Meromorphic projective structures, grafting and the monodromy map
MMEROMORPHIC PROJECTIVE STRUCTURES,GRAFTING AND THE MONODROMY MAP
SUBHOJOY GUPTA AND MAHAN MJA bstract . A meromorphic projective structure on a punctured Riemann surface X \ P is determined, after fixing a standard projective structure on X , by a mero-morphic quadratic di ff erential with poles of order three or more at each puncturein P . In this article we prove the analogue of Thurston’s grafting theorem forsuch meromorphic projective structures, that involves grafting crowned hyper-bolic surfaces. This also provides a grafting description for projective structureson C that have polynomial Schwarzian derivatives. As an application of ourmain result, we prove the analogue of a result of Hejhal, namely, we show thatthe monodromy map to the decorated character variety (in the sense of Fock-Goncharov) is a local homeomorphism. C ontents
1. Introduction 22. Background 52.1. Projective structures 52.2. Grafting 62.3. Measured laminations 82.4. Thurston parametrization 83. Meromorphic projective structures and crowned hyperbolic surfaces 113.1. Meromorphic projective structures and their markings 113.2. Crowned hyperbolic surfaces 133.3. Measured laminations on crowned hyperbolic surfaces 154. Proof of Theorem 1.1 204.1. Linear di ff erential systems and asymptotics of the solutions 204.2. Exponential map and infinite-grafting 224.3. Grafting and the Schwarzian derivative 234.4. Inverse of the grafting map 275. Projective structures on C a r X i v : . [ m a t h . G T ] A p r SUBHOJOY GUPTA AND MAHAN MJ
1. I ntroduction
Let S be a closed oriented surface of genus g ≥
2. A marked complex projec-tive structure on S is a geometric structure modelled on C P , that is, it comprisesan atlas of charts to C P with transition maps that are restrictions of elements ofPSL ( C ). Passing to the universal cover, this yields a developing map f : (cid:101) S → C P that is ρ -equivariant where ρ : π ( S ) → PSL ( C ) is the holonomy of the projectivestructure.Complex-analytically, a projective structure on S is obtained by fixing a refer-ence projective structure, and solving the Schwarzian equation(1) u (cid:48)(cid:48) + qu = S , where q is the lift of a quadratic di ff erential on S that is holomorphic with re-spect to a choice of complex structure. In particular, the developing map is obtainedas the ratio of a pair of linearly independent solutions, and the holonomy homomor-phism records the monodromy of the solutions around homotopically non-trivialloops on the surface.Conversely, given a projective structure, the Schwarzian derivative of the de-veloping map yields a quadratic di ff erential on (cid:101) S (cid:27) D that is invariant under theFuchsian group Γ determined by the choice of complex structure on S ; this givesback the holomorphic quadratic di ff erential q on the quotient surface. (See Propo-sition 2.1.)The space of marked projective structures P g then forms a bundle over Te-ichm¨uller space T g that is a ffi ne with respect to the vector bundle Q g of quadraticdi ff erentials.A more geometric description of a projective structure was provided by Thurston,who showed that one can obtain projective structures by starting with a hyperbolicsurface (a Fuchsian projective structure), and grafting along a measured geodesiclamination. Indeed, the resulting grafting map (2) Gr : T g × ML → P g is then a homeomorphism. See §2.4 for references, and a sketch of the proof.Recently, Allegretti and Bridgeland [AB] introduced the space of meromorphic projective structures where the quadratic di ff erential (in Equation (1)) is allowed tohave higher order poles. Such meromorphic projective structures can be thought ofas arising from certain degenerations of projective structures in P g ; indeed, mero-morphic quadratic di ff erentials naturally arise in a compactification of the bundle Q g (see for example [BCG + ]). Our aim in this article is to extend Thurston’s geo-metric description to include such structures, and also provide parametrizations ofthe corresponding new spaces that we need to define (see Equation (3)).If there are k ≥ k -tuple n = ( n , n , . . . , n k ) whereeach n i ≥
3, then we denote the corresponding space of marked meromorphicprojective structures by P g ( n ). Here, the marking records a real “twist” parameterat each pole (see §3.1 for details). EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 3
The replacement of Fuchsian structures on closed surfaces (in Thurston’s de-scription) are hyperbolic surfaces with “crown ends”, where each crown end com-prises a collection of bi-infinite geodesics enclosing boundary cusps . For any fixedtuple of integers n as above, let T g ( n ) be the space of marked hyperbolic surfacesof genus g and k crowns, with their respective numbers of boundary cusps given by( n i −
2) for 1 ≤ i ≤ k . Once again, the marking not only provides a labeling of thecrown ends, and the boundary cusps of each, but also “twist” data for each crownend. It can be shown that T g ( n ) (cid:27) R χ where χ = g − + k (cid:80) i = ( n i +
1) (see [Gup]).A measured lamination on a crowned hyperbolic surface could have weightedgeodesic arcs going out towards a boundary cusp, in addition to components thatare compactly supported, and we shall always include the geodesic sides of eachcrown end, each of infinite weight. The space of such measured laminations ML g ( n ) is also homeomorphic to R χ – see Theorem 3.8 in §3.4, which relies on acombinatorial argument that we defer to the Appendix.Our main result is: Theorem 1.1 (Meromorphic grafting theorem) . Fix integers g ≥ , k ≥ such that g + k > and a k-tuple n = ( n , n , . . . , n k ) where each n i ≥ . Any meromor-phic projective structure P ∈ P g ( n ) can be obtained by starting with a crownedhyperbolic surface ˆ X ∈ T g ( n ) , and grafting along a measured geodesic lamination λ ∈ ML g ( n ) .This construction is uniquely determined by the projective surface P. Moreover,the grafting map (3) (cid:99) Gr : T g ( n ) × ML g ( n ) → P g ( n ) is a homeomorphism.Remark. From the definitions of the spaces (see §3, and the preceding discussion)together with Lemma 3.4 and Theorem 3.8 it shall follow that the two sides areindeed homeomorphic to cells of the same dimension.Note that Thurston’s construction of the inverse map to Equation (2) can becarried out for the equivariant projective structure (cid:101) P on the universal cover of thesurface; we give details of the procedure in §2.5, following [KT92], [Tan97], and[KP94]. In particular, this yields some measured lamination on the Poincar´e disk,invariant under some Fuchsian group, grafting along which yields (cid:101) P (see Theorem2.1). Theorem 1.1 precisely determines the geometry of the hyperbolic surface andmeasured laminations we obtain in the quotient, when we start with a meromor-phic projective structure in the space P g ( n ). The proof in §4 shall crucially dependon the asymptotics of the developing map in the neighborhood of the poles, culledfrom classical work in the theory of linear di ff erential systems.In §5 we recall work of Sibuya ([Sib75]) concerning solutions to the Schwarzianequation for polynomial quadratic di ff erentials on the complex plane. The proofof Theorem 1.1 also applies to this setting, and yields the following description SUBHOJOY GUPTA AND MAHAN MJ of the space of the corresponding projective structures on C , which could be ofindependent interest (see §5.1): Theorem 1.2.
For d ≥ let P ( d ) be the space of meromorphic projective structureson C that correspond to polynomial quadratic di ff erentials of degree d. Then thereis a grafting parametrization (4) (cid:99) Gr C : Poly ( d ) × Diag ( d ) → P ( d ) where • Poly ( d ) is the space of hyperbolic ideal polygons with ( d + vertices, and • Diag ( d ) is the space of weighted diagonals on an ideal polygon with ( d + vertices, together with the geodesic sides of the polygon, each with infiniteweight. In §5 we provide more detailed definitions of the spaces appearing in the abovetheorem. It was known from the work of Sibuya and others (see Corollary 4.1)that the developing maps above will have ( d +
2) asymptotic values, where d is thedegree of the polynomial. Moreover, Sibuya had observed that the corresponding crown-tip map Ψ from P ( d ) to the appropriate space of ( d + C P (see Equation (19)) is not injective. As an application of Theorem 1.2, weprovide a characterization of the fibers of Ψ , that is, the set of projective structuresin P ( d ) that determine the same ordered tuple of asymptotic values (called ‘crowntips’) – see Theorem 5.1 for the complete statement.For closed surfaces, the grafting description for projective structures has beenuseful in the study of the monodromy (or holonomy) map(5) hol : P g → χ g from P g to the PSL ( C )-character variety of surface-group representations (see, forexample, [Bab17] and [BG15]).Here, we define a monodromy map Φ (see Equation 23) from the space of mero-morphic projective structures P g ( n ) to the decorated character variety (cid:98) χ g , k ( n ) thatrecords, in addition to the PSL ( C )-representation of the punctured surface, the ad-ditional data of the crown-tips at each pole. See §6.1 for a definition, that followsthat of the moduli stack of framed local systems of Fock-Goncharov in [FG06] (seealso §4 of [AB]).As an application of our main result, Theorem 1.1, we shall prove (see §6.2): Theorem 1.3.
The monodromy map Φ : P g ( n ) → (cid:98) χ g , k ( n ) is a local homeomor-phism. Note that it was shown in [AB] that this monodromy map is holomorphic, withrespect to natural complex structures that these spaces acquire. Theorem 1.3 thusimplies that in fact Φ is a local biholomorphism. This proves the analogue ofHejhal’s result for P g (see [Hej75], [Ear], [Hub81]) and confirms a conjecture of[AB] in our setting, where the order of each pole is greater than two. Note that the EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 5 case when the order of each pole is not greater than two was handled in [Luo93].Theorem 1.3 can be thought of as an extension of the Ehresmann-Thurston prin-ciple, to our non-compact setting. Indeed, our proof in §6.2 shall use this principlein the usual context of compact manifolds, possibly with boundary (see TheoremI.1.7.1 of [CEG06]). To be more specific, we shall apply this principle to projectivestructures on the surface-with-boundary obtained by removing the crowns. For thecrown ends, we shall exploit the fact that there are only finitely many leaves ofthe measured lamination entering them, that can be completed to a triangulationof the crowned surface. This shall allow us to use a theorem of Fock-Goncharov(Theorem 1.1 of [FG06]) which implies, in our setting, that the weights on theseleaves are uniquely determined by the decorated monodromy.In the case of a closed surface, the image of the monodromy map (see Equation5) was characterized in [GKM00]. Their work can be thought of as the solutionof the
Riemann-Hilbert problem for the Schwarzian equation on a closed Riemannsurface. In a sequel we shall address the analogous problem for punctured surfaces,where meromorphic projective structures will play a role.
Acknowledgments.
SG thanks Kingshook Biswas, Shinpei Baba and Dylan Al-legretti for illuminating conversations, and acknowledges the SERB, DST (Grantno. MT / / / sgps / / / / ackground We recall basic facts on projective structures, and of the Thurston parametriza-tion, that will play a crucial role in the rest of the paper. Throughout this section, S g would be a closed oriented surface of genus g ≥
2, whereas S will denote aclosed oriented surface, with possibly finitely many punctures.2.1. Projective structures.
As mentioned in §1, a marked projective structure on S is a maximal atlas of charts to C P such that the transition maps are restrictionsof M¨obius transformations. We had also mentioned that an equivalent definitionis obtained by passing to the universal cover of the surface (cid:101) S , where the localcharts can be patched together to define a globally defined developing map . Thus,a (marked) projective structure on S consists of two pieces of data:(1) a developing map f : (cid:101) S → C P , and SUBHOJOY GUPTA AND MAHAN MJ (2) a holonomy (or monodromy) homomorphism ρ : π ( S ) → PSL ( C ),such that f is ρ -equivariant, with respect to the action of π ( S ) by deck-transformationson the universal cover (cid:101) S , and the action of the M¨obius group ρ ( π ( S )) on C P .Two projective structures ( f , ρ ) and ( g , σ ) are said to be equivalent if the repre-sentations ρ and σ are conjugate by some element A ∈ PSL ( C ), and the pair ofmaps A ◦ f , g are equivariantly homotopic to each other.For a closed surface S g , the space of equivalence classes is then the space ofmarked projective structures, denoted P g .Since a projective structure on S g automatically also defines a complex structureon the underlying surface, there is a forgetful map π : P g → T g , where T g is theTeichm¨uller space of S g .An example of a projective structure is a Fuchsian structure, where the devel-oping map is injective with image a hemisphere of C P (that can be identified with D ) and the holonomy representation ρ is discrete, faithful with image in PSL ( R ).Since any Riemann surface has such a uniformizing Fuchsian structure the fibersof the above projection map π are never empty. In fact, it is well-known that thefibers are parametrized by holomorphic quadratic di ff erentials (see, for example,§2 of [Hub81]): Proposition 2.1.
Let X be a compact Riemann surface of genus g ≥ . The spaceof marked projective structures on X forms an a ffi ne space for the vector spaceQ ( X ) of holomorphic quadratic di ff erentials on X.Proof sketch. The di ff erence of two projective structures C and C is given by aholomorphic quadratic di ff erential q , namely if f : U → C P is the transition mapbetween the two structures, then(6) q = (cid:32) f (cid:48)(cid:48) f (cid:48) (cid:33) (cid:48) − (cid:32) f (cid:48)(cid:48) f (cid:48) (cid:33) where the right hand side is the Schwarzian derivative of f .Conversely, it is not hard to check that if u and u are two linearly independentsolutions of Equation (1), then the ratio f : = u / u has Schwarzian derivative q .Then, given a projective structure C , with developing map f , the new projectivestructure C has a developing map given by f ◦ f . (cid:3) By the Riemann-Roch theorem, we know the dimension of Q ( X ), and we im-mediately obtain: Corollary 2.2.
The space P g of marked projective structures on S g is homeomor-phic to R g − .Remark. In fact, P g is a complex manifold of dimension 6 g −
6; see [Hub81].2.2.
Grafting.
