TTHE MASKIT EMBEDDING OF THE TWICEPUNCTURED TORUS
CAROLINE SERIES
Abstract.
The Maskit embedding M of a surface Σ is the space ofgeometrically finite groups on the boundary of quasifuchsian space forwhich the ‘top’ end is homeomorphic to Σ, while the ‘bottom’ end con-sists of two triply punctured spheres, the remains of Σ when two fixeddisjoint curves have been pinched. As such representations vary in thecharacter variety, the conformal structure on the top side varies over theTeichm¨uller space T (Σ).We investigate M when Σ is a twice punctured torus, using themethod of pleating rays. Fix a projective measure class [ µ ] supportedon closed curves on Σ. The pleating ray P [ µ ] consists of those groups in M for which the bending measure of the top component of the convexhull boundary of the associated 3-manifold is in [ µ ]. It is known that P is a real 1-submanifold of M . Our main result is a formula for theasymptotic direction of P in M as the bending measure tends to zero, interms of natural parameters for the 2-complex dimensional representa-tion space R and the Dehn-Thurston coordinates of the support curvesto [ µ ] relative to the pinched curves on the bottom side. This leads toa method of locating M in R . MSC classification: Introduction
Pictures of various slices and embeddings of one dimensional Teichm¨ullerspaces into C have become familiar in recent years. A common feature is thecomplicated fractal boundary which has been studied by various authors forexample [24], [25]. Such examples are always based on the once puncturedtorus and its close relatives. This paper presents for the first time a methodwhich makes viable the prospect of plotting a deformation space associatedto a higher genus surface. The project immediately introduces many diffi-culties: such a deformation space will intrinsically have at least 2 complexdimensions and the underlying combinatorics of the curve complex is notthat of the Farey tesselation associated once punctured torus.The example we choose is the Maskit embedding of the twice puncturedtorus, in which the representation variety is smooth of complex dimension 2.The key idea is explicitly to locate the pleating rays , that is, the loci in therepresentation variety along which the projective class of the bending mea-sure of the convex hull boundary is fixed. These lines are certain branches Date : October 26, 2018. a r X i v : . [ m a t h . G T ] A ug CAROLINE SERIES of the solution set of a family of equations where traces of various elementsin the group take real values. To explain in more detail, we first considerthe analogous embedding for the once punctured torus Σ , .The Maskit embedding of Σ , was initially explored experimentally byMumford and Wright, see [29, 25]. The detailed study [12] by the authorand Linda Keen introduced the concept of pleating rays which justified thesecomputational results. As explained in more detail below, these rays wereused to plot Figure 1. The lined region, which repeats periodically withperiod 2 in both directions, indicates all representations ρ : π (Σ , ) → SL (2 , C ) whose image G is free and discrete and for which one fixed es-sential non-peripheral simple curve γ ∞ ∈ π (Σ , ) is accidentally parabolic.The parameter µ ∈ C is essentially the trace of another fixed curve γ whichtogether with γ ∞ generates π . After suitable normalisation, this is enoughto determine a representation ρ . The resulting hyperbolic 3-manifold H /G is geometrically finite and homeomorphic to Σ , × R . Its end invariants ω ± are both Riemann surfaces, representing the conformal structures on thequotients of the components of the regular set by G . One end ω − is con-formally a triply punctured sphere, corresponding to the surface Σ , withthe fixed curve γ ∞ pinched. The other end ω + is a Riemann surface home-omorphic to Σ , and can thus be viewed a point in T (Σ , ). By standardAhlfors-Bers theory, each point in T is represented up to conjugation byexactly one such group G . The Maskit embedding is the map
T → C whichtakes a surface to the µ -parameter of the group G which represents it. InFigure 1, the parameter iµ = Tr γ has been chosen so that the embedding isas close to the identity map T (Σ , ) = H (cid:44) → C as possible. The embeddingrepeats periodically under translation µ (cid:55)→ µ + 2. Figure 1.
