13.13), we can compare (47) with the computationsperformed in Lemmas 11.1 and 11.2 to see that Proposition 15.10.
The map O takes the P action of (47) on P M g to the standard P action on Q M g .While we have defined them via coordinates, it is not hard to see that dilation rays are geometricallymeaningful families of surfaces. Generally, we obtain paths of surfaces along which the length of λ scalesnicely, and we can identify some dilation rays as directed lines in Thurston’s asymmetric metric on T ( S ).Mirzakhani observed (see [Mir08, Remark p. 33]) that for a maximal lamination µ , the dilation ray t (cid:55)→ X tµ corresponds to the stretch path directed by µ defined by Thurston in [Thu98, Section 4]. Veryroughly, stretch paths are obtained by gluing together certain expanding self-homeomorphisms of the idealtriangles comprising X \ µ along the leaves of µ . Lemma 15.11 (Proposition 2.2 of [Thu98]) . Let P n be a regular ideal hyperbolic n -gon. For any K ≥ K -Lipschitz self-homeomorphism P n → P n that maps each side to itself and expands arclengthalong the boundary by a constant factor of K . Proof.
The orthogeodesic foliation O ( P n ) is measure equivalent to a partial foliation by horocycles centeredat the spikes of P n . The desired K -Lipschitz homeomorphism P n → P n is constructed by fixing the centralhorocyclic n -gon and mapping each horocyclic arc at distance s from the central region to the horocyclic arcat distance Ks in the same spike linearly so that the length of each horocycle shrinks by a factor of K − . (cid:3) Any partition κ = ( κ , ..., κ n ) of 4 g − PT reg g ( κ ) of pairs ( X, λ ), where thecomplement of λ in X is a union of regular ideal ( κ i + 2)-gons. Then P M reg g ( κ ) is the moduli space of pairswhere (cid:96) X ( λ ) = 1.Gluing together the expanding maps of regular polygons provides an explicit model of dilation rays in P M g ( κ ) and identifies them with geodesics for the Thurston metric. The following proposition was inspiredin part by recent work of Horbez and Tao, in which they investigate the minimally displaced sets in theThurston’s metric using a similar construction [HT]. Proposition 15.12.
For any (
X, λ ) ∈ PT reg g ( κ ), the dilation ray { X tλ : t ∈ R } ⊂ PT reg g ( κ ) is a directedunit-speed geodesic in Thurston’s asymmetric Lipschitz metric. Proof.
Since λ is regular on X , σ λ ( X ) ∈ SH + ( λ ) lies in the fiber over the empty arc system. Scaling σ λ ( X )preserves this arc system, so X tλ is regular for all t . It suffices to prove that the optimal Lipschitz constantfor a map X → X tλ in the homotopy class determined by markings is e t for all t ≥ H λ ( X ) denote the (partial) foliation of X by horocyclic arcs that is measure equivalent to O λ ( X ). Themaps of Lemma 15.11 assemble to an e t -Lipschitz homeomorphism X \ λ → X λt \ λ such that H λ ( X ) mapsto H λ ( X tλ ) = e t H λ ( X ) on each component (as measured foliations). Now, using the fact that σ λ ( X tλ ) = e t σ λ ( X ), we can adapt the argument of [Bon96, Lemma 11] (as sketched in Proposition 13.12) to show thatthis map is locally Lipschitz hence extends across λ to an e t -Lipschitz homeomorphism X → X tλ .Thus e t provides an upper bound for the optimal Lipschitz constant in the homotopy class determinedby markings. On the other hand, (cid:96) X tλ ( λ ) = ω SH ( σ λ ( X tλ ) , λ ) = ω SH ( e t σ λ ( X ) , λ ) = e t (cid:96) X ( λ ) , so e t is also a lower bound for the optimal Lipschitz constant. This completes the proof of the proposition. (cid:3) Remark 15.13.