Let ( f , ρ ) be a Fuchsian projective structure P on S . In what fol-lows Γ < PSL ( R ) shall be the Fuchsian group realized as the image of the holo-nomy map ρ . Note that the image of the developing map can be taken to be theupper hemisphere U of C P . The operation of grafting deforms this to a di ff erentprojective structure, as we shall now describe. EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 7
Fix a simple closed curve a ∈ π ( S ), let g : = ρ ( a ). Let α be an arc in U preservedby the infinite-cyclic subgroup of PSL ( R ) generated by g ∈ Γ . Let Γ · α be thecollection of arcs stabilized by conjugates of (cid:104) g (cid:105) under Γ .Then for a positive real parameter t >
0, a t -grafting of P along α is obtainedby rotating one side of each arc in Γ · α relative to the other, by angle equal to t . A lune is the resulting region between α and its rotated copy. (See Figure 1.)Let Ω be the new domain on C P obtained from U by the insertion of this Γ -invariant collection of lunes, one of angle t at each translate γ · α where γ ∈ Γ .Then Ω is the developing image of a new projective structure P on S .Recall that C P can be thought of as the boundary at infinity of hyperbolic 3-space H . For any domain on C P invariant under a M¨obius group, there is aninvariant geometric object in the interior of H , namely the boundary of the geo-desic convex hull (see [Thu80], and our later discussion in §2.4). In particular, theboundary of the convex hull of the upper hemisphere U is the equatorial plane, andthat of the new domain Ω is a “pleated” plane that is bent along the geodesic axis γ α joining the endpoints of α , and its Γ -translates. Here, a “bending” is a relativerotation of one side of the geodesic axis γ α by angle t that corresponds to an ellipticelement E t α in PSL ( C ).The deformation of ρ to a new holonomy homomorphism ρ (cid:48) : π ( S ) → PSL ( C )is best described in terms of a “bending cocycle”. We sketch the construction below– for details, see §5.3 of [Dum09], or II.3.5 of [EM87].To start, we “straighten” the arcs Γ · α to their geodesic representatives, namelyconsider the collection ˜ γ of the geodesic axes of the hyperbolic element g and itsconjugates. The bending cocycle is then a map(7) β : H \ ˜ γ × H \ ˜ γ → PSL ( C )where β ( x , y ) defined as follows: consider the oriented geodesic arc σ from x to y , and let g , g , . . . g n be the geodesics from ˜ γ that intersect σ , in that order, eachoriented so that y lies to its right. Then β ( x , y ) : = E ◦ E ◦ · · · E n where E i is theelliptic element that fixes the axis g i and rotates clockwise by an angle equal to t .Note that if σ ∩ ˜ γ = ∅ , then we set β ( x , y ) : = Id . If we fix a basepoint x ∈ H \ ˜ γ ,then the new representation ρ (cid:48) is defined by:(8) ρ (cid:48) ( c ) = β ( x , c · x ) ◦ ρ ( c )for any c ∈ π ( S ). Indeed, the domain Ω is invariant under the new M¨obius group Γ (cid:48) = ρ (cid:48) ( π ( S )); the element ρ (cid:48) ( γ ) (resp. its conjugates), acts by translations alongthe lune inserted at α (resp. its Γ -translates) and the new projective surface Ω / Γ (cid:48) is obtained by grafting a projective annulus at γ on the original hyperbolic surface U / Γ . Straight lunes.
In the grafting construction the resulting projective structures areisotopic if the grafting arc α is changed by an isotopy; in particular, they remainunchanged in P g . In particular, any lune can be isotoped to a straight lune whichis bounded by circular arcs in C P , for example one obtained by grafting along ageodesic line α . SUBHOJOY GUPTA AND MAHAN MJ F igure
1. Grafting in a lune of angle t at an arc α .2.3. Measured laminations.
Given a hyperbolic structure on S , a geodesic lam-ination is a closed subset that is foliated by disjoint, complete geodesics. A col-lection of disjoint simple closed geodesics is certainly an example, but a geodesiclamination could also have dense leaves, that is infinite geodesics which accumu-late on to the entire lamination. (A lamination, all whose leaves are dense, is alsocalled minimal .) A geodesic lamination is measured if it is equipped with a trans-verse measure, that is, a positive measure on arcs transverse to the leaves, that isinvariant under transverse homotopy.Such a measured lamination can in fact be recovered from transverse measuresof finitely many closed curves (which are also called their “intersection numbers”).A measured lamination is thus a topological object that can be defined independentof a hyperbolic metric, as long as the surface has a marking. The space ML g ofsuch measured laminations on S g is homeomorphic to R g − (see [FLP12]), wherethe topology is induced by the transverse measures.Note that if the hyperbolic structure is given by a Fuchsian group Γ , a geodesiclamination determines a closed set F ⊂ G : = ∂ U × ∂ U \ ∆ , where U is the upperhemisphere of C P , identified with H , and ∆ is the diagonal, and a transversemeasure is a measure supported on this subset. Any such measured laminationis then a limit of a sequence of weighted multicurves, which correspond to finitesums of Dirac measures converging in the weak- ∗ topology.We can then define grafting of the Fuchsian structure along a measured lamina-tion: a new domain Ω ⊂ C P is obtained as a limit of the construction described in§2.2, where at each stage we insert lunes corresponding to the weighted geodesicsin the finite approximation of the lamination, mentioned above. A similar limit-ing construction defines the bending cocycle Equation (7) that determines the newholonomy representation ρ (cid:48) exactly as in Equation (8).Together, these define a new projective structure.2.4. Thurston parametrization.
In the previous subsections, we have discussedhow a Fuchsian structure X can be grafted along a measured geodesic lamination λ to define a new complex projective surface. As mentioned in the Introduction,Thurston showed that this provides a unique construction of any projective struc-ture on a closed surface S g (see Equation (2)). EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 9
In this section we discuss a slightly more general statement, that can be culledfrom the work of Kulkarni-Pinkall ( c.f.
Theorem 10.6 of [KP94]) and Kamishima-Tan ([KT92]). For a recent exposition, see [Bab]. As usual, a Riemann surfaceequipped with a complex projective structure will be called a projective surface.
Definition 2.3.
A maximal disk on a projective surface ˜ X is an embedded disk Usuch that the restriction of f to U is a di ff eomorphism onto a round disk f ( U ) in C P ; moreover U is not strictly contained in another disk with the same property. Theorem 2.1.
Let ˜ X be a simply-connected projective surface that is not projec-tively isomorphic to C , or the universal cover of C P \ { , ∞} . Then there exists aunique measured lamination L on the Poincar´e disk D such that ˜ X is obtained bygrafting D along L.The map associating L to ˜ X is equivariant, i.e. if ˜ X is the universal cover ofa projective surface S , and the developing map ˜ X → C P is π ( S ) − equivariantvia a representation ρ C : π ( S ) → PSL ( C ) , then L is invariant under a naturallyassociated representation ρ R : π ( S ) → PSL ( R ) . Moreover, the image Γ of ρ R isdiscrete, and the quotient D / Γ is homeomorphic to S .Finally, the map ˜ X → L is continuous.Sketch of the proof.
We follow the exposition in [KT92] with some di ff erences interminology arising from the fact that their work concerns conformally flat struc-tures on manifolds of possibly higher dimension, of which projective structures onsurfaces is a special case.The goal is to construct a pleated surface [Thu80, Chapter 8] canonically asso-ciated to ˜ X . Let f : ˜ X → C P be the developing map of the projective structure.Since ˜ X is not projectively equivalent to the standard structure on C , it follows thateach point of ˜ X is contained in a proper maximal disk (see Proposition 1.1.3 of[KT92]).A maximal disk U acquires a natural Poincar´e metric; define the set U ∞ to bethe subset of ∂ ∞ U that does not lie in ˜ X , and let C ( U ∞ ) denote its projective convexhull in U . Maximality guarantees that there are at least two points in U ∞ , so thatthe convex hull is non-empty. Moreover, each point of ˜ X lies in the projectiveconvex hull of a unique maximal disk [KT92, Theorem 1.2.7].Note that the image of a maximal disk U under the developing map is a rounddisk f ( U ) on C P = ∂ H . The disk f ( U ) admits a canonical projection Φ U to atotally geodesic copy of H ⊂ H . Thus, Φ U ( U ) is the convex hull of ∂ ∞ U in H .Note that Φ U ( C ( U ∞ )) is an ideal totally geodesic hyperbolic polygon containedin Φ U ( U ). We have assume in our hypotheses in the Theorem that the projectivesurface ˜ X is not the universal cover of C P \ { , ∞} ; this guarantees that there existsat least one such polygon Φ U ( C ( U ∞ )) that is not degenerate, i.e. has at least threesides. The pleated surface below is constructed from the collection of Φ U ( C ( U ∞ ))’sas follows.Define a map Ψ : ˜ X → H by Ψ ( x ) = Φ U ( f ( x )) if x ∈ C ( U ∞ ). It is easy toverify that Ψ is continuous, and the image of Ψ is a pleated plane P , in the senseof Thurston [Thu80, Chapter 8]. Note that Ψ may not even be locally injective;indeed, a “straight lune” in ˜ X (see §2.2.) arises when a family of maximal disks which have a pair of common ideal boundary points collapses to a single geodesicline γ , giving a bi-infinite geodesic in the pleating locus [Thu80, Chapter 8]. If γ is isolated in L , then the ideal polygons or plaques on either side of γ lie on apair of totally geodesic half-planes that can be thought of as being obtained from a(larger) totally geodesic polygon in H after bending along γ by a positive angle. Itis possible that γ is not an isolated geodesic in the pleating locus L , in which casethe angle of bending is defined as a transverse measure on the pleating locus. Thetransverse measure is called the bending measure and is denoted as µ .Straightening the pleated plane P determines a hyperbolic plane H (or thePoincar´e disk D ). The pleating locus gives a geodesic lamination L on D . Thelamination L equipped with the transverse measure µ gives a measured lamination L . This proves the first statement of the Theorem.We now observe equivariance. It su ffi ces to show that Ψ : ˜ X → H taking ˜ X to apleated surface is equivariant. To see this, note that for U a maximal disk in (cid:101) X , sois g . U for any g ∈ π ( S ). Hence Ψ g . U ( C ( g . U ∞ )) = ρ C ( g )( Ψ U ( C ( U ∞ ))) , where the ρ C ( g ) − action on the RHS is via hyperbolic isometries. It follows thatthe totally geodesic hyperbolic polygons in P are equivariant with respect to theaction of ρ C ( π ( S )). Hence the pleating locus L , realized as a family of geodesicsin H is also equivariant with respect to the action of ρ C ( π ( S )). Next, note that thetransverse measure on L is given by the bending measure µ . The latter determinesand is determined by the straight lunes that occur in ˜ X . Since the developing map f is equivariant under ρ C , the bending measure µ is invariant under the induced π ( S ) − action on P . Hence the measured lamination L is invariant under the in-duced π ( S ) − action on H , where the latter is obtained from P by straightening.Consequently, we obtain a representation ρ R : π ( S ) → PSL ( R ), such that L is Γ -invariant, where Γ is the image of ρ R . Moreover, since the lunes that get col-lapsed by the map Ψ are contractible, one can show that Ψ induces a homotopyequivalence between the quotient spaces S and D / Γ . It is a standard topologicalfact that in this case this implies S and D / Γ are homeomorphic; in particular therepresentation ρ R is discrete and faithful. This proves the second statement of theTheorem.Lastly, we observe the continuity of the map ˜ X → L . As in the previous para-graph, it su ffi ces to note the continuity of the map associating the projective surface˜ X to the pleated plane P . This follows from the fact that the pleated plane P de-pends continuously on the family of totally geodesic polygons Ψ U ( C ( U ∞ )), whilethe latter depends continuously on the family of projective polygons C ( U ∞ )). Thisproves the third statement of the Theorem. (cid:3) Remark.
We refer the reader to [Thu80, Chapter 8] for more details on pleatedsurfaces and to [Thu80, Chapter 9] for realizability of measured laminations viapleated surfaces. We also note that the map ˜ X → L associating a measured lami-nation L on H to a projective surface ˜ X is exactly the inverse of the grafting mapthat obtains the projective surface ˜ X from H by grafting according to the measured EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 11 lamination L . Together with the equivariance statement of Theorem 2.1, this provesThurston’s theorem, namely, the map Gr in Equation (2) is a homeomorphism.We shall also use the following terminology: Definition 2.4.
Given a projective structure P, a grafting lamination L for P on ahyperbolic surface X is a measured lamination such that grafting X along L yieldsP.
3. M eromorphic projective structures and crowned hyperbolic surfaces
In this section we shall provide a more detailed exposition of some of the objectsand their spaces already introduced in §1, in particular, those appearing in thestatement of Theorem 1.1 (see the map defined by Equation 3).3.1.
Meromorphic projective structures and their markings.
For a Riemannsurface with punctures, [AB] considered projective structures obtained by solu-tions of Equation (1) when q is holomorphic away from the punctures, and haspoles of finite order, greater than two, at the punctures. Poles of order one alreadyappear in classical Teichm¨uller theory: for Fuchsian structures they arise when theuniformizing structure has a finite-volume cusp at the puncture. Examples of pro-jective structures corresponding to meromorphic quadratic di ff erentials with polesof order two include branched structures; see [Luo93].Recall from the proof of Theorem 2.1 that the “di ff erence” of two projectivestructures, given by the Schwarzian derivative of the transition maps between chartsin the two structures, is a holomorphic quadratic di ff erential. Following the defini-tion in §3.3 of [AB], we say: Definition 3.1.
A meromorphic projective structure is a projective structure on apunctured Riemann surface X \ P such that the di ff erence (in the sense describedabove) with the restriction of a standard (holomorphic) projective structure on X isgiven by a holomorphic quadratic di ff erential on X \ P that extends to a meromor-phic quadratic di ff erential q with poles of order greater than two at each p ∈ P.If, in a choice of a coordinate disk around a pole, q has the expression (9) q = (cid:18) a n z n + a n − z n − + · · · + a z + h ( z ) (cid:19) dz where h ( z ) is a holomorphic function, then the polar part of the di ff erential is de-fined to be q − h ( z ) dz .Remarks.
1. We shall assume the standard projective structure on X is the uni-formizing one, which in case the Euler characteristic χ ( X ) < χ ( X ) = C , else is the projective surface C P itself.2. Unlike in [AB], our definition above disallows poles of order two (or “regular”singularities); this shall make our defining spaces of structures simpler, as our pro-jective structures shall automatically have no “apparent singularities”. Recall that the horizontal directions of a quadratic di ff erential q at a point arethe tangent directions in which the di ff erential takes real and positive values. In aneighborhood of a pole of order n ≥
3, as in Equation (9), the quadratic di ff eren-tial q has ( n −
2) equispaced directions at the pole that horizontal trajectories areasymptotic to (see Theorem 7.4 of [Str84]).
Example.