The Maskit embedding for the once puncturedtorus, showing one period in the upper half µ -plane. Theimage of T (Σ , ) is filled by the light gray pleating rays.Picture courtesy David Wright. HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 3
This paper lays the foundation for computing the analogous picture ofthe Maskit embedding when Σ = Σ , is a twice punctured torus. Therelevant component of the representation variety R (Σ) is smooth of complexdimension 2. Thus our eventual aim is to locate the image M of the Maskitembedding of the Teichm¨uller space T (Σ) in C . As for Σ , , we will do thisby locating the pleating rays, which in this case are real 1-submanifolds of M along which the projective class of the bending measure of the component ∂ C + /G of the convex hull boundary facing ω + is supported on a fixed pairof disjoint closed curves on Σ. In general, the pleating ray is a connectednon-singular branch of the real algebraic variety along which the traces ofthe support curves take real values, see Theorem 3.4. The main resultsof this paper identify the correct branch by determining its direction as theparameters of the representation tend to infinity, equivalently as the bendingmeasure tends to zero, see Theorem A. The idea is that M can then beplotted by following these real trace branches until one of the supportingcurves becomes parabolic, see Section 5.Before stating our main theorems, we briefly review our previous resultsfrom [12]. As is well known, the simple closed curves on Σ , can be enu-merated by the rationals Q ∪ ∞ . Normalize so that the exceptional pinchedcurve γ ∞ is represented by ∞ . There is one ray P p/q for each p/q ∈ Q , rep-resenting all curves γ p/q whose images in M are loxodromic. At any pointon this ray, ∂ C + /G is a pleated surface bent along γ p/q while the component ∂ C − /G facing ω − can be viewed as a copy of Σ bent along γ ∞ with bendingangle π . If a closed curve γ p/q is a bending line, then its complex length isreal, so that γ p/q is purely hyperbolic on P p/q . The trace, hence also thecomplex length, of γ p/q has no critical points on P p/q . It follows that P p/q isa totally real 1-submanifold embedded in M and that the hyperbolic length l p/q of γ p/q is strictly monotonic along P p/q with range (0 , ∞ ). As l p/q → P p/q we approach the boundary ∂ M , arriving at an algebraic limitwhich is the doubly cusped group in which l p/q = 0. As l p/q → ∞ on theother hand, there is no algebraic limit and the sequence of representationsdiverges. One of the main results of [12] is that P p/q is asymptotic to theline (cid:60) µ = 2 p/q as l p/q → ∞ , identifying it uniquely among branches ofTr γ p/q ∈ R .Turning now to the twice punctured torus Σ = Σ , , suppose we havea geometrically finite free and discrete representation for which M = Σ × R . Fix elements S , S ∈ π (Σ) corresponding to disjoint non-homotopicclosed curves σ , σ which form a maximal pants decomposition of Σ andneither of which individually disconnects Σ. We consider groups for whichthe conformal end ω − is a union of triply punctured spheres glued acrosspunctures corresponding to σ , σ , while ω + is a marked Riemann surfacehomeomorphic to Σ. In Section 2.2 we give an explicit parameterisation ofa holomorphic family of representations ρ : π (Σ) → G ( τ , τ ) , ( τ , τ ) ∈ C ,such that, for suitable values of the parameters, G ( τ , τ ) has the above CAROLINE SERIES geometry. The Maskit embedding is the map which sends a point X ∈T (Σ , ) to the point ( τ , τ ) ∈ C for which the group G ( τ , τ ) has ω + = X .Denote the image of this map by M = M (Σ , ).Given a projective measured lamination [ ξ ] on Σ, the pleating ray P [ ξ ] is the set of groups in M for which the bending measure β ( G ) of ∂ C + /G is in the class [ ξ ]. We restrict to pleating rays for which [ ξ ] is rational ,that is, supported on closed curves, and for simplicity write P ξ in place of P [ ξ ] , although noting that P ξ depends only on [ ξ ]. From general resultsof Bonahon and Otal [4] (see Theorem 3.1 in Section 3), for any pantsdecomposition γ , γ such that σ , σ , γ , γ are mutually non-homotopic andfill up Σ, and any pair of angles θ i ∈ (0 , π ), there is a unique group in M for which the bending measure of ∂ C + /G is (cid:80) , θ i δ γ i . (This extends tothe case θ i = 0 provided σ , σ , γ j fill up Σ, see Section 3, also for the case θ = π .) Thus given ξ = (cid:80) , a i δ γ i , there is a unique group G = G ξ ( θ ) ∈ M with bending measure β ( G ) = θξ for any sufficiently small θ > P ξ as θ →
0, see Corollary 6.5. Intuitively this is becausethe groups G ξ ( θ ) want to limit on a Fuchsian group, which is however impos-sible since the bending angles on the parabolic pinched curves are fixed as π .Our main result is a formula for the asymptotic direction of P ξ in M ⊂ C interms of the global linear coordinates for measured laminations on Σ , setup in [15]. These coordinates, called here canonical coordinates , assign to ameasured lamination ξ a point i ( ξ ) = ( q ( ξ ) , p ( ξ ) , q ( ξ ) , p ( ξ )) ∈ ( R + × R ) ,see Section 4. The coordinates of a simple closed curve are integral; they areessentially its Dehn-Thurston coordinates relative to σ , σ and are a closeanalogue of the p, q coordinates for curves on Σ , above. In particular, q i ( γ ) = i ( γ, σ i ) where i ( · , · ) is the usual geometric intersection number. If ξ = (cid:80) , a i δ γ i , the above Bonahon-Otal condition on σ , σ , γ , γ is equiva-lent to q i ( ξ ) > , i = 1 ,
2. We call such laminations admissible , see Section 3.We call a pair of curves γ , γ exceptional if q ( γ ) q ( γ ) = q ( γ ) q ( γ ), andwe say ξ = (cid:80) , a i δ γ i is exceptional if a i > , i = 1 , γ , γ isexceptional. The main result of this paper is: Theorem A.
Suppose that ξ = (cid:80) , a i δ γ i is admissible and not exceptional.Then as the bending measure β ( G ) ∈ R + ξ tends to zero, the pleating ray P ξ approaches the line (cid:60) τ i = 2 p i ( ξ ) /q i ( ξ ) , Arg τ i = π/ , (cid:61) τ / (cid:61) τ = q ( ξ ) /q ( ξ ) . This theorem is stated more precisely as Theorem 7.3. To actually locatethe pleating rays, note (Lemma 3.3) that Tr ρ ( γ ) is real whenever γ is abending line. We prove: Theorem B.