As in the last line of the proof of Proposition 15.12 we always have (cid:96) X tλ ( λ ) = e t (cid:96) X ( λ ) forarbitrary λ ∈ ML ( S ). Thus the distance from X to X tλ in Thurston’s metric is at least t . However, we donot always know how to build e t -Lipschitz proper homotopy equivalences X \ λ → X tλ \ λ (in the correcthomotopy class) that expand arclength along ∂X \ λ by a constant factor of e t . Remark 15.14.
Our dilation rays are different from Thurston’s stretch rays defined with respect to oneof the finitely many maximal completions of λ when λ is not maximal. This follows from the fact that O λ ( X ) (cid:54) = O λ (cid:48) ( X ), where λ (cid:48) is a maximal completion of λ .The map PT reg g ( κ ) × R → PT reg g ( κ ) defined by the rule ( X, λ, t ) (cid:55)→ ( X tλ , e − t λ ) is called the stretch flow .The stretch flow is Mod( S )-equivariant and (cid:96) X tλ ( e − t λ ) = (cid:96) X ( λ ) , hence descends to P M reg g ( κ ). Corollary 15.15.
Let ν be a P -invariant ergodic probability measure on P M g . • For ν -almost every ( X, λ ), the dilation ray t (cid:55)→ X tλ is a unit speed geodesic in Thurston’s asymmetricmetric. • On a set of full ν -measure, the action of the diagonal subgroup of P is identified with the stretchflow and O conjugates stretch flow to Teichm¨uller geodesic flow.In particular, the stretch flow is ergodic with respect to ν . Proof.
By Corollary 1.1, ν -almost every point is regular (with respect to the same topological type oflamination), so the first statement of the theorem is immediate from Proposition 15.12.The second statement is essentially a restatement of Theorem B combined with the previous statement.Alternatively, in the Gardiner-Masur parameterization of QT g (Theorem 2.1), the Teichm¨uller geodesic flowat time t is given by ( η, λ ) (cid:55)→ ( e t η, e − t λ ), so unraveling the definitions and using commutativity of Diagram(2) (Theorem 13.13) gives the result.For ergodicity, we apply Theorem C which asserts, in particular, that O ∗ ν is a SL R -invariant probabilitymeasure on Q M g ( κ ). By the Howe–Moore Ergodicity Theorem (see, e.g., [FK02, Theorem 3.3.1]), theTeichm¨uller geodesic flow is ergodic for ν . So O maps any stretch flow–invariant set B of positive ν -measureto an O ∗ ν Teichm¨uller geodesic flow–invariant set of positive measure, which must have full measure byergodicity. Thus ν ( B ) = 1, demonstrating ergodicity of the stretch flow. (cid:3) Recently, Allessandrini and Disarlo [AD20] constructed Lipschitz maps between some pairs of degenerateright angled hexagons that stretch alternating boundary geodesics by a constant factor. Recall from Section6 that the Teichm¨uller space of an ideal quadrilateral is 1 dimensional and can be described as the the coneover a pair of points corresponding to the two arcs α and β that join opposite sides of Q . Lemma 15.16.
Let Q be an ideal quadrilateral with weighted filling arc system sδ , where δ ∈ { α, β } . Let Q t be the quadrilateral with arc system e t sδ . There is an e t -Lipschitz surjection Q → Q t that multipliesarclength along the boundary of Q by a factor of e t . Moreover, the projection of the compact edge of thespine of Q is mapped to the projection of the compact edge of the spine of Q t . Proof.
Every ideal quadrilateral has an orientation preserving isometric involution swapping opposite sides.Thus the orthogeodesic representative of δ cuts Q into 2 isometric pieces, each of which is a right angledhexagon with two degenerate sides. On each piece, we can apply [AD20, Lemma 6.9] to obtain maps whichglue together along δ to give a map with the desired properties. (cid:3) We immediately obtain some new geodesics for Thurston’s metric.
Proposition 15.17. If S \ λ consists of ideal triangles and quadrilaterals, then for any X ∈ T ( S ), t (cid:55)→ X tλ is a directed, unit speed geodesic for Thurston’s asymmetric metric. Proof.