For the quadratic di ff erential q = z − n dz where n ≥
3, these horizontaldirections at the pole are at the points { exp(2 π j / ( n − | ≤ j < n − } on the unitcircle on the tangent plane obtained by a real blow-up at the pole.We also define: Definition 3.2.
A marking of a meromorphic projective structure on X \ P is achoice of a homeomorphism (up to homotopy) with a surface S with boundary C,where each component of C has (a positive number of) labelled marked points onit. The homeomorphism takes the horizontal directions at each pole to the markedpoints on a corresponding boundary component. Here we consider two homeo-morphisms the same if they are homotopic relative to the boundary (that is, by ahomotopy that keeps the boundary fixed pointwise).As mentioned in §1, P g ( n ) shall denote the space of marked meromorphic pro-jective structures with k poles of orders given by the tuple n = ( n , n , . . . , n k ) ,where each n i ≥ . We shall assume g + k > , that is, the underlying surface hasnegative Euler characteristic.Remark. Note that under the above notion of equivalence of two marked sur-faces, two markings that di ff er by a Dehn twist around the boundary componentare distinct.It is useful to also consider an “appended” Teichm¨uller space of the underlyingmarked Riemann surfaces : Definition 3.3.
Let S be an oriented surface of genus g and k punctures, havingnegative Euler characteristic, and let n be a k-tuple of integers as above. Then thespace ˆ T g , k shall denote the space of marked complex structures on S , together withan additional real parameter r i at the i-th puncture, for ≤ i ≤ k. Note that amarking includes a labeling of the punctures, and is considered up to a homotopyas in Definition 3.2. The real parameter r i serves to record: (a) A set of ( n i − equispaced points on a circle obtained as a real blowup ofthe i-th puncture, where the first point is at exp ( i π r i ) , and (b) The integer parameter (cid:98) r i (cid:99) that denotes the number of Dehn twists about aboundary circle obtained from a real blowup of the i-th puncture.Remark. Recall that for the Teichm¨uller space of a punctured surface, the punc-ture is thought of as a boundary component of length zero. The “appended” Te-ichm¨uller space defined above can be thought of as adjoining an extra Fenchel-Nielsen twist parameter about this boundary curve.
EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 13
Note that there is a projection π : P g ( n ) → ˆ T g , k that maps a meromorphicprojective structure to the punctured Riemann surface underlying it, which at eachpuncture has • a set of equispaced points on the circle obtained as its real blowup, givenby the horizontal directions of the meromorphic quadratic di ff erential, and • a marking that remembers the twist parameter; in particular, the number ofDehn-twists around the corresponding boundary component.We then have: Lemma 3.4.
The space P g ( n ) is homeomorphic to R χ where χ = g − + k (cid:80) i = ( n i + .Proof. Clearly ˆ T g , k (cid:27) R g − + k since the usual Teichm¨uller space of a genus- g surface with k punctures is homeomorphic to R g − + k ( c.f. the remark followingDefinition 3.3). Fix a marked Riemann surface X ∈ ˆ T g , k , and a coordinate chart U i around the i -th puncture (where 1 ≤ i ≤ k ). Then the fiber π − ( X ) of meromorphicprojective structures that project to X , consists of meromorphic quadratic di ff eren-tials that have a pole of order n i at the i -th puncture, with horizontal directions asprescribed by the corresponding real parameter r i on X . The horizontal directionsat a pole are determined by the argument Arg( a n ), where n : = n i , and a n is theleading order coe ffi cient of the polar part (Equation (9)) as expressed in the chart U i .This leaves the positive real number | a n | , together with the remaining coe ffi cients a , a , . . . a n − ∈ C of the polar part, a total of (2 n −
1) parameters. The holomorphicquadratic di ff erentials on a closed surface of genus g , by Riemann-Roch, is a com-plex vector space of dimension 3 g −
3. Hence the fiber π − ( X ) is homeomorphic toa cell of (real) dimension 6 g − + k (cid:80) i = (2 n i − P g ( n ) is homeomorphic to a cell of dimension (cid:32) g − + k (cid:80) i = (2 n i − (cid:33) + (cid:32) g − + k (cid:80) i = (cid:33) = χ . (cid:3) Remark.
In fact, P g ( n ) can be shown to be a complex manifold of dimension χ (see Proposition 8.2 of [AB]).3.2. Crowned hyperbolic surfaces. A hyperbolic crown is an annulus equippedwith a hyperbolic metric such that one of the boundary components is a closed ge-odesic (the crown boundary ), and the other comprises a finite chain of bi-infinitegeodesics, each adjacent pair of which encloses a boundary cusp . The bi-infinitegeodesics shall be called the geodesic sides of the crown. A marking on the hy-perbolic crown is a labeling of the boundary cusps together with a choice of ahomotopy class of an arc from the crown boundary to a boundary cusp. The lat-ter is an integer parameter that records the number of twists around the boundarycomponent. F igure
2. A hyperbolic crown with basepoint p on the boundary.A crowned hyperbolic surface ˆ S is obtained by gluing a hyperbolic crown to ahyperbolic surface with geodesic boundary γ , such that the boundary componentof the crown is identified with γ . The hyperbolic crown is then a subsurface of ˆ S that we refer to as its crown end .Topologically, a crowned hyperbolic surface is a surface with boundary, togetherwith a collection of marked points on the boundary. A marking on a crownedhyperbolic surface is a choice of homotopy class of an identification with such asurface, where the homotopy fixes the boundary pointwise. The latter conditionamounts to fixing some boundary data (see below) that are additional parametersfor specifying such a surface.The “wild” Teichm¨uller space T g ( n ) introduced in §1 (see also [Gup]) is thespace of such marked crowned hyperbolic surfaces corresponding to the tuple n ;each surface in this space has k crown ends, each having ( n i −
2) boundary cusps.See also [Pen04] for a broader context.
Boundary twist data.
A crown end of a crowned hyperbolic surface has an addi-tional real parameter associated with it that we now describe. Let γ be the bound-ary of the crown with length l . Let α be a fixed choice of a directed arc betweenboundary cusps on the crowned hyperbolic surface ˆ S , such that α is non-trivial inhomotopy (relative to its end-points), and not peripheral in the sense that it can-not be homotoped into the crown end. (In particular, α intersects γ twice.) First,note that the marking of the crowned surface determines an integer twist data thatrecords the number t ∈ Z of twists around γ that α makes. We shall also assumethat all the twisting round γ that α makes, takes place inside the crown.Next, a hyperbolic crown with m boundary cusps determines a basepoint on theboundary γ : namely consider the geodesic side of the crown between the cuspslabelled m and 1, and consider the foot p of the perpendicular that realizes the dis-tance of that geodesic side from γ . We shall refer to this as the canonical basepoint for the crown.The real twist parameter of the crown end is then measured relative to thiscanonical basepoint: let d be the distance along γ from p to the point where α intersects γ first (in the orientation of γ acquired from the crown). Recall α com-pletes t complete twists around γ ; then the twist parameter associated with thecrown end is defined to be τ = t · l + d . EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 15 F igure
3. A measured lamination on a crowned hyperbolic surface.Alternatively, instead of the choice of a directed arc α , the twist parameter canbe thought as comprising an integer twist data t , together with a choice of a base-point p on γ at a distance d ∈ [0 , l ) from p on the (oriented) boundary of thecrown. As before, this can be recorded as the real number τ = t · l + d .For the proof of the following fact, already mentioned in the introduction, seeLemma 2.16 of [Gup]: Proposition 3.5.
The space T g ( n ) (cid:27) R χ where χ = g − + k (cid:80) i = ( n i + . The pa-rameters include g − + k real numbers that specify the hyperbolic surface withgeodesic boundary obtained by removing the crown ends, together with parame-ters determining each crown, including the boundary twist parameters as definedabove. Measured laminations on crowned hyperbolic surfaces.
As described in§1, a measured lamination on a crowned surface could have non-compact support,with finitely many leaves that exit through the boundary cusps of the crown end. Inthis paper, such a lamination will also include the geodesic sides of the crown end,each of which is assigned weight ∞ . (See Figure 3.)Suppose that the crowned surface is in T g ( n ). Thus it has k crown ends, wherethe number of boundary cusps of crown ends is given by the k -tuple n . Then thespace of such measured laminations is ML g ( n ). Just as for ML g in §2.3, thisspace can be thought of as parametrizing topological objects.Note that the space ML g of measured laminations on a closed surface of genus g ≥ ML g acquires its topology via this parametrization. One way of parametrizing the space ML g ( n ) (and equipping it with a topology) would be to use weighted train-tracks with stops , as introduced in §1.8 of [PH92]. (Note that [PH92] considers a singlestop on each boundary component, but this can easily be extended to the case ofmultiple stops.) In what follows, we provide an alternate parametrization, by dividing a mea-sured lamination λ on a crowned hyperbolic surface into its intersections with thecrown-ends, and with the surface with boundary that is the complement of thecrowns. (We shall always assume that the twisting of leaves entering a crown endaround the corresponding crown boundary takes place in the crown.)This approach takes advantage of the fact that the parametrization of measuredlaminations on a surface with boundary is well-known (see, for example, Proposi-tion 3.9 of [ALPS16]). In what follows we shall first prove a similar parametriza-tion of measured laminations on a hyperbolic crown (Proposition 3.7), and param-etrize ML g ( n ) by combining these two parametrizations (Proposition 3.8). Part ofthe proof is to show that when we attach crown ends to a surface-with-boundary,then measured laminations on the pieces can be matched up to produce a measuredlamination on the crowned hyperbolic surface – the details of this are deferred tothe Appendix.We shall implicitly assume that ML g ( n ) acquires a topology via this parametriza-tion.We start with the following observation: Lemma 3.6.
The intersection of the measured lamination λ ∈ ML g ( n ) with acrown end C is a collection of (isolated) weighted arcs, each of infinite length,that either run from a boundary cusp to the crown boundary, or between two non-adjacent boundary cusps.Proof. In the universal cover, the boundary cusp points corresponding to a lift ˆ P ofthe crown C have precisely two accumulation points: the endpoints of the geodesicline that is the lift of the crown boundary γ . No leaf of λ can be asymptotic tothese two points. This is because such a leaf would have to spiral infinitely manytimes around the closed curve γ , and therefore could not have positive transversemeasure. Hence the restriction of a lift of the lamination λ to ˆ P is a collection ofgeodesic lines, each having (one or both) endpoints at a set of isolated points onthe ideal boundary.Recall that γ = ∂ C is the closed geodesic that is the crown boundary. We notefinally that there can be at most finitely many geodesics in λ that intersect C . Tosee this, observe that the intersection λ ∩ γ is a closed subset of γ . For each com-plementary interval I i in γ \ ( λ ∩ γ ), there is an polygon B i in C bounded by I i onone side, two geodesic leaves of λ that exit the boundary cusps, and possibly somegeodesic sides of the crown. If λ ∩ γ is infinite, there are infinitely many such dis-tinct (and necessarily disjoint) B i ’s forcing the total area of the crowned hyperbolicsurface to be infinite–a contradiction. (cid:3) In what follows, we define a measured lamination on a hyperbolic crown tobe a collection of finite weighted geodesics as above. (that we refer to as arcs ).Note that the closed geodesic that is the crown boundary, could also be part ofthe lamination. We also require that there is at most one arc from a boundarycusp to the crown boundary; thus, arcs obtained by “splitting” (see Appendix) will
EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 17 F igure
4. A collection of weighted diagonals on a hyperboliccrown (left) determines a dual metric graph (right). This figure de-picts a case when the resulting graph is of a generic type: varyingthe weights parametrizes a cell of top dimension in the resultingcell-complex.be considered the same arc (with a total weight equal to the sum of individualweights).
Proposition 3.7.
For a hyperbolic crown C with ( n − boundary cusps, the spaceof measured geodesic laminations on C is parametrized by R n − . The parametersinclude the transverse measure l of the boundary γ of the crown, and the boundarytwist parameter τ ∈ R , which together parametrize R .Proof. Recall that the bi-infinite geodesic sides of a crown end are also part ofthis lamination on the crown, each equipped with infinite weight. A collection ofdisjoint weighted geodesics G on C can then be represented by a dual metric graph G , that we define as follows:The vertices of G are one for each complementary region of G , and each edgeof G is either(a) transverse to an arc in G and having length equal to its weight, and connectingthe vertices in the complementary regions on either side, or(b) has infinite length, from a vertex to a geodesic side of the crown, in case thecomplementary region is bounded by such a side.Note that there are ( n −
2) edges of infinite length corresponding to the n geodesicsides of the crown, and if the crown boundary has positive measure, there is aunique cycle of edges corresponding to the boundary, that we shall denote by c .See Figure 4.Moreover, the requirement of at most one arc from a boundary cusp to crownboundary, ensures that each vertex of G is at least trivalent.For any fixed positive transverse measure l of the boundary, the space of suchmetric graphs is homeomorphic to R n − (see Proposition A.4 of [DGT]). The ideais that for a fixed topological-type of the graph, varying the lengths of the edgesparametrizes a cell, and the di ff erent cells fit together to give a cell-complex that ishomeomorphic to a ball. In fact, just as in §3.4 of [ALPS16], one can interpret a non-positive transversemeasure l the following way: in such a case, there will be no geodesic arcs incidenton the crown boundary, but instead the crown boundary itself, which is a closedgeodesic, will be part of the measured geodesic lamination, and will be given aweight | l | .The dual metric graph in such a case will be a tree, with ( n −
2) edges of infinitelength as before, but now with an additional finite-length edge corresponding tothe closed boundary geodesic, instead of a cycle. Once again, for any fixed l ≤ R n − ( c.f. Theorem 16 of[GW18]).It remains to verify that the total space (as we vary the transverse measure in R ) is also homeomorphic to a ball having two additional dimensions; one of theseparameters is the transverse measure itself, that we denote by l , and the other is theboundary twist parameter τ .However, note that the twist parameter for the crown C will only a ff ect the mea-sured lamination on it only in the case that l >
0, for only then will there begeodesic leaves incident on the crown boundary. If τ = t · l + d , we shall call t the integer part of the twist parameter. This integer records the number of Dehn twistssuch leaves make around the crown boundary. The real part of the twist parameter τ is the real number d ∈ [0 , l ), and it determines the position of the basepoint on thecycle c of the dual metric graph, relative to the canonical basepoint on the crownboundary (see §3.2).In the remainder of this proof we shall describe how the parameters ( l , τ ) stilldetermine copy of R ( c.f. the proof of Proposition 5.5 in [DGT]). Together withthe previous discussion, this would imply that the space of measured geodesiclaminations on C is R n − × R (cid:27) R n − .First, consider the upper half-plane H ⊂ R where the height is given by thetransverse measure l , and the twist parameter τ determines the horizontal coordi-nate. At a fixed height l >
0, when the cycle c has total length l , the range of values0 ≤ τ < l will correspond to 0 integer twist, the range l ≤ τ < l will correspondto the integer part t =
1, and so on. This partitions H into wedge-shaped regions V j = { j · l ≤ τ ≤ ( j + · l } for j ∈ Z , that represent the di ff erent integer parts of thetwist parameter.Next, we include the points where the transverse measure l ≤ R of the upper half-plane H with an identification of the positive and negative half-rays; that is, both thepoints ( − l ,
0) and l ,
0) represents the same point, where the transverse measure is l ≤ V j accumulate onto these half-rays as j → ±∞ . This fitsinto the tiling of the interior of the upper half-plane described earlier:If one fixes the twist parameter τ , and then decreases the transverse measure ofthe boundary (that is, go vertically down to the boundary in H ), then the leavesof the lamination intersecting the crown boundary will have an increasing number EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 19 of twists around the boundary, but have proportionately smaller weights, and willlimit (as a measured lamination) to the boundary geodesic with a weight.It is easy to now verify that the closed upper half-plane H with the identifica-tion on the boundary half-rays as described above, is homeomorphic to R , as weclaimed. (cid:3) Proposition 3.8.