Suppose that ξ = (cid:80) , a i δ γ i is admissible and that the pair γ , γ is not exceptional. Then any point on the ray P ξ satisfies the equations (cid:61) Tr ρ ( γ i ) = 0 , i = 1 , and these equations have a unique solution as τ i → ∞ HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 5 in the direction specified by Theorem A. If the curve γ is admissible, thenthere exists γ disjoint from γ such that the pair γ , γ is not exceptional,and thus P γ is determined by the above result applied to ξ = 1 · δ γ + 0 · δ γ . In the exceptional case we obtain only partial results detailed in Theo-rem 7.5. We believe the above theorems to be still true in this case, butas discussed in Section 7.2, the result appears to be beyond the scope ofthis paper. The lack of a complete result will not affect the plotting of theasymptotic arrangement of pleating rays and planes.As explained in Section 5, these results in principle enable one to compute M , modulo the unproven conjecture that the rational pleating rays aredense. We hope to explore how to actually implement the computations inpractice elsewhere.Theorems A and B are proved together. The proofs have two main parts.First (Section 6) we show that asymptotically, the lengths of the geodesicrepresentatives σ +1 , σ +2 of σ , σ on ∂ C + /G tend to 0 while at the same timebecoming orthogonal to the bending lines. (This should be compared to thesituation in [28], where in the limit as the bending angles go to zero, thebending lines on ∂ C + /G and ∂ C − /G become ‘orthogonal’ in the sense thataverage of the cosine of the angle between them goes to zero.) From this wededuce (Theorem 6.1) that as θ → τ , τ → ∞ in such a way thatArg τ i → π/ , (cid:61) τ / (cid:61) τ → q ( ξ ) /q ( ξ ) . Second, we use a formula for trace polynomials from [15]. Note that thetrace Tr ρ ( γ ) is a polynomial on the parameter space C . The formula,see Theorem 4.1, expresses the top terms of this polynomial in terms of itscanonical coordinates i ( γ ). We also make use Thurston’s symplectic form onthe space of measured laminations M L , which turns out to have the standardform relative to our canonical coordinates (Section 4.2). To complete theproofs of Theorems A and B, in Section 7 we use the asymptotics of the tracepolynomials together with Thurston’s form and some simple linear algebrato extract the unique possible asymptotic directions of the pleating rays.One might also ask for the limit of the hyperbolic structure on ∂ C + /G as the bending measure tends to zero. The following result is an immediateconsequence of the first part of the proof of Theorem A: Theorem C.
Let ξ = (cid:80) , a i δ γ i be as above. Then as the bending measure β ( G ) ∈ R + ξ tends to zero, the induced hyperbolic structure of ∂ C + /G along P ξ converges to the barycentre of the laminations σ and σ in the Thurstonboundary of T (Σ) . This should be compared with the result in [28], that the analogous limitthrough groups whose bending laminations on the two sides of the convexhull boundary are in the classes of a fixed pair of laminations [ ξ ± ], is aFuchsian group on the line of minima of [ ξ ± ]. CAROLINE SERIES
Although this paper is written in the context of the twice punctured torus,the results of Section 6 and hence also Theorem C should apply to the Maskitembedding of a general surface. The top terms formula is needed only todetermine the asymptotic value of (cid:60) τ i . Y. Chiang obtained an analogoustop terms formula for the five times punctured sphere in [6]. We believethere is a more general result and hope to explore this elsewhere.The plan of the paper is as follows. In Section 2 we describe our holomor-phic family of groups which realise the Maskit embedding and give estimateson the rough shape of M . In Section 3 we briefly review facts about con-vex hull boundaries, bending measures and pleating rays. In Section 4 wereview canonical coordinates for simple curves and the top terms formulafrom [15], and discuss Thurston’s symplectic form. In Section 5 we discussin more detail how Theorem A may be used to compute pleating rays andillustrate the theorem with some very simple examples which can be com-puted by hand. The remaining two sections contain the main work of thepaper as described above. Theorem C is proved at the end of Section 6 andTheorems A and B are proved in Section 7. Acknowledgments
This paper was begun in the early 1990’s as joint project with Linda Keenand John Parker. I would like to thank them for allowing me to include someof our preliminary results and to continue alone. Most of our joint results arecontained in [15]; other work done in draft only we refer to here as [16]. Weconjectured a partial version of Theorem A but proofs were incomplete, inparticular we lacked the orthogonality idea, the use of Thurston’s symplecticform, the use of Minsky’s twist and the major general results in [4, 8].I would like to thank MSRI and the organisers of the program on
Te-ichm¨uller space and Kleinian groups for their hospitality: during the pro-gram the research for this paper was completed.2.
The Maskit embedding and plumbing parameters
Let Σ be a twice punctured torus. Figure 2 shows a fundamental domain∆ for a Fuchsian representation of Σ, on which some definite hyperbolicmetric has been fixed. The sides of ∆ are identified by hyperbolic isometries S , S , T which we can view as free generators for π (Σ).2.1. The Maskit embedding.
Let R (Σ) be the representation variety of π (Σ), that is, the set of representations ρ ( π (Σ)) → SL (2 , C ) modulo con-jugation in SL (2 , C ) (but see e.g. [10] p.61) with the algebraic topology. Let M ⊂ R be the subset of representations ρ for which(i) The image G = ρ ( π (Σ)) is free and discrete and the images of S i , i = 1 , G ) are simply connected andthere is exactly one invariant component Ω + ( G ). HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 7 S S Ts S s S s T s T − s S − s S − Figure 2.