The proof is nearly identical to the proof of Proposition 15.12, so we only provide a brief outline.Construct an e t -Lipschitz surjective map X \ λ → X λt \ λ from the units of Lemma 15.11 and Lemma15.16. For the same reason as before, this map extends continuously across the leaves of λ and provides an e t -Lipschitz homotopy equivalence X → X tλ in the homotopy class determined by markings. Thus e t is an HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 83 upper bound for the Lipschitz constant among homotopy equivalences X → X tλ in correct homotopy class.This is clearly an upper bound for the ratio max µ ∈ML ( S ) (cid:96) µ ( X tλ ) (cid:96) µ ( X ) . But e t is also a lower bound for this ratio, because the length of λ is scaled by a factor of e t .By a theorem of Thurston [Thu98, Theorem 8.5], there is a e t -Lipschitz homeomorphism X → X tλ homo-topic to the map constructed above. This completes the proof of the proposition. (cid:3) Remark 15.18.
The proof of Proposition 15.17 clearly supplies a more general statement: If λ is fillingand cuts X ∈ T ( S ) into a regular polygons and quadrilaterals of any shape, then t (cid:55)→ X tλ is a geodesic forThurston’s metric.There are other cases in which we can glue Lipschitz maps between degenerate right angled hexagons thatcan be found in the literature (e.g., [AD20, PY17]). However, these other cases require additional symmetrythat is not always present in our setting. We suspect that there is a different approach that would provethat dilation rays can always be identified with Thurston geodesics, so that O conjugates a kind of Thurstongeodesic flow to Teichm¨uller geodesic flow.16. Future and ongoing work
There is much more to understand about the correspondence between hyperbolic and flat geometry de-scribed in this paper. In addition to using the orthogeodesic foliation to import tools from Teichm¨ullerdynamics into the world of hyperbolic geometry (and vice versa), the authors expect this link to provideretroactive explanations for analogous phenomena in the two settings.We describe a number of future directions and potential applications of the correspondence below, someof which will be addressed in a forthcoming sequel.
Continuity and equidistribution.
Theorem D states that for a fixed lamination O λ : T ( S ) → MF ( λ ) isa homeomorphism, but as Mirzakhani already observed [Mir08, p. 33], O cannot be continuous on PT g . Atfault is the basic fact that the support of a measured lamination does not vary continuously in the relevanttopology.In forthcoming work [CF], the authors investigate the continuity properties of O restricted to specificfamilies of ( X, λ ) with constrained geometry and topology. On these families, the support of λ is forced tovary continuously in the Hausdorff topology as the pair varies (in the usual topology on PT g ). For example,each of the regular loci has this property. With this extra geometric control in hand, we prove that O restricts to a homeomorphism PT reg g ( κ ) ↔ QT nsc g ( κ ) on each regular locus.By imposing a stronger (yet still geometrically meaningful) topology on ML ( S ), we ensure the continuityof O varying over all pairs: let ML ( S ) denote the set of measured laminations with the “Hausdorff +measure” topology so that measured laminations are close in ML ( S ) if they are close both in measure andtheir supports are Hausdorff close. We prove a general phenomenon that O : T ( S ) × ML ( S ) → QT ( S ) islocally H¨older continuous with respect to a nice family of locally defined metrics in geometric train trackcoordinates.Our continuity arguments depend on a detailed analysis of the geometric structure of small foliatedtrain track neighborhoods of a lamination on a hyperbolic surface. This analysis is sufficiently robust toproduce “enough continuity” to deduce that O is a Borel-measurable isomorphism, a fact which is pivotalfor applications. The results of Section 1.2 then live in a more natural setting, as well.Combined with this work, the conjugacy of Theorems A and B allows us to import techniques of flatgeometry to the hyperbolic setting. In particular, while O is not continuous, its discontinuity is controlledenough that we can translate between equidistribution in P T g and equidistribution in Q T g . Symplectic structure.