The space of measured laminations ML g ( n ) is homeomorphic to R χ where χ = g − + k (cid:80) i = ( n i + .Proof. It is well-known that the space of measured laminations on a surface S ofgenus g and k boundary components is a cell of dimension 6 g − + k (see, forexample, Proposition 11 of [GW18]). The parameters are the transverse measure,and a twist parameter, for each interior pants curve for a pant decomposition of thethe surface-with-boundary, together with the transverse measures of the k boundarycomponents. The case that the transverse measure of such a component is non-positive, say l ≤
0, can be interpreted as in the proof of Proposition 3.7. Namely,in that case the boundary itself is a leaf of the lamination, with weight | l | .To the i -th boundary component, where 1 ≤ i ≤ k , we can now attach a crownend with ( n i −
2) boundary cusps. By Proposition 3.7, a measured lamination onsuch a crown end is determined by ( n i −
1) parameters, and in the Appendix wedescribe how such a lamination is matched with the measured lamination on thesurface-with-boundary, to obtain a measured lamination on the crowned hyper-bolic surface ˆ S . However, for this gluing, the transverse measures on the commonboundary induced by the two laminations need to match. So, the total number ofreal parameters is χ , as desired.From Lemma 3.6, it is not hard to see that any measured lamination on thecrowned hyperbolic surface arises as a result of such a construction, completingthe proof. (cid:3) Remark.
Alternatively, such measured laminations can be shown to be equiv-alent to measured foliations with pole singularities, as defined in [GW18] – see,for example, §11.8-9 of [Kap01] for a proof of this equivalence in the case ofclosed surfaces. The latter space of measured foliations with pole singularities isparametrized in Proposition 10 of [GW18], and shown to be homeomorphic to R χ . Grafting.
The operation of grafting a crowned hyperbolic surface ˆ S along a mea-sured lamination λ on it makes sense. As described in §2.2 and §2.3, we first passto the universal cover and perform the relative bending for each of the lifts of theleaves of λ or its finite approximations, and then take a limit. The infinite graft-ing for each geodesic side of the crown end (which have infinite weight) can bethought of as grafting in an infinite concatenation of lunes: conformally this yieldsa half-plane. This gives us a new projective structure on the punctured Riemannsurface; we shall see later (see §4.3) that this is in fact a meromorphic projectivestructure as in Definition 3.1.
4. P roof of T heorem (cid:99) Gr : T g ( n ) × ML g ( n ) → P g ( n ) . Recall that n = ( n , n , . . . , n k ) records the orders of the poles (each greater thantwo) at the k labelled punctures, and for each 1 ≤ i ≤ k , we denote m i = ( n i −
2) tobe the number of boundary cusps of the corresponding crown end for a surface in T g ( n ).In this section, we complete the proof that the map (cid:99) Gr is a homeomorphism.For ease of notation, we shall assume throughout that k =
1, that is, the un-derlying Riemann surface has a single puncture, or equivalently, the underlyinghyperbolic surface has a single crown end. Thus there is an integer n ≥ n −
2) isthe number of boundary cusps of the crowned hyperbolic surface. The proofs inthe section only involve a local analysis around the pole, and are exactly the samefor multiple punctures / crown ends.4.1. Linear di ff erential systems and asymptotics of the solutions. We beginwith some key results from classical work on linear di ff erential equations on thecomplex plane.Recall that the developing map for a meromorphic projective structure is theratio of two linearly independent solutions of the Schwarzian equation (1), wherethe quadratic di ff erential q is of the form Equation (9) on a coordinate disk U around the pole.Using a change of coordinate z (cid:55)→ w : = c / z for a suitable c , one can considerthe (transformed) quadratic di ff erential to be of the form(11) q ( w ) = − · ( w d + α d − w d − + · · · α w + α + α − w − + · · · )where d = n −
4, so that it has a pole of order n at ∞ . Our choice of the factor ( −
2) ismerely in order to match with the classical literature (see, for example, Equations1.1 and 1.2 of [HS66]).The Schwarzian equation restricted to U is then the equation(12) u (cid:48)(cid:48) ( w ) + q ( w ) u ( w ) = ∞ in C .Taking X ( w ) = (cid:0) uu (cid:48) (cid:1) , the equation (12) can be written as the linear system ofrank two:(13) X (cid:48) ( w ) = A ( w ) X ( w ) where A ( w ) = (cid:32) − q ( w ) 0 (cid:33) . EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 21
In what follows we shall assume that d is even, that is, the pole is of even order.The case of an odd order pole can be reduced to this by taking a double coverbranched at the pole – see §5.3 of [AB] for details.Note that the gauge transformation X = (cid:32) w d / (cid:33) Z converts the system to(14) Z (cid:48) ( w ) = w d / ∞ (cid:88) k = B k w − k Z ( w )with the advantage that the leading order term B = (cid:32) − (cid:33) is diagonalizable.An analysis of this linear system can then be carried out as in the work of Hsieh-Sibuya ([HS66]); though they consider the case where q ( w ) is a polynomial, theiranalysis extends to our setting where q ( w ) is holomorphic in a heighborhood of ∞ with a finite order pole at ∞ .In fact, this more general setting is handled for linear systems of arbitrary rankby work of Balser-Jurkat-Lutz in [BJL79] (see also Chapter XIII of [HS99]. Thefollowing theorem can be culled from Theorem A of [BJL79]; see also Theorem6.1 of [Sib75], and the exposition in [Bak77] and §5.3 of [AB]. Theorem 4.1.
There are ( d + sectors S k in C bounded by the rays at angles π d + · (2 k ± where k ∈ { , , , . . . , d + } , and ( d + uniquely determined fundamentalsolutions Y k ( w ) to Equation (12) that (a) are holomorphic in a neighborhood U of ∞ , (b) have an asymptotic expansion (15) Y k ( w ) = cw ρ (cid:16) + O ( w − / ) (cid:17) e ( − k + E ( w ) in S k − ∪ S k ∪ S k + , where c and ρ are some constants (that may dependon k), and E ( w ) is a polynomial of degree ( d / + . (c) are related by (16) Y k ( w ) = Y ( ω k w ) where ω = e π id + .Remark. The sectors are often called
Stokes sectors and the rays between thesectors are called
Stokes rays ; we shall also refer to the rays at angles π d + · k for k ∈ { , , , . . . , d + } to be the anti-Stokes rays , that bound the anti-Stokes sectors that we denote by (cid:98) S k . (The latter would later feature in aspects of the correspond-ing projective structures: in particular, the developing map would be asymptoticto the crown-tips along the anti-Stokes rays, and its restriction to the anti-Stokessectors would correspond to infinite-grafting on the geodesic sides of the crown.) Note that one consequence of the asymptotics of Equation (15) is that for each k ∈ { , , , . . . , d + } , the fundamental solution Y k ( w ) → w → ∞ in S k ,whereas Y k ( w ) → ∞ as w → ∞ in S k ± . The solution Y k is said to be subdominant in the Stokes sector S k .In particular, this shows that Y ( w ) and Y ( w ) are linearly independent solutionsof Equation (12), so we can consider the developing map for the correspondingprojective structure it defines to be the ratio(17) f ( w ) = Y ( w ) Y ( w ) . In what follows, recall that an asymptotic value of a meromorphic function asabove, defined in a neighborhood of ∞ ∈ C , is the limiting value in C P (if it exists)of the function along a curve diverging to ∞ .As an immediate corollary of Theorem 4.1 and Equation (17) we then obtain: Corollary 4.1.
The developing map f : U → C P as defined above is holomorphicin a neighborhood of ∞ , and has ( d + asymptotic values c , c , . . . , c d + ∈ C P ,one in each sector S k , for ≤ k ≤ d + .Moreover, in each anti-Stokes sector, there is the asymptotic expansion: (18) f ( ξ ) ∼ e − ξ in the coordinate ξ = w ( d + / .(Here, Equation (18) means that ξ m (cid:16) f ( ξ ) − e − ξ (cid:17) → as ξ → ∞ , for eachm ≥ .)Proof. The asymptotics in Equation (18) is an immediate consequence of Equa-tions (15), (16) and (17); note that the growth in a sector is dominated by exp( E ( w ))and the leading order term of E ( w ) is w ( d + / , so the change of coordinate is w (cid:55)→ ξ = w ( d + / . In particular, along the anti-Stokes rays, we have Im( w ( d + / ) = w ( d + / ) → ∞ .We remark that Equation (18) can also be derived from §5.6 (Theorem 5.6.1) of[Hil69] – see also Theorem 3.2 of [Ari17]. There, the coordinate ξ is described tobe the natural coordinate for the quadratic di ff erential q in Equation (11), that is, ξ = w (cid:82) √ q . (cid:3) Exponential map and infinite-grafting.
The proofs in the following sub-section shall rely on the following notions that include a well-known geometricinterpretation of the exponential function ( c.f. [BPM]).Consider the entire function f : C → C P given by the exponential map f ( z ) = e π iz . As is well known, this is the uniformizing function for the logarithm function g ( z ) = ln( z ). Such an entire function has 0 as an asymptotic value, as can be seenby restricting f to the imaginary axis. Let R be an embedded arc between 0 and ∞ .The domain C can be thought of as obtained from C P by taking countably infinitecopies of C P \ R indexed by Z , and identifying one side of the slit R on the i -thcopy with the other side of the slit R in the ( i + EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 23 referred to as attaching a logarithmic end between 0 and ∞ . The point 0 (or ∞ ) in C P is said to be a logarithmic singularity , or equivalently, the map f is a branchedcover over C P with infinite ramification (or branching) over the branch points 0and ∞ . Note that these branch-points are precisely the asymptotic values of themap f in the domain.We remark that a 2 π -grafting along an embedded arc α ⊂ C P is obtained byattaching a copy of C P \ α along a slit at α , or alternatively, grafting in a lune ofangle 2 π (see §2.2). Given a projective structure on a surface S , such an operationalong an embedded arc in the developing image does not change the holonomyrepresentation (see [Gol90]).4.3. Grafting and the Schwarzian derivative.
We verify that (cid:99) Gr is well-defined,that is: Proposition 4.2.
The grafting operation on a crowned hyperbolic surface in T g ( n ) along a measured lamination λ on it (as defined in §3.4) results in a projectivestructure P on a punctured Riemann surface X with a Schwarzian derivative thatlies in P g ( n ) .Remark. The fact that the grafting operation results in some projective structureis a consequence of the definitions (see §2.2); the above proposition identifies thespace in which the resulting projective structure lies.Let ˆ S be a crowned hyperbolic surface in T g ( n ) . Recall that we have assumed,at the beginning of the section, that ˆ S has a single hyperbolic crown end with ( n − C . (See §3.3 for a description ofa hyperbolic crown – in particular, note that it is conformally an annulus of finitemodulus.)The proof of Proposition 4.2 is a local argument involving the grafting operationfor this crown, and is an immediate consequence of the following two lemmas. Lemma 4.3.
The projective surface obtained by grafting ˆ S along λ is conformallya punctured Riemann surface X.Remark. Note that X is in fact of genus g and a single puncture, that is, hasthe same topology as the crowned hyperbolic surface (see the second statement ofTheorem 2.1). Proof of Lemma 4.3.