The fundamental domain ∆(iii) The quotient Ω( G ) /G has 3 components; Ω + ( G ) /G is homeomorphicto Σ and the other two components are triply punctured spheres.In this situation, see for example [18] Section 3.8, the corresponding 3-manifold M ρ = H /G is topologically Σ × (0 , G is ageometrically finite cusp group on the boundary (in the algebraic topology)of the set of quasifuchsian representations of π (Σ). The ‘top’ componentΩ + /G of the conformal boundary may be identified to Σ ×{ } and is homeo-morphic to Σ. On the ‘bottom’ component Ω − /G , identified to Σ × { } , thetwo curves σ , σ corresponding to the generators S , S have been pinched,making Ω − /G a union of two triply punctured spheres. The conformalstructure on Ω + /G , together with the pinched curves σ , σ , are the endinvariants of M in the sense of Minsky’s ending lamination theorem. Sincea triply punctured sphere is rigid, the conformal structure on Ω − /G is fixedindependent of ρ . The structure on Ω + /G varies; it follows from standardAhlfors-Bers theory using the measurable Riemann mapping theorem (seeagain [18] Section 3.8), that there is a unique group corresponding to eachpossible conformal structure on Ω + /G . Formally, the Maskit embedding ofthe Teichm¨uller space of Σ is the mapΦ : T (Σ) → C which sends a point X ∈ T (Σ) to the unique group G ∈ M for which Ω + /G has the marked conformal structure X .2.2. A concrete realisation of M . Groups in M may be manufacturedby the plumbing construction of Kra [17], see Section 2.3 below. Here wesimply write down a suitable holomorphic family of representations andverify directly that groups thus constructed have the required properties.Groups in the family depend on two complex parameters τ , τ ∈ C . CAROLINE SERIES
To define ρ ( π (Σ)) → SL (2 , C ), it suffices to give the images of the threefree generators S , S , T of π (Σ). Following [15], for τ , τ ∈ C define ρ = ρ ( τ , τ ) by: ρ ( S ) = (cid:18) (cid:19) , ρ ( S ) = (cid:18) (cid:19) , ρ ( T ) = (cid:18) τ τ τ τ (cid:19) . Denote the image of ρ ( τ , τ ) by G ( τ , τ ). Note that the holomorphic fam-ily G = { G ( τ , τ ) : τ i ∈ C } has complex dimension 2, the dimension ofTeich(Σ). Not all groups G lie in M , in particular any representation with (cid:61) τ i = 0 , i = 1 , M . We also need to restrict τ , τ so as to have only one copy of each group up to conjugation. Byabuse of notation, for W ∈ π (Σ), we shall use W also to denote the image ρ ( W ) ∈ G ( τ , τ ), and write W = W ( τ , τ ) as needed to avoid confusion.By direct computation (see Appendix 1) we find Tr [ S i , T − ] = τ j + 2 where j = 1 + i mod 2. Thus τ , τ are invariant functions on R , so that thecomponent C of C \ (cid:61) τ i = 0 with (cid:61) τ i > , i = 1 ,
2, consists entirely ofnon-conjugate groups.The following propositions from [16] justify our use of the family G . Proposition 2.1.
Let G ( τ , τ ) ∈ G be as above. If (cid:61) τ i > , i = 1 , and (cid:61) τ (cid:61) τ > then G ( τ , τ ) ∈ M . Moreover the limit set Λ( G ) is containedin the two strips ≤ (cid:61) z ≤ / , (cid:61) τ − / ≤ (cid:61) z ≤ (cid:61) τ , together with thepoint at ∞ .Proof. A fundamental domain for G = G ( τ , τ ) is shown in Figure 3, inwhich the disks B , B have equal radius 1 / (cid:61) τ . The formal proof that G is free and discrete is a straightforward application of Maskit’s secondcombination theorem, [20] p.160. To see that G ∈ M , one checks from theproof of the combination theorem that the lower half plane and the halfplane above the line (cid:61) z = (cid:61) τ project to the two triply punctured sphereswhich together form Ω − /G , while the simply connected component Ω + iscontained in the strip 0 < (cid:61) z < (cid:61) τ . The claim about Λ also follows. Forfurther details, see Appendix 1. (cid:3) As explained above, a standard argument using the measurable Riemannmapping theorem now shows that any group in M can be represented bya group in G . Complementing Proposition 2.1 we have the following result,also proved in Appendix 1. Proposition 2.2.
Suppose that G ( τ , τ ) ∈ M . Then (cid:61) τ i ≥ / , i = 1 , and (cid:61) τ (cid:61) τ ≥ . We shall mainly do our calculations with G normalised as above, so that S ( z ) = z + 2. On occasion it is convenient to normalise S in this way, inwhich case we interchange the roles of τ and τ . More precisely we have: Lemma 2.3.
The involution K ( z ) = − / ¯ z conjugates S to S . Moreover KT ( τ , τ ) K − = T ( − ¯ τ , − ¯ τ ) − . HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 9 S S S B B l S ( l ) − τ − τ τ + 1 Figure 3.