For a maximal lamination λ , Bonahon and S¨ozen identified the Goldman symplecticform on the Teichm¨uller component of Hom( π S, PSL R ) / PSL R (also the Weil-Petersson symplectic form)as σ ∗ λ ω H in shear coordinates [SB01]. For arbitrary λ ∈ ML ( S ) and X ∈ T ( S ), the shape-shifting cocyclesbuilt in Section 14 provide an open set of deformations of the hyperbolization [ ρ : π S → PSL R ] of X (Theorem 15.1). Taking derivatives (as in [SB01]) identifies the tangent space to [ ρ ] with the vector space of Ad ρ -invariant Lie algebra valued 1-cocycles, yielding a reasonably explicit formula for a vector in the tangentspace at [ ρ ]. Using this formula, it is then possible to compute an expression for the Goldman symplectic form in shear-shape coordinates. From the formula, it immediately follows that O preserves the (degenerate)symplectic structure on strata.It may be possible to promote the intersection pairing ω SH : SH ( λ ) × H ( λ ) → R to a geometricallymeaningful symplectic pairing ω SH : SH ( λ ) × SH ( λ ) → R that interacts nicely with the natural Poissonstructure on B ( S \ λ ). The fact that SH ( λ ) is not in general a vector space complicates the discussion. Inparticular, it is not always clear how to make sense of the tangent space to a point in SH ( λ ) due to thepolyhedral structure of B ( S \ λ ). Measures.
As discussed in Section 9.3, the piecewise-integral-linear (PIL) structure on SH ( λ ) endows itwith an integer lattice and distinguished measure in the class of Lebesgue. For each filling λ , the integer latticein SH + ( λ ) restricts to an integer lattice on the fiber H + ( λ ) over the empty arc system due to integrality ofthe equations defining the piecewise-linear structure of B ( S \ λ ). The empty arc system corresponds to theset of X on which λ is regular, and so the PIL structure induces a measure (in the class of Lebesgue) on thisregular locus. Using our expression for the Goldman symplectic form in shear-shape coordinates, it may bepossible to identify this measure explicitly in terms of some geometrically meaningful quantity coming eitherfrom symplectic or Weil-Petersson geometry. For example, for maximal λ the main result of [SB01] impliesthis measure is (a constant multiple of) the Weil-Petersson volume element.Using snug train tracks, one can define a Thurston measure on the space ML ( κ ) of polygonal measuredlaminations of a given topological type. While this is essentially Lebesgue measure in train track coordinatesfor the “measure + Hausdorff” topology, it is not locally finite in the usual topology on ML ( S ). That said,integrating the snug Thurston measure against the one on the regular locus defined above should still yield a“Lebesgue” measure ν ( κ ) on P M g ( κ ), just as one can recover the Lebesgue measure on C n from Lebesguemeasure on R n and ( i R ) n .The Masur-Veech measures on stratum components are Lebesgue by definition and O preserves integerpoints; the authors therefore expect that O takes ν ( κ ) to the Masur-Veech measures of strata (up to somenormalization factor). The discussion above also indicates that Masur–Veech measure should disintegrateinto Weil–Petersson on the regular locus and Thurston on ML ( κ ), paralleling [Mir08, Theorem 1.4]. Counting.
The integral points of the PIL structure on SH + ( λ ) correspond to integer multicurves trans-verse to λ , so when λ is itself a multicurve, integral points correspond to square-tiled surfaces. Usingour coordinates for F uu ( λ ), one can recover the leading coefficient for the polynomial counting the num-ber of square-tiled surfaces with given horizontal curve of bounded area (which was originally computedin [AH20a, DGZZ20]). In particular, since renormalized lattice point counts equidistribute to Lebesguemeasure, the coefficient in question can be identified as the Lebesgue measure of (a torus bundle over) theportion of the combinatorial moduli space B ( S \ λ ) / Mod( S \ λ ) with controlled boundary lengths. Using theconvergence of the renormalized Weil-Petersson form to the Kontsevich form [Mon09b] (which on maximalcells induces the Lebesgue volume), this framework can also be manipulated to relate counts of curves onhyperbolic surfaces and intersection numbers on M g,n .Moreover, since our coordinates also record the singularity type of the associated differential, the authorsexpect that the same strategy can also be used to count square-tiled surfaces in a given stratum with fixedhorizontal multicurve. The leading coefficient should then be the Lebesgue volume of (a torus bundle over)a compact part of the appropriate subcomplex of the combinatorial moduli space. The relationships withhyperbolic geometry and intersection theory in these settings are more subtle, however, due in part to thenon-transversality of the metric residue condition cutting out B ( S \ λ ) and the combinatorial conditionsspecifying strata. The authors hope to investigate these properties more fully in future work. Expanding horospheres.