It su ffi ces to check that the crown end C (that is topologicallyan annulus) is conformally a punctured disk after the grafting; in the rest of theproof we shall focus entirely on this crown end.Recall that the grafting lamination λ intersects C along finitely many isolatedleaves of finite weight, that are either between the boundary cusps of the crown, orfrom a boundary cusp to the closed geodesic boundary of C . We denote the col-lection of these geodesic leaves of finite weight intersecting C by λ C . More impor-tantly for us, λ includes the geodesic sides γ , γ , . . . γ n − of the crown boundary,each with infinite weight. F igure
5. The surface (cid:102) A ∞ is obtained by infinite grafting along thecircular arcs in the lift of the crown in C P . This infinite graftingadds in a topological half-disk, denoted by the dotted lines.It shall be useful to pass to the universal cover. That is, consider the universalcover (cid:101) C of the hyperbolic crown as a Z -invariant domain in D ⊂ C P , and performa grafting along the lifted lamination (cid:102) λ C , together with the Z -invariant collectionof the lifts G = { ˜ γ i , ˜ γ i , . . . , ˜ γ in − } i ∈ Z each with weight ∞ .The advantage is that the grafting here admits a synthetic-geometric descriptionsimilar to that in the previous section:Namely,(i) along each of the leaves of (cid:102) λ C we insert a “straight lune” of angle equal to thecorresponding weight (see §2.2), and(ii) for each arc ˜ γ ∈ G , we take countably infinite copies of C P , each with a slitalong ˜ γ , determining sides ˜ γ j + and ˜ γ j − on the j -th copy, where j ∈ Z ≥ , and identify˜ γ (on the original domain) with ˜ γ + , and then successively identify ˜ γ j − with ˜ γ j + + foreach j .(The fact that it is an infinite chain indexed by non-negative integers, instead ofa bi-infinite chain, is used in the proof of the final claim.)Topologically, this appends an open half-disk along each boundary arc ˜ γ ∈ G in the original domain, and results in a conformal (immersed) domain (cid:102) A ∞ in C P .See Figure 5. As in §2.2, (cid:102) A ∞ is invariant under a cyclic subgroup of PSL ( C )generated by a new M¨obius transformation, and yields a conformal annulus A ∞ inthe quotient.We have to show that the conformal modulus of A ∞ is infinite, i.e. A ∞ is bi-holomorphic to D ∗ . Equivalently, we need to show that (cid:102) A ∞ is biholomorphic to theupper half-plane.Divide the strip (cid:102) A ∞ into topological rectangles by circular arcs from the crown-tips (starting points of ˜ γ ij for j = , , . . . ( n −
2) and i ∈ Z . We denote theserectangles by R i , R i , . . . R in − . EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 25
It su ffi ces to show: Claim. The conformal modulus of the union of any pair of adjacent rectanglesin the above subdivision, call it R j ∪ R j + , is infinite.Proof of claim. Here, we appeal to the grafting description for the exponential mapdescribed in §4.2. Recall that here, a logarithmic end comprising a bi-infinite chainof copies of C P \ γ is attached along a choice of an embedded arc γ between thetwo branch-points of infinite order, 0 and ∞ , on C P . Moreover, we know thatthe resulting Riemann surface is bi-holomorphic to C . Let D be a round disk in C P properly containing the arc γ and its endpoints, and let D c = C P \ D be thecomplementary disk. It follows that the Riemann surface is obtained by attachingthe logarithmic end along γ to D is biholomorphic to C \ D c , which is conformallya punctured disk.Observe that attaching a logarithmic end along γ is equivalent to introducing aslit along γ , and then performing an infinite-grafting along the resulting two sides γ + and γ − . Here, we use the fact that an infinite-grafting along a side adds on achain of copies of C P index by non-negative integers; thus, infinite-grafting alongthe two sides of the slit introduces a bi-infinite chain of C P s, i.e. a logarithmicend. See Figure 6.F igure
6. The surface obtained by infinite grafting along the twosides of a slit along γ on a disk D . The rectangle R (cid:48) = D \ β isquasiconformally related to R j ∪ R j + (see Figure 5).Pick two circular arcs from 0 and ∞ respectively, to the boundary of D , intersect-ing ∂ D orthogonally. If we slit along one of the arcs, call it β , we get a topologicalrectangle R (cid:48) , that is sub-divided into two rectangles R a and R b by the other arc.From the above discussion, the rectangle R (cid:48) has infinite modulus, as the surfaceobtained by identifying the sides of the rectangle (the two sides of the slit β ) isconformally a punctured disk.Finally, note that one can easily build a quasiconformal map from R a ∪ R b to R j ∪ R j + ; in fact, we can do so by a map that is conformal on the ends obtainedby the infinite grafting on the sides. Thus, R j ∪ R j + is also a rectangle of infinitemodulus, as claimed. (cid:3) This completes the proof of the Lemma. (cid:3)
Now let U (cid:27) D ∗ be a neighbourhood of the puncture on X that corresponds tothe crown end C after grafting. In what follows we shall think of U as a region {| z | > } ⊂ C . The developing map f : (cid:101) X → C P for the projective structure P ,when restricted to a lift ˜ U of U , is a map equivariant with respect to the action of π ( U ) = Z on the domain, and the cyclic monodromy around the puncture, in thetarget.To complete the proof of Proposition 4.2, it su ffi ces to show: Lemma 4.4.
The Schwarzian derivative of f | ˜ U descends to a meromorphic qua-dratic di ff erential on U with a pole of order n at the puncture.Proof. Our proof is an adaptation of the “rational approximation” argument ofNevanlinna – see §3.4 of [Nev70], and also the proof of Theorem 40.1 in [Sib75].Consider the sequence of conformal annuli A N for N ≥ C along λ C , together with a 2 π N -grafting on each of the geodesic sides γ , γ , . . . γ n − of the crown boundary.It follows from the proof of Lemma 4.3 that A N form an exhaustion of A ∞ , thatis, A N ⊂ A N + for each N , such that mod( A N ) → ∞ as N → ∞ and A ∞ = (cid:83) N ≥ A N .In particular, for any compact subset Ω ⊂ ˜ U there is a su ffi ciently large integer N such that Ω is strictly contained in (cid:102) A N for all N ≥ N . Then the restriction f | Ω is then the uniform limit of a subsequence of the corresponding developing maps f N | Ω : Ω → (cid:102) A N where N ≥ N . Recall that each (cid:102) A N is conformally immersed in C P , and by our construction f N is a conformal immersion to C P with order-2 N branching at the Z -invariant collection of points where the lifts of two adjacentsides of the crown end meet. A simple calculation then shows that the Schwarzianderivative of f N is then of the form φ N ( z ) dz where φ N is a meromorphic func-tion with poles of order at most two at the ( n −
2) critical points that map to thebranch-points of finite order. Thus, the restriction of f to the interior of A N , and inparticular f N | Ω for N ≥ N , is a locally univalent holomorphic function since thecritical points lie on the boundary of A N . Moreover, since the number of poles oforder two does not depend on N , this holomorphic function is of fixed polynomialgrowth that does not depend on N .By the uniform convergence f N → f on Ω , these Schwarzian derivatives con-verge uniformly to the Schwarzian derivative of f | Ω , which is then of the form φ ( z ) dz where φ is a holomorphic function on Ω of a fixed polynomial growth thatdoes not depend on Ω .By the usual invariance of the Schwarzian derivative under postcomposition byM¨obius maps, this Schwarzian derivative of f | ˜ U descends to a meromorphic qua-dratic di ff erential on U . The polynomial growth condition then implies that it hasat most a finite order pole at the puncture.The fact that the order of the pole is exactly n follows from the discussion in§4.1:From our description of (cid:102) A ∞ in the proof of Lemma 4.3, each fundamental regiondetermines exactly ( n −
2) infinitely-branched points, and thus the developing map f | ˜ U has exactly ( n −
2) asymptotic values. From Corollary 4.1, in the expression
EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 27 F igure
7. A circular arc between a pair of consecutive asymptoticvalues of f (see figure on right) has infinitely many pre-images in asector (see figure on left). Each region between these pre-images(one shown shaded) map to a copy of C P \ ˜ γ ; the image of thesector thus wraps infinitely many times around C P . φ ( z ) dz for the Schwarzian derivative as expressed in U = {| z | > } , the rationalfunction φ ( z ) would have polynomial growth of order exactly ( n − n at the puncture. (cid:3) Inverse of the grafting map.
Let P be a meromorphic projective structure in P g ( n ), and let (cid:101) P denote the universal cover. Recall that the Thurston construction(see Theorem 2.1) applied to (cid:101) P would yield the Poincar´e disk D and a measuredlamination L on it. Recall that L is the grafting lamination for (cid:101) P .In this subsection we shall prove the following Proposition, which says that the image of the inverse of the grafting map lands in T g ( n ) × ML g ( n ). Proposition 4.5.
For (cid:101)
P as above, the pair ( D , L ) obtained from Theorem 2.1 is theuniversal cover of a pair ( X , λ ) ∈ T g ( n ) × ML g ( n ) .Proof. For ease of notation, we shall continue with our assumption of a singlepuncture, in which case n is just a single integer n ≥ (cid:101) P ,it follows that ( D , L ) would be invariant under some Fuchsian group Γ such that D / Γ is homeomorphic to the underlying surface of P – a once-punctured surfaceof genus g .Restrict the projective structure (cid:101) P to the lift of a neighborhood U of the puncture.We need to verify that the grafting lamination for the restriction (cid:103) P | U includes acyclically ordered chain of geodesics on D with infinite weight on each, such thatthe chain is invariant under a hyperbolic monodromy around the puncture.By Corollary 4.1, the developing map for (cid:103) P | U descends to a meromorphic func-tion f on U = {| z | > } having ( n −
2) asymptotic values in equi-angled sectors S , S , . . . S n + . We denote these asymptotic values by c , c , . . . , c n + ∈ C P re-spectively. Moreover, by Corollary 4.1 the restriction of f to each anti-Stokessector of angle 2 π/ ( n −
2) has the same asymptotic expansion as an exponentialmap in suitable coordinates for the sector (see Equation (18)). In particular, thedeveloping image of a sector is identical to that of the exponential map. See Figure7. By our geometric interpretation of the exponential map in §4.1, this developingimage can be described as follows: for each 0 ≤ j ≤ ( n −
3) choose a circular arc γ j in C P between c j and c j + such that γ j is contained in the image of f . We obtaina Riemann surface A ∞ by attaching a chain of copies of C P slit along γ j , indexedby non-negative integers, along each γ j . We shall call this a half-logarithmic end ,which can be thought of as conformally immersed in C P . The map f then mapsinto A ∞ , and in particular, its restriction to a sector surjects on to the correspondinghalf-logarithmic end.As a consequence of this geometric description for each pair of successive points { c j , c j + } , there is a family of round disks embedded in C P parametrized by non-negative reals, such that each disk in the family touches c j and c j + , and their unionexhausts the corresponding half-logarithmic end. In the immersed surface in C P ,this family of disks starts from a disk D that has the circular arc γ j as part ofits boundary, and then rotates around C P , such that D t (where t ∈ R ≥ ) has acorresponding boundary arc that makes an angle t with γ j .The construction in Theorem 2.1 then shows that the corresponding pleated sur-face will have as bending line the geodesic line in H with endpoints { c j , c j + } ∈ C P = ∂ H . Moreover, in the construction of the associated pleated surface inTheorem 2.1, the entire family of disks along the half-logarithmic end will col-lapse onto this line. In other words, the domain projective surface has an “infinite”lune, and hence the corresponding leaf in the grafting lamination will have infiniteweight.Passing to the universal cover, one obtains a chain of such geodesic lines in D that will be invariant with respect to the (Fuchsian) holonomy around the puncture.To show that this monodromy is actually a hyperbolic element, we only need to ruleout the case that it is parabolic, since we already know that D / Γ is homeomorphicto the underlying surface S of P :Suppose the holonomy around the puncture is a parabolic transformation h . Ifthe chain of geodesics { ˜ γ i } i ∈ Z in D is invariant under the infinite cyclic group (cid:104) h (cid:105) ,then their endpoints limit to the same point p ∈ ∂ D as i → ±∞ , where the fixedpoint Fix( h ) = { p } . If we pick another element g ∈ π ( S ) then the conjugate sub-group g (cid:104) h (cid:105) g − would leave invariant another such chain of geodesics correspondingto another lift of a loop around the puncture. We note that the grafting laminationcomprises disjoint leaves and Fix( ghg − ) = { g . p } . If g . p (cid:44) p , then the two chainsof geodesics based at p , g . p must intersect, contradicting the fact that no two leavesof the grafting lamination intersect. Hence g . p = p , and since this is true for everyelement g ∈ π ( S ), we conclude that Γ is elementary, which is impossible as D / Γ is homeomorphic to a surface with non-abelian fundamental group. EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 29
Thus, the above chain of geodesics in D is invariant under this hyperbolic mon-odromy, and in the quotient X = D / Γ , it descends to a crown end for the hyperbolicsurface. From our construction, in each fundamental region, there are exactly ( n − n −
2) boundarycusps. Thus, the quotient hyperbolic surface X lies in T g ( n ).Moreover, the grafting lamination L on D is invariant under Γ , and descendsto a measured lamination λ on such a crowned hyperbolic surface, and thus, bydefinition, lies in ML g ( n ) ( c.f. §3.4). (cid:3) We can finally show:
Proposition 4.6.
The grafting map (cid:99)
Gr is a homeomorphism.Proof.
Recall from Theorem 2.1 that the Thurston construction for a projectivesurface obtained by grafting recovers the original hyperbolic surface and measuredgeodesic lamination.By Proposition 4.5, the Thurston construction then defines an inverse map to (cid:99) Gr . Moreover, by the same proposition, (cid:99) Gr is surjective.Since the domain of the map is homeomorphic to R χ (see Proposition 3.8) and (cid:99) Gr is continuous (see the last statement of Theorem 2.1, we conclude that (cid:99) Gr is ahomeomorphism (by invariance of domain). (cid:3) This completes the proof of Theorem 1.1.5. P rojective structures on C The proof of Theorem 1.1 also applies in the case when g = k =
1, andwe obtain a grafting description for a certain space of projective structures on thecomplex plane C – see Theorem 1.2 from §1. After defining the spaces appearingin Theorem 1.2 in §5.1, we provide a proof, and give an application of Theorem1.2 in §5.2.5.1. Definitions and the proof of Theorem 1.2.
We start with a more detaileddescription of the spaces in Theorem 1.2:A projective structure in P ( d ) is determined by a conformal immersion f : C → C P (the developing map) such that the Schwarzian derivative of f (see Equation(6)) is a polynomial quadratic di ff erential on C of degree d , that is, it can be ex-pressed as q = ( z d + a d − z d − + · · · + a z + a ) dz where the coe ffi cients ( a , a , . . . , a d − ) ∈ C d − . Note that, up to a conformal auto-morphism of C , any polynomial quadratic di ff erential can be assumed to be monicand centered as above.In this section there will be no additional real twist parameter at ∞ ; indeed, thereare no non-trivial Dehn-twists around ∞ since C is simply-connected, and the nor-malization as above fixes the horizontal directions of q to be at angles 2 π j / ( d + where j = , , . . . ( d + f has exactly ( d +
2) asymptotic valuesthat we call the crown tips . As usual, we shall consider two projective structureson C to be equivalent if the developing maps are isotopic such that the isotopykeeps the crown tips fixed. Recall from Corollary 4.1 that the asymptotic valuesare achieved along rays in the horizontal directions of q which are at equal anglesof 2 π/ ( d +
2) starting from the horizontal direction; this gives a cyclic ordering tothe set of crown tips.Sibuya showed (see Chapter 8 of [Sib75]), using the methods from the theory oflinear di ff erential systems , that in fact the cyclically ordered collection of (possiblynon-distinct) crown-tips C = { c , c , · · · c d + } on C P satisfy (a) c k (cid:44) c k + and (b)there are at least three distinct points in C .Let C ( d ) be the space of ordered ( d + C P that satisfy (a) and (b)above, up to the action of PSL ( C ). (In particular, we can arrange so that the firstthree points are 0 , ∞ and 1.)We can define the “crown-tip map”(19) Ψ : P ( d ) → C ( d )that assigns to a projective structure on C , the ordered tuple of crown-tips that itdetermines.Next, Poly( d ) is the space of hyperbolic ideal polygons with ( d +
2) vertices a , a , · · · , a d + up to isometry, together with a cyclic ordering of the vertices.Assume, without loss of generality, that a , a , · · · , a d + gives this cyclic order-ing. Suppose further that after acting by a suitable isometry, the vertices a , a , a are placed at − , , i . The cross-ratios of successive quadruples { a j , · · · , a j + } , j = , , · · · , d − d −
1) real parametersthat uniquely determine the ideal polygon. Thus, the space Poly( d ) is homeomor-phic to R d − .Finally, the space Diag( d ) is the space of weighted diagonals in an ideal ( d + R d − – see, for example, Theorem 3.3 of [MP98] andthe discussion in section 3.2 of [GW18]. (Note that the geodesic sides of infiniteweight do not contribute any parameters.)We shall assume that each of these spaces acquire a natural topology via theparametrization we have described for them. EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 31
Proof of Theorem 1.2.