A fundamental domain for G ( τ , τ ).To get some further feeling for M , note that the right Dehn twist D σ around σ induces the automorphism S (cid:55)→ S , S (cid:55)→ S and T (cid:55)→ S T of π (Σ). Since S T ( τ , τ ) = T ( τ + 2 , τ ), it follows that D σ induces the map( τ , τ ) (cid:55)→ ( τ + 2 , τ ) on M , and similarly for D σ .By abuse of notation, from now on we use M to denote the set of ( τ , τ )with (cid:61) τ i > G ( τ , τ ) has properties (i), (ii) and(iii) above. The above two propositions give rough bounds on the shape of M .2.3. The plumbing construction.
The parameters τ i have geometricalmeaning as the plumbing parameters of Kra [17]. The idea is to make a pro-jective structure on Σ by starting from two triply punctured spheres withpunctures identified in pairs. One ‘plumbs’ across the punctures by identi-fying punctured disk neighbourhoods D, D (cid:48) of the two paired cusps usingthe formula zw = τ where z, w are holomorphic parameters in D and D (cid:48) . Inthe hyperbolic metric, this corresponds to identifying the punctured disksby twisting by (cid:60) τ and scaling by a factor of (cid:61) τ . Making this construc-tion in the above setting, with plumbing parameter τ i across the puncturecorresponding to σ i , results in the family G .3. Bending lines and pleating rays
Let M = H /G be a hyperbolic 3-manifold, and let C /G be its convexcore, where C is the convex hull in H of the limit set of G , see [9]. If M is geometrically finite then there is a natural homeomorphism betweeneach component of ∂ C /G and of Ω /G . Each component F of ∂ C /G inheritsan induced hyperbolic structure from M . Moreover F is a pleated surface ,meaning that it is the isometric image under a map f : H → M of a hy-perbolic surface H which restricts to an isometry into M on the leaves of a geodesic lamination L on H , and which is also an isometry on each comple-mentary component of L . (Strictly, the pleated surface is the pair ( H, f );this becomes important when considering the induced marking on F .) Wecall L the bending lamination of F and the images of the complementarycomponents of L , the flat pieces of F . The bending lamination carries atransverse measure called the bending measure which describes the anglesbetween the flat pieces, see [9] for details. By a bending line of F , we willmean any complete geodesic on ∂ C /G which is either completely containedin L , or in the interior of a flat part. We also use the term bending line tomean any lift of a bending line of F to a complete geodesic in H .We shall be interested in manifolds for which the bending lamination is rational , that is, supported on closed curves. Denote the space of homo-topy classes of simple closed essential loops on F by S ( F ) and the spaceof measured laminations on F by M L ( F ). The subset of rational lamina-tions is denoted M L Q and consists of measured laminations of the form (cid:80) i a i δ γ i , abbreviated (cid:80) i a i γ i , where the curves γ i ∈ S ( F ) are disjoint andnon-homotopic, a i ≥
0, and δ γ i denotes the transverse measure which givesweight 1 to each intersection with γ i . If (cid:80) i a i γ i is the bending measure of apleated surface F , then a i is the angle between the flat pieces adjacent to γ i ,also denoted θ γ i . In particular, θ γ i = 0 if and only if the flat pieces adjacentto γ i are in a common totally geodesic subset of ∂ C /G , equivalently havelifts which lie in the same hyperbolic plane in H .We take the term pleated surface to include the case in which a closed leaf γ of the bending lamination maps to the fixed point of a rank one paraboliccusp of M . In this case, the image pleated surface is cut along γ and thusmay be disconnected. Moreover the bending angle between the flat piecesadjacent to γ is π . This is because for a geometrically finite group, thepunctures on the two components of ∂ C /G are paired and the associatedflat pieces lift to two tangent half planes in H , see e.g. [18] Chapter 3 fordetails.The key results about the existence of hyperbolic manifolds with pre-scribed bending laminations are due to Bonahon and Otal. We need thefollowing special case of their main theorem: Theorem 3.1 ([4] Theorem 1) . Suppose that M is -manifold homeomor-phic to Σ × (0 , , and that ξ ± = (cid:80) i a ± i γ ± i ∈ M L Q (Σ) . Then there exists ageometrically finite group G such that M = H /G and such that the bendingmeasures on the two components ∂ C ± /G of ∂ C /G equal ξ ± respectively, ifand only if a ± i ∈ (0 , π ] for all i and { γ ± i , i = 1 , . . . , n } fill up Σ , equivalentlyif i ( ξ + , γ ) + i ( ξ − , γ ) > for every γ ∈ S . If such a structure exists, it isunique. Specialising now to the case of interest to this paper, let ρ = ρ ( τ , τ )be a representation π (Σ) → SL (2 , C ) in the family G from Section 2, andsuppose that the image G = G ( τ , τ ) ∈ M . The boundary of the convexcore C /G has three components, one ∂ C + /G facing Ω + /G and homeomorphic HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 11 to Σ, and two triply punctured spheres whose union we denote ∂ C − /G . Theinduced hyperbolic structures on the two components of ∂ C − /G are rigid,while the structure on ∂ C + /G varies. We denote the bending lamination of ∂ C + /G by β ( G ) ∈ M L . Following the discussion above, we view ∂ C − /G asa single pleated surface with bending lamination π ( σ + σ ), indicating thattwo triply punctured spheres are glued across the annuli whose core curves σ and σ correspond to the parabolics S i ∈ G . Corollary 3.2.