Counting problems for square-tiled surfaces/curves on hyperbolic surfaces areintricately related to the equidistribution of L -level sets for the intersection number with/hyperbolic lengthof laminations as one takes L → ∞ . When λ is a multicurve, the equidistribution of such “expandinghorospheres” to the Masur–Veech measure on the principal stratum of Q M g / the pullback by O of thismeasure on P M g (sometimes called Mirzakhani measure ) was established in [Mir07, AH20b, Liu20] using thegeometry of the (symmetrized) Lipschitz metric, the non-divergence of the earthquake flow, and a no-escape-of-mass argument. On the other end of the spectrum, the equidistribution of expanding horospheres formaximal λ to Q M g can be proven using a standard “thickening plus mixing” argument from homogeneousdynamics; in the flat setting this is implicit in the work of Lindenstrauss and Mirzakhani [LM08], and was HEAR-SHAPE COCYCLES FOR MEASURED LAMINATIONS 85 recently generalized in [For20, Theorem 1.6] using different methods. Equidistribution in the hyperbolicsetting then follows from Mirzakhani’s conjugacy.Using our extension of Mirzakhani’s conjugacy (and the continuity results described above), the authorsbelieve that the same “thickening plus mixing” technique can be used to prove that expanding horospheresbased at any λ equidistribute to the Mirzakhani measure on P M g . Moreover, we also expect an analogousresult for strata: intersections of expanding horospheres based at λ and the regular locus should equidistributeto the pullback to P M g of the Masur–Veech measure for a component of Q M g ( κ ). Veering triangulations.
The map O provides yet another connection between hyperbolic geometry, topol-ogy, and singular flat geometry by relating two descriptions of the “veering triangulations” of hyperbolicmapping tori. Using measured train track splitting sequences, Agol produced in [Ago11] a canonical idealtriangulation of the mapping torus of a pseudo-Anosov mapping class with “fully punctured” fiber. Gu´eritaud gave an alternate characterization of this triangulation in terms of Delaunay cellulations of qua-dratic differentials (see [Gu´e16, Section 2]).Our analysis allows us to build a hyperbolic geometric model of Agol’s construction of the veeringtriangulation that is mapped to Gu´eritaud’s singular flat geometric model via O . More explicitly, let( X, λ ) ∈ PT reg g ( κ ) and suppose that O λ ( X ) is minimal and filling. By taking a small parameter t to 0,we obtain family of nested train track neighborhoods N t ( λ ) ⊂ X . Then we can extract a trivalent traintrack splitting sequence τ i (cid:31) τ i +1 (cid:31) ... whose weight spaces terminate to the simplex of measures supportedon λ . The dual triangulations T i can be represented geometrically on X as geodesic segments joining theprong singularities of O λ ( X ).Each triangulation T i is also realized as a triangulation of q = q ( O λ ( X ) , λ ) by saddle connections, anda computation using the shearing coordinates σ λ ( X ) near each switch of τ i shows that no triangle of T i is“monotonic,” i.e., every edge is contained in a singularity free rectangle of q . Equivalently, T i is a layer ofthe veering triangulation as defined by Gu´eritaud. The veering triangulation joining λ to O λ ( X ) is seen tobe an invariant of the dilation/stretch line produced by Proposition 15.12 directed (in forward and backwardtime) by λ at X as well as the Teichm¨uller geodesic line defined by q .The authors believe that this correspondence merits additional exploration. For example, recent work ofMinsky and Taylor [MT17] indicates the strong relationship between combinatorics of veering triangulationsand geometry of hyperbolic 3-manifolds. It would be interesting to relate the geometry of the universalbundle over a stretch line with the combinatorics of the veering triangulation. References [AD20] D. Alessandrini and V. Disarlo,
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