The fact that the grafting map described in Equation (4),which is: (cid:99) Gr C : Poly( d ) × Diag( d ) → P ( d )is well-defined follows from Lemmas 4.3 and 4.4 in §4.3.This is also implied by the work of Sibuya in [Sib75], as we now describe:Let P ∈ Poly( d ) be an ideal polygon, thought of as conformally embedded in D ⊂ C P , with ideal vertices a , a , . . . , a d + along the equatorial (real) circle. Itis easy to verify that grafting P along a set of diagonals with finite weight takesthese vertices to an ordered tuple of points c , c , . . . c d + that lies in the space C ( d )defined above.Given such an ordered set C of points in C P satisfying (a) and (b) above, Sibuyaconsidered the Riemann surface R by attaching an infinite chain of copies of C P ( c.f. §4.2) to arcs chosen between successive points. In our grafting terminology,this is equivalent to performing, in addition to the grafting along the diagonals in P of finite weight, an infinite grafting along the geodesic sides of P .A theorem of Nevanlinna ([Nev32]) then asserts that the resulting surface R isparabolic, i.e. R is conformally equivalent to C . (This is the analogue of Lemma 4.3from §4.3.) Moreover, Theorem 40.1 of [Sib75] shows that the map f : C → C P ,i.e. the composition of the biholomorphism from C to R , followed by the branchedcover to C P , has a Schwarzian derivative that is a polynomial quadratic di ff erentialof degree d . (This is the analogue of Lemma 4.4 from §4.3.) By construction, theasymptotic values of f are the infinite-order branch-points at C . Thus, f definesa projective structure P ∈ P ( d ), with the crown-tips C . See Chapter 8 §40, 41 of[Sib75] for details.Then, the proof in §4.4 carries through, to show that (cid:99) Gr C admits an inverse map.Recall that this uses the Thurston construction – see Proposition 4.5. In fact, thepresent discussion would be easier than the work required in the proof of Proposi-tion 4.5, since the punctured surface C is simply-connected, and we need not pass tothe universal cover. Theorem 2.1 applies directly, and the argument in Proposition4.5 (that uses Corollary 4.1) shows that the grafting lamination includes a closedchain of ( d +
2) geodesic lines in D , each of infinite weight, that thus bounds anideal polygon P ∈ Poly( d ). The remaining geodesic leaves of the grafting lamina-tion must be pairwise disjoint, and hence must constitute a collection of weighteddiagonals in P .Thus, this inverse map has image in Poly( d ) × Diag( d ) when we start with anyprojective structure in P ( d ).In particular, this proves that (cid:99) Gr C is a bijection. Since the spaces in the domainand range of (cid:99) Gr C in Equation (4) are homeomorphic to R d − , we conclude, fromthe invariance of domain, that (cid:99) Gr C is a homeomorphism. (cid:3) Fibers of the crown-tip map.
In this section we use the grafting descriptionin Theorem 1.2 to characterize the fibers of the map Ψ in Equation (19), i.e. theset of all projective structures in C that have the same ordered set of crown-tips (as defined in §5.1).The work of Bakken in [Bak77] showed that Ψ is in fact a local biholomor-phism. However, it was known, due to examples of Sibuya (see §42 of [Sib75])and Bakken (see §7 of [Bak77]), that Ψ is not globally injective.We shall now prove: Theorem 5.1.
Fix an ordered tuple C ∈ C ( d ) . For any disjoint collection of diag-onals (20) D = { l , l , . . . , l d − } in an abstract ( d + -gon, there exists a unique ideal polygon P ∈ Poly ( d ) and aunique collection of non-negative weights { w , w , . . . , w d − } , w i ∈ [0 , π ) , on thediagonals such that (21) (cid:99) Gr C ( P , L ) ∈ Ψ − ( C ) whenever L ∈ Diag ( d ) is a weighted diagonal assigning weight ( w i + π n i ) to thediagonal l i , for a tuple ( n , n , . . . n d − ) ∈ Z d − ≥ , together with the geodesic sides ofP, each with infinite weight. We write this as: (22) L = ( w + π n ) · l + ( w + π n ) · l + · · · + ( w d − + π n d − ) · l d − Moreover, any element of the fiber Ψ − ( C ) is given via a grafting construction(Equation (21) ) by the following data: (1) a choice of diagonals as in Equation (20) , (2) the unique associated ideal polygon P and ( d − − tuple of weights asabove, with w i ∈ [0 , π ) , (3) a ( d − − tuple of integers n i ∈ Z ≥ as in Equation (22) . Before giving the proof of Theorem 5.1, we describe the operation of graftingfor an ideal quadrilateral that will play a role; note in particular that grafting idealpolygons along a collection of weighted diagonals can be described completelyin two dimensions, that is, on the complex plane C , without reference to three-dimensional hyperbolic geometry as in §2.2. Grafting an ideal quadrilateral.
Consider an ideal quadrilateral defined by the(cyclically ordered) tuple of ideal vertices ∞ , − , , λ where λ ∈ R + . A grafting(or “bending”) by angle t along the diagonal between 0 and ∞ can be seen on theupper half-plane as follows: the diagonal line in this model is the vertical geodesic α from 0 to ∞ ; this divides the upper half-plane into the two regions R − and R + thatare the quarter-planes defined by Re( z ) < z ) > ff ected by a map that is the identity on R − and the rotation z (cid:55)→ e − i θ on R + ; the image is a new domain that is obtained from the upper half-plane bygrafting in a lune of angle t , at the vertical geodesic α . Clearly, the grafting fixesthe points − , , ∞ and takes λ to the new point λ e − it ∈ C P . (See Figure 1 in §2.2.)Note that the resulting tuple of points ( ∞ , − , , λ e − it ) could also have been ob-tained by grafting along the diagonal line α (cid:48) between − λ . A cross-ratio calcu-lation shows that there is a conformal map that realizes the permutation ( ∞ , − , , r ) (cid:55)→ EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 33 F igure
8. Any configuration of points on C P (right) can be ob-tained by grafting an ideal polygon along a collection of weighteddiagonals (left).( − , , / r , ∞ ), and hence a graft by an angle 2 π − t along α (cid:48) , results in the sameconfiguration of four points. Proof of Theorem 5.1.
Let { c , c , . . . , c d + } be the ordered tuple C ∈ C ( d ).An ideal ( d + a , a , . . . a d + would be triangulatedby the collection of diagonals determined by D . Each diagonal line δ determinesan ideal quadrilateral Q comprising the two ideal triangles adjacent to δ . Thisdetermines a collection Q of overlapping quadrilaterals: each pair of quadrilateralsin Q is either disjoint, or overlaps along an ideal triangle. Note that there is a dualtree T determined by this configuration of diagonals – the vertices of T correspondto the ideal quadrilaterals, and there is an edge between vertices whenever thecorresponding quadrilaterals overlap.It is easy to check by an inductive proof based on the tree T , that the ideal( d + Q ,where vertices of each are taken in the induced cyclic order.Now for each quadrilateral Q ∈ Q we can choose the ideal vertices { a j , a k , a l , a m } of Q such that it has a cross-ratio | λ Q | , where λ Q the cross-ratio of the four points c j , c k , c l , c m . Let P be the ideal polygon that this data uniquely determines.To assign weights to these diagonals, note that Q has a diagonal d Q ∈ D ; the toyexample preceding the lemma describes how one can graft Q along this diagonal δ by an angle w ( Q ) ∈ [0 , π ) such that the images of the vertices { a j , a k , a l , a m } arethe four points c j , c k , c l , c m (in the ordered tuple C ) with cross-ratio equal to λ Q .We equip that diagonal d Q with weight w ( Q ).Thus by construction, grafting each diagonal d Q in D by an angle w ( Q ), weobtain (cid:99) Gr C ( P , L ) where (see Equation (22)) L = (cid:88) Q w ( Q ) d Q takes the vertices of P to the tuple of points c , c , . . . c d + , as desired. Thus, (cid:99) Gr C ( P , L ) is a projective structure in P ( d ) with crown-tips exactly the ordered tu-ple C ∈ C ( d ). (The infinite grafting on the geodesic sides of P does not a ff ect thepositions of these crown-tips.) Note that adding 2 π to the weights of the diagonals, i.e. performing integer-2 π grafting (see §4.1) does not change the configuration of crown tips C .On the other hand, by Theorem 1.2, any projective structure in Ψ − ( C ) is ofthe form (cid:99) Gr C ( P (cid:48) , L (cid:48) ) for some ideal polygon P ∈ Poly( d ) and weighted diagonals L (cid:48) ∈ Diag( d ). By the previous discussion, the collection of diagonals D underlying L (cid:48) uniquely determines P (cid:48) , and the weights of L (cid:48) modulo 2 π . (cid:3)
6. T he monodromy map and T heorem Φ : P g ( n ) → (cid:98) χ g , k ( n )where the target is the decorated character variety that we shall define in §6.1.In §6.2, we shall prove Theorem 1.3; this shall use the grafting description formeromorphic projective structures that Theorem 1.1 provides.6.1. Decorated character variety.
For an oriented surface S g , k of genus g and k (labelled) punctures and negative Euler-characteristic, the usual PSL ( C )- charactervariety is χ g , k = Hom( π ( S g , k ) , PSL ( C )) // PSL ( C )where the geometric-invariant-theory (GIT) quotient on the right, yields a quasi-projective variety of (complex) dimension 6 g − + k .In what follows, we shall denote the representation variety as R g , k : = Hom( π ( S g , k ) , PSL ( C )) . Thus, R g , k is the space of representations, prior to the quotient. Given ρ ∈ R g , k , themonodromies around the k punctures shall be denoted by ρ , ρ , . . . , ρ k ∈ PSL ( C )respectively.Fix a k -tuple n = ( n , n , . . . , n k ) where each n i ≥ P ∈ P g ( n ) is a pro-jective structure on a surface ˆ S of genus g and k boundary components, with m i : = n i − i -th boundary component, where 1 ≤ i ≤ k .In particular, the holonomy of the projective structure determines a representation ρ ∈ χ g , k .In addition to this, we know from the grafting description provided by Theo-rem 1.1, or from Corollary 4.1, that the developing map for P , when restricted toneighborhood of the i -th pole of order n i ≥
3, has m i : = ( n i −
2) asymptotic values,where 1 ≤ i ≤ k . This yields a point in C P for each connected component of a ∂ ˆ S \ { marked points on the boundary } .Passing to the universal cover, we have a family of points on C P that are trans-lates of an (ordered) fundamental set C i = { c i , c i , . . . c im i − } , by the monodromy ρ i around the i -th boundary component of ˆ S .Just as for the set of crown-tips C ( d ) (defined before Equation 19), no two adja-cent points in C i are the same, that is, c ij (cid:44) c ij + for 0 ≤ j ≤ m i −
2. (Their translatesunder ρ i may coincide, for example, when ρ i is an elliptic element of finite order.) EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 35
Let (cid:98) C ( m i , ρ i ) be the space of such ordered m i -tuples of points in C P , togetherwith a choice of a (fixed) monodromy matrix ρ i ∈ PSL ( C ) (which determinestranslates of the ordered set by the cyclic group generated by ρ i ). This definesthe space of decorations at a boundary component of ˆ S , and its quotient by theconjugation-action of PSL ( C ) is exactly the space of “configurations of flags” de-fined by Fock-Goncharov (see pg. 11 of [FG06]), in the present context, since a“flag” in C can be thought of as a point in C P .We then define: Definition 6.1.
The decorated character variety is the space (cid:98) χ g , k ( n ) = (cid:110) ( ρ, C , C , . . . , C k ) | ρ ∈ R g , k and C i ∈ (cid:98) C ( m i , ρ i ) for ≤ i ≤ k (cid:111) // PSL ( C ) where m i = n i − and ρ i is the monodromy around the i-th puncture, for each ≤ i ≤ k.Remarks.
1. This coincides with the notion of the moduli stack of framed rep-resentations (or framed local systems) of Fock-Goncharov – see Definition 2.2 of[FG06] or Definition 2.7 of [Pal13], and also §1.4 of [AB].2. For other notions of a “decorated” character variety, see [Boa14] or [CMR17](see §1.6.3 of [AB] for a discussion.)From the preceding discussion, the meromorphic projective structure P ∈ P g ( n )uniquely determines a decorated monodromy ˆ ρ ∈ (cid:98) χ g , k ( n ). This defines the mon-odromy map Φ (see Equation (23)).Moreover, the work of Allegretti-Bridgeland shows that:(a) the image of the monodromy map Φ lies in an open dense subset (cid:98) χ g , k ( n ) ∗ ⊂ (cid:98) χ g , k ( n ) comprising (in their terminology) those representations having non-degenerateframing – see §6 of [AB],(b) the space (cid:98) χ g , k ( n ) ∗ is a complex manifold – see §9 of [AB].The following is essentially a consequence of the “Decomposition Theorem” ofFock-Goncharov (see Theorem 1.1. of [FG06], and Theorem 2.8 of [Pal13]). Proposition 6.2.