A lamination ξ ∈ M L Q (Σ , ) is the bending measure of agroup G ∈ M if and only if i ( ξ, σ ) , i ( ξ, σ ) > . If such a structure exists,it is unique. We call ξ ∈ M L Q (Σ , ) for which i ( ξ, σ ) , i ( ξ, σ ) > admissible .3.1. Pleating rays.
Denote the set of projective measured laminations onΣ , by P M L and the projective class of ξ = a γ + a γ ∈ M L by [ ξ ]. The pleating ray P = P [ ξ ] of ξ ∈ M L is the set of groups G ∈ M for which β ( G ) ∈ [ ξ ]. To simplify notation we write P ξ for P [ ξ ] and note that P ξ depends only on the projective class of ξ , also that P ξ is non-empty if andonly if ξ is admissible. In particular, we write P γ for the ray P [ δ γ ] . As β ( G )increases, P ξ limits on the unique geometrically finite group G cusp ( ξ ) in thealgebraic closure M of M at which at least one of the support curves to ξ isparabolic, equivalently so that β ( G ) = θ ( a γ + a γ ) with max { θa , θα } = π . We write P ξ = P ξ ∪ G cusp ( ξ ).Likewise for disjoint non-homotopic curves γ , γ ∈ S , we define the pleating plane P γ ,γ of γ , γ to be the set of groups G ∈ M for which β ( G ) = (cid:80) a i γ i with a i >
0. Thus P γ ,γ is the union of the pleating rays P ξ with ξ = (cid:80) a i γ i , a i >
0. The rays P γ , P γ are clearly contained inthe boundary of P γ ,γ ; note that γ i may not be admissible even though ξ = (cid:80) a i γ i is. We call planes for which one or other of the support curves isnot admissible degenerate , as they do not contain the corresponding ray P γ i .We write P γ ,γ = ∪ η P η where the union is over η = (cid:80) a i γ i with a , a ≥ ∂ C /G on eitherside of a bending line are invariant under translation along the bending line,the translation can have no rotational part. Lemma 3.3.
If the axis of g ∈ G is a bending line of ∂ C /G , then Tr( g ) ∈ R . Notice that the lemma applies even when the bending angle θ γ along γ vanishes. Thus if G ∈ P γ ,γ we have Tr g ∈ R , i = 1 ,
2, for any g ∈ G whoseaxis projects either curve γ i .In order to compute pleating planes, we need the following result whichis a special case of Theorems B and C of [8], see also [12]. Recall that acodimension- p submanifold N (cid:44) → C n is called totally real if it is definedlocally by equations (cid:61) f i = 0 , i = 1 , . . . , p , where f i , i = 1 , . . . , n are localholomorphic coordinates for C n . As usual, if γ is a bending line we denote its bending angle by θ γ . Recall that the complex length cl ( A ) of a loxodromicelement A ∈ SL (2 , C ) is defined by Tr A = 2 arc cosh cl ( A ) /
2, see e.g. [27]or [8] for details. By construction, P γ ,γ ⊂ M ⊂ R (Σ). Theorem 3.4.
The complex lengths clγ , clγ are local holomorphic coordi-nates for R (Σ) in a neighbourhood of P γ ,γ . Moreover P γ ,γ is connectedand is locally defined as the totally real submanifold (cid:61) Tr γ i = 0 , i = 1 , of R . Any pair ( f , f ) , where f i is either the hyperbolic length (cid:60) cl ( γ i ) or thebending angle θ γ i , are global coordinates on P γ ,γ . This result extends to P γ ,γ , except that one has to replace (cid:60) cl ( γ i ) byTr γ i in a neighbourhood of a point for which γ i is parabolic. In fact asdiscussed in [8] Section 3.1, complex length and traces are interchangeableexcept at cusps (where traces must be used) and points where a bendingangle vanishes (where complex length must be used). The parameterisationby lengths or angles extends to P γ ,γ .Notice that the above theorem gives a local characterisation P γ ,γ as asubset of the representation variety R and not just of M . In other words,to locate P , one does not need to check whether nearby points lie a priori in M ; it is enough to check that the traces remain real and away from andthat the bending angle on one or other of θ γ i does not vanish. As we shallsee, this last condition can easily be checked by requiring that further tracesbe real valued.4.
Canonical coordinates for simple curves
Our main result Theorem A involves the explicit coordinatisation of thespace
M L = M L (Σ , ) of measured laminations on Σ , introduced in [15].The coordinates, called π , -coordinates in [15] and canonical coordinates inthis paper, are essentially Dehn-Thurston coordinates relative to the curves σ , σ . They are global coordinates for M L which take values in ( R + × R ) ,the sign of a given coordinate in a given chart being constant. At the sametime, they can be viewed as giving a piecewise linear cone structure to M L ,the charts being laminations supported on a particular set of train trackson Σ. Modulo their boundaries, these charts partition
M L ; which chartsupports a given lamination being determined by simple linear inequalitiesbetween their coordinates. The coordinates are set up so as to maintain asclose an analogy as possible with the once punctured torus Σ , . The chartsare also very closely related to the π -train tracks of [2].In more detail, the canonical coordinates i ( γ ) = ( q ( γ ) , p ( γ ) , q ( γ ) , p ( γ )) ∈ ( Z + × Z ) of a curve γ ∈ S (Σ , ) are defined as follows. We set q i ( γ ) = i ( γ, σ i ) ≥ σ , σ together bound a pair of pants we note:(1) q + q ∼ = 0 mod 2 . This equation will be important in Section 7.
HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 13
The definition of p i ( γ ) (which is more complicated and can be omitted atfirst reading) is made relative to the fundamental domain ∆ of Section 2.2.The reader may find the discussion for Σ , in Appendix 2 enlightening. B B s v v s S s S s T s T − s S − s S − p p q − p χ χχ + q χ + q | p | − q Figure 4.
Typical configuration for canonical coordinates.The cell shown is defined by the five inequalities p ≥ , p ≤ , q ≥ p , q ≤ − p , q ≤ q .Referring to Figure 4, label each side of ∆ by the generator which carriesit to a paired side, so that the bottom side is labelled s T since T carries it tothe top side s − T ; similarly the top left side is labelled s S since S carries itto the right side s S − ; and the lower left and right sides are labelled s S , s S − respectively. Since each side joins a puncture to a puncture, the intersectionnumbers of a curve γ ∈ S with these sides are well defined. Let s be thearc joining the vertices v , v in the middle of the vertical sides of ∆, whichboth project to the puncture S S − on Σ , . This partitions ∆ into two foursided ‘boxes’ B , B with sides s , s S i , s S i − and s T − or s T respectively.The geodesic representative of γ on ∆ intersects each box B i in a numberof pairwise disjoint arcs, none of which runs from one side of B i to itself.We observe, see [15] Lemma 4.1, that in at least one box B i , there is eitherno ‘corner arc’ joining s to s S i , or no corner arc joining s to s S i − . This isbecause if strands of γ included all four corner arcs in both boxes, the ‘inner-most’ such arcs would link to form a loop round the puncture S S − which is the projection to Σ , of v and v . This is impossible. In a similar way, therecannot be corner arcs surrounding all of the four vertices w , . . . , w markedin Figure 4, since their innermost strands would link to form a loop round thecommon projection of the w i , the puncture S T S − T − . Using this togetherwith the ‘switch conditions’ i ( γ, s S i ) = i ( γ, s S − i ) and i ( γ, s T ) = i ( γ, s T − ),one checks (see [15] Section 4) that i ( s , γ ) = i ( s , s T ) = i ( s , s T − ). It alsofollows that there are equal number of arcs joining each pair of diagonallyopposite corners of each B i .Suppose for definiteness that the box with missing corner arcs is B .In this situation, it is not hard to see that q ( γ ) = i ( γ, σ ) = i ( γ, σ ).In analogy with the case of Σ , as described in Appendix 2, we define | p | = i ( γ, s S ) = i ( γ, s S − ). Since i ( γ, s ) ≥ i ( γ, σ ) = q ( γ ), we have q ≥ q . Set χ = | q − q | /
2. (Note that χ is integral by (1).) One verifies that q ( γ ) = i ( γ, s ) − χ ≥ | p | = i ( γ, s S ) − χ = i ( γ, s S − ) − χ .If B is the box with missing corner arcs, make similar definitions with theroles of p , p reversed.To fix the sign of p , take p > γ joining s S to s T − and p ≤ p > w > χ arcsof γ joining s S to s and p ≤ q ≤ q we make similardefinitions interchanging the indices 1 , B i as a fundamental domain for a once punctured torus Σ i , with oppositesides identified. With such an identification, the arcs of γ in B i would glueup to form a multiple loop on Σ i , . As is well known, closed curves onΣ , are in bijective correspondence with lines of rational slope in R . Thecoordinate ( q i , p i ) indicates that the ‘slope’ of γ in B i is p i /q i , see Appendix2. The number χ is the number of ‘corner strands’ joining adjacent sidesof B i which, were the box actually Σ , , would link to form χ copies of aloop round the puncture. Such corner strands occur in box B if and only if q > q ; this is why we subtracted χ from the B -intersection numbers only.Note that, in contrast to Σ , , the numbers q i , p i are no longer in generalrelatively prime. Up to the choice of a base point for the twist, p i /q i is theDehn-Thurston twist coordinate of γ relative to σ i , see [23] Lemma 3.5 andSection 6.4 below.The connection between this definition and the more standard cell de-composition of M L by weighted train tracks is as follows. Collapse all thearcs of γ joining one side of B i to another, into a single strand joining themidpoints of the same two sides. The collapsed strands join across s toform a train track on Σ , , whose branches are the strands and all of whoseswitches are at the midpoints of the sides (including s ). In [15] we calledthese special tracks, π , -train tracks; here we refer to them as canonical .The non-negative weights on this collection of train tracks form a cell decom-position of M L (Σ , ). However the coordinates ( | p i | , q i ) are not in generalequal to the actual weights on these rather complicated configurations of HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 15 branches. Rather, i ( γ ) has global meaning. Inequalities among the coeffi-cients of i ( γ ) determine which track in the cell decomposition supports agiven curve γ and, given the cell, the coefficients determine the weights onthe track.Canonical coordinates extend naturally by linearity and continuity toglobal coordinates for M L (Σ , ). The cell structure is best understood byreferring to Figure 9 in Appendix 2. This shows the four cells for Σ , whichglue to form P M L (Σ , ) = S . Each of the four configurations in Figure 9can occur in either B or B , and for each such configuration there are 2further options for the box in which q i > q j , This makes in all 32 cells, theimages of which in P M L glue along their faces to make S , see [15].An important feature of canonical coordinates is that it is easy to readoff the coordinates of a curve γ W ∈ S represented by a word W in thegenerators S , S , T of π (Σ). First, write W as a cyclically shortest word e e . . . e n and set e n +1 = e . Draw arcs on ∆ from s e i to s e − i +1 , i = 1 , . . . , n .Suppose that γ W is simple on Σ. Then by [15] Theorem 3.1, these arcs canbe arranged so as to be pairwise disjoint and the weighted canonical trackthey define gives precisely the canonical coordinates of γ W . This method,similar to the method of π -train tracks developed at length in [2], wascrucial in the proof of the top terms formula below. Some examples aregiven in Section 5.4.1. Top terms formula.