The space (cid:98) χ g , k ( n ) ∗ is a complex manifold of dimension χ = g − + k (cid:80) i = ( n i + .Proof. By (b) above, it is enough to verify there is an open set of real dimension R χ contained in (cid:98) χ g , k ( n ) ∗ .Pick a crowned hyperbolic surface X ∈ T g ( n ) and a measured lamination in ML g ( n ) that comprises a maximal set L of disjoint weighted bi-infinite geodesicsbetween the various boundary cusps. Note that the maximality implies that L divides X into ideal triangles, and an easy combinatorial count using the Euler-characteristic of the punctured surface implies that the cardinality |L| = χ . In fact,from Proposition 3.8 we obtain an open set V ⊂ ML g ( n ) containing L by varying the weights of the geodesic lines underlying L . Moreover, it is easy to check,from shear-coordinates in Teichm¨uller theory (see, for example, [BBFS13]), thatby varying the real “shear” parameter on each of the geodesic lines in L , we obtainan open set U ⊂ T g ( n ) containing X .By Theorem 1.1, we know P (cid:48) = (cid:99) Gr ( X (cid:48) , L (cid:48) ) ∈ P g ( n ) for any pair ( X (cid:48) , L (cid:48) ) ∈ U × V , and by part (a) of the discussion preceding the theorem, the decoratedmonodromy ˆ ρ for such a structure is a point in (cid:98) χ g , k ( n ) ∗ .By Propositions 3.5 and 3.8, both U , V are homeomorphic to R χ , and by (b)above (cid:98) χ g , k ( n ) ∗ is a manifold. Thus, it su ffi ces, by the invariance of domain, toshow that Φ ◦ (cid:99) Gr is injective on U × V .Let ρ ∈ χ g , k be the holonomy of P (cid:48) forgetting the decorations ( c.f. Definition6.1). By the construction in the proof of Theorem 2.1 we know that via the develop-ing map for this projective structure, we obtain a ρ -equivariant map Ψ : (cid:101) X → H .The map Ψ gives a pleated plane P that is pleated (or bent) at the lifts of theleaves of L (cid:48) . Note that Ψ is well-defined up to post-composition by an elementof PSL ( C ). By the maximality of L (cid:48) , these lifts triangulate P into a ρ -invariantcollection of ideal triangles. Recall that after “straightening” P , we obtain a to-tally geodesic copy of the hyperbolic plane, and the pleating locus determines alamination invariant under a Fuchsian group Γ .As in the proof of Theorem 5.1, the collection of geodesic lines (cid:101) L (cid:48) determines acollection of “bent” quadrilaterals Q by assigning, to each line l ∈ (cid:101) L (cid:48) , the quadri-lateral Q formed by the two ideal triangles in P adjacent to l .Theorem 1.1 of [FG06], applied to our setting ( G = PSL ( C )) implies that thecomplex cross ratios of each Q ∈ Q is determined by the decorated monodromy ˆ ρ .(See also the Example on page 11 of [FG06].) Then, from the discussion in §5.2on grafting an ideal quadrilateral, the weights on the leaves of (cid:101) L (cid:48) , as well the realcross-ratios of the “straightened” quadrilaterals, are determined uniquely by thesecomplex cross-ratios. The real cross-ratios, in turn, determine the ideal quadrilat-erals (overlapping along ideal triangles) that constitute the fundamental domain ofthe Γ -action on the “straightened” pleated plane. This uniquely determines the hy-perbolic surface X (cid:48) that we obtain in the quotient. Moreover, the Γ -invariance ofthe pleating locus determined by (cid:101) L (cid:48) shows that the weighted geodesics constitutingthe lamination L (cid:48) are uniquely determined.In other words, the decorated monodromy ˆ ρ recovers the pair ( X (cid:48) , L (cid:48) ) ∈ U × V .This completes the proof of the injectivity of the monodromy map Φ ◦ (cid:99) Gr on U × V ,and hence of the Proposition. (cid:3) Remarks.
1. Alternatively, one can show that at a generic point, the space ofconfigurations (cid:98) C ( m i , ρ i ), for fixed ρ i , is of complex dimension m i , for each 1 ≤ i ≤ k . Adding these contributions to dim C ( χ g , k ) = g − + k , we again get complexdimension χ .2. Our proof of Proposition 6.2 in fact shows that the monodromy map Φ is a EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 37 homeomorphism on the open set U × V . The argument in the next section to showthat Φ is a local homeomorphism everywhere will follow a similar strategy, withan additional di ffi culty arising from the fact that the grafting lamination may notbe just a maximal set of weighted geodesic lines.6.2. Proof of Theorem 1.3.
In this section, we shall prove Theorem 1.3, namely,that Φ in Equation 23 is a local homeomorphism.Throughout the section, we shall fix a base meromorphic projective structure P ∈ P g ( n ) that has monodromy (cid:98) ρ ∈ (cid:98) χ g , k ( n ).By Theorem 1.1, we know that P = (cid:99) Gr ( X , λ ) for some pair ( X , λ ) ∈ T g ( n ) ×ML g ( n ).Our task is to show that there is a small neighborhood U of X in T g ( n ) and aneighborhood V of λ in ML g ( n ), such that if(24) Φ ◦ (cid:99) Gr ( X (cid:48) , λ (cid:48) ) = Φ ◦ (cid:99) Gr ( X , λ ) = (cid:98) ρ for a pair ( X (cid:48) , λ (cid:48) ) ∈ U × V , then we have(25) X = X (cid:48) and λ = λ (cid:48) . Let X = X S ∪C where X S is a hyperbolic surface with geodesic boundary compo-nents. and C is the collection of crown ends. Similarly, we have the decompositionof any crowned hyperbolic surface X (cid:48) = X (cid:48) S ∪ C (cid:48) .Given a measured lamination λ on X , let λ = λ S ∪ L where λ S is supported in acompact part of the surface X S (away from the crown ends), and L consists of thefinitely many leaves of λ that intersect the crown ends C . (Here, we shall ignorethe geodesic sides of the crowns, each of which have infinite weight.)Similarly, we have the disjoint union λ (cid:48) = λ (cid:48) S ∪ L (cid:48) on the crowned hyperbolicsurface X (cid:48) .In what follows we shall call L (resp. L (cid:48) ) a triangulation of the crowned surface X (resp. X (cid:48) ), if there is no geodesic line between the boundary cusps of C (resp. C (cid:48) ) that is disjoint from the leaves already in the collection. Note that if L (resp. L (cid:48) ) is not a triangulation, then we can choose an extension to a triangulation byadding geodesics of zero weight between boundary cusps of the crowns. We shalldenote the resulting triangulation by L + (resp. L (cid:48) + ).The first step of the proof is to show: Proposition 6.3.
Suppose Equation (24) holds, where the pair ( X (cid:48) , λ (cid:48) ) is su ffi -ciently close to ( X , λ ) in T g ( n ) × ML g ( n ) , then the crown ends of X (cid:48) and X areisometric, and L (cid:48) = L .Proof. Choose a neighborhood V of λ in ML g ( n ) such that for any λ (cid:48) ∈ V , thereare triangulations L (cid:48) + (resp. L + ) of C (cid:48) (resp. C ), such that the homotopy classes ofthe arcs in the triangulations are identical, and the corresponding weights are close.(For the notation used here, see the paragraph preceding this Proposition.) This is possible since the space ML g ( n ) is a cell-complex where the finitelymany cells correspond to the di ff erent topological types of the dual metric graphs( c.f. the proofs of Propositions 3.7 and 3.8). The homotopy classes of the arcsin the triangulation determines this topological type of the dual metric graph; oncethis is fixed, two laminations being close in ML g ( n ) implies that the correspondingweights on the arcs (that determine the lengths of the finite edges) are close.Throughout we shall assume that ( X (cid:48) , λ (cid:48) ) is close enough to ( X , λ ) such that λ (cid:48) ∈ V .Consider the lifts of the crown ends C to the universal cover of X , together withthe lifts of the arcs in L + . These are invariant under a Fuchsian group Γ ; we choosea fundamental domain for this action on the combined set of crown ends and liftsof arcs. Namely, we get(i) a finite collection crown ends (cid:101) C , (cid:101) C , . . . , (cid:101) C N and a corresponding collec-tion of fundamental domains F , F , . . . , F N for the Z -action on each ofthese crowns, and(ii) a finite collection of arcs A + (that are lifts of arcs of L + ) between the idealvertices determined by the boundary cusps of F , F , . . . , F N ,such that any other lift of an arc in L + is taken to an arc in A + by a unique elementof Γ .Similarly, we have a finite collection of arcs A (cid:48) + in the universal cover of X (cid:48) ,that is the fundamental domain for the action of a Fuchsian group Γ (cid:48) on the lifts of L (cid:48) + .Moreover, since L + (resp. L (cid:48) + ) is a triangulation, the collection A + (resp. A (cid:48) + )is maximal, in the sense that we cannot add any other geodesic line to the col-lection that are between a pair of ideal vertices determined by F , F , . . . F N (resp F (cid:48) , F (cid:48) , . . . F (cid:48) N ) and are disjoint to the ones already in A + (resp. A (cid:48) + ). In particular,the arcs in A + and A (cid:48) + bound ideal triangles.Now, as in the proof of Theorem 5.1, we consider a collection of ideal quadri-laterals Q determined by the geodesic lines in A + , namely, for each line in A + thetwo adjacent ideal triangles determine an ideal quadrilateral Q ∈ Q . Note that eachpair of quadrilaterals in Q are either disjoint, or overlap along an ideal triangle.Recall that by Theorem 1.1 of [FG06], the decorated monodromy ˆ ρ uniquelydetermines the (complex) cross-ratios of quadruples { c j , c k , c l , c m } of crown tipsdetermined by the image of the lifts of the crown ends by the developing map.In particular, the (complex) cross-ratio λ Q of an ideal quadrilateral in Q ∈ Q after grafting is determined by ˆ ρ . Just as in the proof of Theorem 5.1, this uniquelyspecifies the real cross-ratio | λ Q | as well as the weight w ( Q ) of the correspondingarc (which is a diagonal of Q ). In particular, the weights on the arcs in A ⊂ A + and A (cid:48) ⊂ A (cid:48) + , and hence L and L (cid:48) , are uniquely determined and are equal. Thus L = L (cid:48) . EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 39
Moreover, the (real) cross-ratio | λ Q | for each of the quadrilaterals Q ∈ Q uniquelydetermines the ideal vertices of F , F , . . . , F N as well as F (cid:48) , F (cid:48) , . . . , F (cid:48) N ; thisshows that the crown ends in C are isometric to those in C (cid:48) . (cid:3) To complete the proof, we need to show:
Proposition 6.4.
Suppose Equation (24) holds, where the pair ( X (cid:48) , λ (cid:48) ) is suf-ficiently close to ( X , λ ) in T g ( n ) × ML g ( n ) , then the hyperbolic surfaces-with-boundary X S and X (cid:48) S are isometric, and λ S = λ (cid:48) S .Proof. Let L S and L (cid:48) S be the geodesic arcs of L ∩ X S and L (cid:48) ∩ X (cid:48) , respectively.We already know from Proposition 6.3 that the arcs in L S and L (cid:48) S have identicalweights and determine the same homotopy classes.Let ρ ∈ χ g , k be the representation in the usual PSL ( C )- character variety ofthe punctured (or bordered) surface obtained by “forgetting” the decorations at thepunctures ( c.f. Definition 6.1). Thus, ρ is the image of ˆ ρ under the forgetful map p : (cid:98) χ g , k ( n ) → χ g , k . We then apply the Ehresmann-Thurston principle for manifolds with boundary(see, for example, Theorem I.1.7.1 of [CEG06] or Proposition 1 of [Dan13]):Let D ( S g , k , C P ) be the space of developing maps for projective structures onthe surface-with-boundary S (homeomorphic to S g , k ) that are fixed on a collarneighborhood of the boundary. This space is equipped with the usual compact-open topology. Then the Ehresmann-Thurston principle implies that there is aneighborhood W of the developing map ¯ f for the restriction of P (that we fixedat the beginning of the section) to S , such that any developing map in W that hasthe same holonomy ρ as P , is equivariantly isotopic to ¯ f .In particular, in the space of projective structures on S , the restrictions of P and P (cid:48) = (cid:99) Gr ( X (cid:48) , λ (cid:48) ) are equivalent. Thus the Thurston construction in §2.4 whenapplied to the corresponding developing maps, yields isometric hyperbolic surfaces X S and X (cid:48) S , and identical grafting laminations λ ∩ X S and λ (cid:48) ∩ X (cid:48) S . Since we alreadyknow the arcs in L ∩ X S and L (cid:48) ∩ X (cid:48) have identical weights and determine the samehomotopy classes, we conclude that λ S = λ (cid:48) S .This completes the proof. (cid:3) By Propositions 6.3 and 6.4, we conclude that if ( X (cid:48) , λ (cid:48) ) is su ffi ciently close to( X , λ ), and Equation (24) holds, then in fact Equation (25) is true, namely X = X (cid:48) and λ = λ (cid:48) . This shows that the monodromy map Φ is locally injective. Sincewe already know that (a) Φ is continuous, (b) the image lies in the smooth part ofthe decorated character variety, and (c) the dimensions at a point P of P g ( n ) and asmooth point of (cid:98) χ g , k ( n ) are identical (see Proposition 6.2), we conclude that Φ is alocal homeomorphism from the invariance of domain.This proves Theorem 1.3. F igure
9. A pair of properly homotopic arcs on a crown (left) anda surface with boundary (right) obtained by splitting a single arc.A ppendix
A. M atching laminations
The proof of Theorem 3.8 for parametrizing the space ML g ( n ) relies on the factthat we know parametrizations of the space of measured laminations on a surface-with-boundary, and a crown separately. In this appendix we give details of how wematch two such laminations together to obtain one on the entire crowned hyper-bolic surface. Splitting arcs.
We start with the following notion:
Definition A.1 (Properly homotopic arcs) . Two arcs α , α on a hyperbolic crownare said to be properly homotopically equivalent if both have one end-point on theboundary geodesic γ , both arcs end at the same crown-tip, and both complete thesame number of integer twists around γ .Similarly, two arcs on a crowned hyperbolic surface are said to be properlyhomotopically-equivalent if (1) their restrictions to the hyperbolic crown end are properly homotopically-equivalent in the sense above, and (2) their restrictions to the hyperbolic surface-with-boundary in the comple-ment to the crown are homotopic arcs, where the homotopy is allowed tomove endpoints on the boundary geodesic. In what follows, a splitting of a weighted isolated arc in a measured laminationshall refer to a replacement of the arc by several properly homotopic arcs (seeFigure 9) with weights that have the same sum.Note that for a crown end with more than one boundary cusp, this replacementcan be done simultaneously for finitely many geodesic arcs that cross the crownboundary and proceed to the boundary cusps. Moreover, to have the correct mark-ing on the crown, we also need to maintain the (integer) number of twists of thearcs around the boundary component.
Determining the matching.
We shall denote the surface-with-boundary by S thehyperbolic crown by C , and the boundary of S by γ . For simplicity of exposition,we assume here that k =
1, that is, there is a single crown end; in the case k >
1, wecan consider γ to be a collection of closed geodesics, and C to a disjoint collection EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 41 of hyperbolic crowns, and our argument holds for such a disconnected surface orboundary as well.The identification of S and C along γ by a hyperbolic isometry yields a hyper-bolic crowned surface that we shall denote by ˆ S . Note that the boundary twistparameter (see §2.4) is crucial to uniquely specify this identification.Given measured laminations on S and C , such that the transverse measure of γ induced by them are identical, we wish to construct a combined lamination on ˆ S .The issue is that the leaves incident on the (common) boundary γ might notmatch – indeed, the numbers of arcs incident on γ from either side, or their end-points, need not be same.To resolve this, our strategy then would be to split these arcs on the subsurfacesand match the resulting arcs, such that the restriction of the resulting arcs on ei-ther subsurface still defines the same collection of homotopy classes of arcs. Inthis matching we also need to distribute the weights; for this, we shall need thefollowing lemma. Lemma A.2.
Let R be a rectangle, with n disjoint weighted arcs with weightsa , a , . . . a n (from left to right) incident on the top edge from outside R, and mdisjoint weighted arcs with weights b , b , . . . b m (from left to right) incident on thebottom edge from outside R. Suppose the total weights of the arcs incident on thetop and bottom edges are the same, that is, n (cid:80) i = a i = m (cid:80) j = b j .Then there is a unique way to split the arcs, and redistribute weights, such that (i) the resulting arcs can be paired by a collection of parallel arcs Γ in R, andpaired arcs have equal weights, and (ii) no two arcs in Γ connect to arcs arising from the same splitting, at both thetop edge and bottom edge.Remarks.
1. A “splitting” of an arc above refers to replacing an arc by finitelymany disjoint copies that then acquire a left-right ordering. Any pair of such copiesis then said to arise from the “same” splitting.2. We shall call the final matching obtained in this Lemma a minimal matching in light of property (ii) above, which ensures there are no unnecessary splittings.
Proof.
Without loss of generality we shall assume n ≤ m . The proof proceeds byinduction on n + m .Note that if n =
1, then there is a unique arc α incident on the top edge. Indeed,then there is a matching: split the arc α into exactly m copies, and assign weights b , b , . . . b m to them (from left to right), and and connect each of the resulting arcsincident on the top edge, with the m arcs incident on the bottom edge. It is easy tosee that (i) and (ii) are satisfied. There is a unique such matching, because none ofthe m arcs incident on the bottom edge can be split; the parallel arcs in R continuingconnecting to them would necessarily connect to arcs obtained by a splitting of α on the top edge, violating (ii).The inductive step is as follows: F igure
10. An example of a minimal matching (Lemma A.2).Consider the first arcs from the left incident on the top and bottom edges, de-noted by α and β respectively. Note that the weight of α is a and the weight of β is b . There are three cases: Case 1: If a = b , then no splitting of arcs is required: we connect the endpointsof α and β by an arc in R . The number of unpaired arcs on the top and bottom edgeseach reduce by 1. Case 2: If a > b , then we split α into two arcs α l and α r and assign weights b and a − b to the left and right arcs, respectively. We pair the left arc with β ,and consider the remaining (unpaired) arcs. Notice that there now n unpaired arcsincident on the top edge, and m − Case 3: If a < b , we split β into two arcs of weights a and b − a , and pairthe left arc with α . This time there are n − m unpaired arcs incident on the bottom edge.In all cases, the total number of unpaired arcs on the top and bottom edges havereduced by at least 1, and the induction is complete.Note that by construction, (i) is satisfied by this matching. We now explain whyproperty (ii) also holds: Recall in Case 2 we split the arc α and pair α l with β , theneven if β had been created in a splitting in the previous step, there are no other arcsfrom that splitting to the right of β . Thus, in the next step, the unmatched arc α r (or a splitting of it) is necessarily paired with an arc on the bottom edge that doesnot arise in the same splitting as β . A similar argument holds in Case 3, where β issplit.It is also easy to see that such a matching is unique: indeed, in any matching,there is a leftmost strand γ through R that connects an arc α incident on the topedge to an arc β of the bottom edge. Note that α might have arisen from a splittingof one of the original arcs incident on the top edge, or β might have arisen from asplitting, but not both, since otherwise (ii) would be violated. We can then concludethe first matching must have come from one of the 3 cases above. Repeating thesame argument for the next strand, we see that the entire matching must have beenobtained by the algorithm above. (cid:3) EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 43
We now return to the situation where we need to match arcs from the measuredlamination on the hyperbolic crown C , and surface-with-boundary S , that share acommon geodesic boundary γ .Let G = { g , g , . . . , g m } be the collection of weighted arcs on the hyperboliccrown C from the boundary cusps (labelled 1 , , . . . , m ) to the crown boundary.Let the measured lamination on S be λ ∪ L , where λ is a measured laminationthat is disjoint from γ , and L is a (non-empty) collection of weighted arcs incidenton γ .Recall from §3.2 that the boundary twist parameter τ of the crown C can bethought of as a choice of a basepoint p on the boundary curve γ , together withan integer twist parameter t that records the number of topological Dehn-twistsaround γ of arcs intersecting it (in our case, the arcs from G and L ). Recall thatthere is also “canonical” basepoint p for the crown. The remaining part of thetwist parameter in fact determines the transverse measure of the arc of γ between p and p .In the proof of the next Proposition, we shall use the point p to cut up γ intoan interval, that is, it shall determine the fundamental domain for the action of theinfinite cyclic group corresponding to γ , on the universal cover of the crown.In case that p coincides with an endpoint of one of the geodesic arcs in G or L , we split the corresponding arc such that the transverse measure of the arc on γ between p and p remains the same, but the new arcs have endpoints distinct from p . We shall assume that p is distinct from the endpoints of the arcs in L ; else, wecan change the arcs by a proper homotopy (as in Definition A.1) by sliding theendpoints along γ .See Definition A.1 for the notion of “properly homotopic” used below. We shallnow prove: Proposition A.3.
There is a unique way to split the weighted arcs in G and L ,redistribute the weights and match the resulting arcs, such that: (a) arcs that are matched have the same weight, (b) a “splitting” of an arc α replaces it by properly homotopic copies (fromboundary to the same boundary cusp in the case α ∈ G , and from boundaryto boundary in case α ∈ L ), (c) after the redistribution of weights, the total weight of all the arcs arisingfrom a splitting of an arc α (in G or L ) equals the original weight of α .This results in a new collection of disjoint arcs ˆ L on the crowned surface ˆ S ,such that no two arcs of ˆ L are properly homotopic. Moreover, ˆ L together with λ ,is a measured lamination on ˆ S .Proof.
We describe the splitting and matching of the arcs in two stages. LemmaA.2 will be used several times. F igure
11. Lemma A.2 is used to determine a preliminary split-ting of the arcs of G in the first stage of the construction.Let P = { p , p , . . . , p N } denote the endpoints of the arcs in L on γ , where thesepoints are ordered in the orientation of γ induced from the orientation on the sur-face, and the basepoint p on γ lies between p N and p (see the discussion above,preceding this Proposition). Note that each arc in L actually determines two end-points in the set P , and this determines a pairing of the elements of P . First stage:
Consider a thin closed annular neighborhood of γ , and cut along ageodesic arc perpendicular to γ and passing through p , to obtain a rectangle R .The arcs of G are incident on the top edge of R , and half-arcs of L are incidenton the bottom edge. (The half-arcs are paired to give the arcs in L , but we do notconsider that fact in this first stage. The weights on the half-arcs are the same asthat of the arc of L they belong to.) The total weight of the arcs incident on thetop and bottom edge of R are the same by our assumption. Hence we can applyLemma A.2, which determines a unique minimal matching involving a splitting ofthe arcs of G , and the half-arcs from L incident on the bottom edge of R .Let G (cid:48) be the set of arcs obtained by this initial splitting of the arcs in G . To eachpoint p i ∈ P we associate a subset G (cid:48) i ⊂ G (cid:48) as follows: if l + i is the half-arc of anarc in L incident on γ at p i , then G (cid:48) i comprises all the arcs of G (cid:48) that are matchedwith a splitting of l + i .Note that property (ii) of the minimal matching (see Lemma A.2) ensures thatfor any fixed 1 ≤ i ≤ N , the arcs of G (cid:48) i are asymptotic to distinct cusps. Second stage:
Now consider two points of P that are paired, say p i and p j . Thatis to say, there is an arc l ∈ L , contained in the surface-with-boundary S that hasendpoints p i and p j on γ . Then consider a rectangular neighborhood R l ⊂ S ofthe arc l , where the top and bottom edges are segments of γ . We can consider thearcs of G (cid:48) i and G (cid:48) j as incident on these top and bottom edges. By property (i) ofthe minimal matching construction in the first stage, the total weight of the arcs in G (cid:48) i is the same as that of the half-arc l + ⊂ l that was incident on p i , that is, equals EROMORPHIC PROJECTIVE STRUCTURES, GRAFTING AND THE MONODROMY MAP 45 F igure
12. In the second stage, Lemma A.2 is used to determinea splitting of l , for each l ∈ L .the weight of l . The same is true for the the total weight of the arcs in G (cid:48) j , sincethat equals the weight of the other half-arc of l . Hence, the total weights of thearcs incident on the top and bottom edges of R l are equal, and Lemma A.2 can beapplied.The arcs in R l of the resulting minimal matching determines a splitting of thearc l , for each l ∈ L . Moreover, it determines a splitting of the arcs in G (cid:48) i for each i ∈ { , , . . . , N } that completes the splitting of arcs in G .At the end of this second stage, we obtain a collection of weighted arcs ˆ L on thecrowned surface ˆ S = S ∪ γ C between the boundary cusps of the crown. It is easyto see that our construction ensures properties (a), (b) and (c) above. Moreover, wecan verify that no two arcs, say ˆ l , ˆ l in ˆ L are homotopic: indeed if they are, theirrestriction to S are homotopic, that is, they are splittings of the same l ∈ L . How-ever, recall that this splitting of l is defined in the second stage, where we determinea minimal matching of the arcs of G (cid:48) i and G (cid:48) j where p i and p j are the endpoints of l .By property (ii) of a minimal matching, a pair of arcs obtained by splitting l mustconnect to distinct splittings of arcs in G (cid:48) i or in G (cid:48) j (or both). However, we notedat the end of the first stage that distinct arcs in G (cid:48) i or G (cid:48) j are asymptotic to distinctcusps. Hence ˆ l and ˆ l are asymptotic to distinct cusps at one end (at least) whichcontradicts the assumption that they are homotopic.We can now homotope each arc in ˆ L to its geodesic representative, and the factthat the homotopy classes are pairwise distinct ensures we obtain a set of weightedgeodesic arcs on ˆ S of the same cardinality as ˆ L . Together with the measuredgeodesic lamination λ on S , they determine a measured geodesic lamination ˆ λ onˆ S . It only remains to show the uniqueness of such a measured lamination. Thisreduces to the uniqueness of the minimal matchings in the first and second stages,as follows:Let ˆ λ be measured lamination on ˆ S that restricts to measured laminations on S and C determined by the same data (i.e. the parameters described in the proof of Theorem 3.8) as that of ˆ λ . Let ˆ L be the part of the measured lamination that is notcompactly supported, comprising weighted arcs exiting the boundary cusps of thecrown. Let λ be the compactly supported part.The intersection of ˆ L with S determines a collection of weighted arcs L (cid:48) withendpoints on the boundary γ . Since the measured lamination L (cid:48) ∪ λ is, up toisotopy, equal to L ∪ λ , we conclude that λ = λ , and the arcs of L (cid:48) must constitutea splitting of the arcs of L . Similarly, the intersection of ˆ L with C determines acollection of weighted arcs G (cid:48) , that is a splitting of the arcs of G .Indeed, we shall now verify that ˆ L is obtained by the splitting-and-matching ofthe arcs G and L exactly as in the two-stage construction above.Consider the arcs ˆ L (cid:48) l of ˆ L (cid:48) that correspond to a splitting of l ∈ L . Then the end-points on γ determine two collection of points I + l and I − l , and we denote the corre-sponding collections of arcs of G (cid:48) incident on these point-sets by G (cid:48) , l , + and G (cid:48) , l , − respectively. Here G (cid:48) , l , ± are splittings of a smaller pair of arc-sets that we denoteby G (cid:48) l , ± , obtained by “combining” arcs that are asymptotic to the same boundarycusp to a single arc (with a weight equal to the total weight of the combined arcs).Note that this ensures that each G (cid:48) l , + and G (cid:48) l , − comprises arcs that are asymptotic to distinct boundary cusps.Since the original arcs ˆ L are pairwise homotopically distinct, a pair of arcs inthe splitting of l cannot connect to arcs of G (cid:48) , l , + and G (cid:48) , l , − that arise in the samesplitting (of a pair of arcs in G ), at both of its ends. Thus, the collection of arcsˆ L (cid:48) l corresponding to a splitting of l is a minimal matching of G (cid:48) l , ± , exactly as in thesecond stage above, which is unique by Lemma A.2.Finally we need to verify that the collections G (cid:48) l , ± as l varies over L , are obtainedby a minimal splitting exactly as in the first stage of the construction above. Thearcs of G (cid:48) l , ± ⊂ G (cid:48) are splittings of the arcs of G , and the arcs connect to splittingsof half-arcs of L . Hence these do constitute a matching, satisfying property (i)of Lemma A.2. Property (ii) also holds, since by construction, the arcs of G (cid:48) l , ± are asymptotic to distinct cusps. Thus (cid:83) l ∈L G (cid:48) l , ± forms a minimal matching of G and half-arcs of L , which is unique by Lemma A.2. This concludes the proof ofuniqueness, and thus the proof of the Proposition. 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