Canonical train tracks and coordinates wereused in [15] to study matrices ρ ( W ) in the family G of Section 2.2, where W ∈ π (Σ , ) corresponds to γ W ∈ S . The matrix coefficients and hencethe trace Tr W are clearly polynomials in τ , τ . Theorem 4.1 ([15] Theorem 6.1) . Let γ be a simple closed curve on Σ with canonical coordinates i ( γ ) = ( q , p , q , p ) . Let γ be represented by W ∈ π (Σ) . Then if q , q > : Tr W = ± | q − q | (cid:0) τ + 2 p /q (cid:1) q (cid:0) τ + 2 p /q (cid:1) q + R (cid:0) q + q − (cid:1) where R ( q + q − denotes a polynomial of degree at most q in τ and q in τ and with total degree in τ and τ at most q + q − . If q = 0 then Tr W is a polynomial in τ only, and Tr W = ± q (cid:0) τ + 2 p /q (cid:1) q + R (cid:0) q − (cid:1) , while if q = 0 there is a similar expression in τ . The Thurston symplectic form.
Thurston defined a symplectic formΩ Th on M L , the symplectic product being defined for curves carried bya common train track τ , see for example [26]. By splitting, we can ar-range that every switch v of τ is trivalent with one incoming branch andtwo outgoing ones. Since Σ is oriented, we can distinguish the right and left hand outgoing branches, the left hand branch being the one to be fol-lowed by a British driver approaching v from the incoming branch, see Fig-ure 5. If n , n (cid:48) are non-negative weightings on τ (representing points in M L ), we denote by b v ( n ) , c v ( n ) the weights of the left hand and right handoutgoing branches at v respectively. The Thurston product is defined asΩ Th ( n , n (cid:48) ) = (cid:80) v b v ( n ) c v ( n (cid:48) ) − b v ( n (cid:48) ) c v ( n ). incoming branch c v ( n ) b v ( n ) vwb w ( n ) c w ( n ) Figure 5.
Weighted branches at a switch.Bearing in mind the interpretation of canonical coordinates as weights ontrain tracks, we can interpret this definition in terms of canonical coordi-nates. It is then not hard to check the following rather remarkable result:
Proposition 4.2.
Suppose that loops γ, γ (cid:48) ∈ S are supported on a com-mon canonical train track with coordinates i ( γ ) = ( q , p , q , p ) , i ( γ (cid:48) ) =( q (cid:48) , p (cid:48) , q (cid:48) , p (cid:48) ) . Then Ω Th ( γ, γ (cid:48) ) = (cid:80) i =1 , ( q i p (cid:48) i − q (cid:48) i p i ) . If γ, γ (cid:48) are disjoint,then Ω Th ( γ, γ (cid:48) ) = 0 . To check the last statement, note first that disjoint curves are necessarilysupported on the same canonical track. Then check that Ω Th is invariantunder splitting and shifting. Then split and shift until the two curves aresupported on disjoint train tracks.5. Computation and examples
Before embarking on the proof of Theorem A, we briefly discuss its im-plications for computation and then give a few examples which it is possibleto work out by hand.5.1.
Computation.
Let us return to the original problem of locating M .Conjecturally, the pleating planes are dense in M . (This was proved for M (Σ , ) in [12].) Thus we concentrate on the problem of locating a givenpleating plane P γ ,γ in the parameter space H ⊂ C . Let V > ( γ i ) , V =2 ( γ i )denote respectively the real analytic varieties in H on which Tr γ i ∈ R \ [ − ,
2] and Tr γ i = ±
2. By Theorem 3.4, P γ ,γ is a connected totally realsubmanifold of V > ( γ ) ∩ V > ( γ ). Its boundary is contained in V =2 ( γ ) ∪V =2 ( γ ) and the two rays P γ and P γ . By Theorem B, as long as thepair γ , γ is not exceptional (for which see Lemma 7.4 and Theorem 7.5), V > ( γ ) ∩ V > ( γ ) is a 2-manifold and has a unique branch near infinitysatisfying the conditions of Theorem A. HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 17
The ray P γ may be determined as follows. At points on P γ , the surface ∂ C + /G cut along γ is flat. Therefore the trace of any curve δ disjoint from γ lies in the variety V > ( δ ). One can show (see Lemma 7.2) that of any twoadmissible curves δ, δ (cid:48) disjoint from γ and distinct from γ , at least one willhave the property that V > ( δ ) ∪ V > ( γ ) is transverse to P γ ,γ , so that theequations (cid:61) γ , (cid:61) δ, (cid:61) δ (cid:48) ∈ R together with the conditions of Theorem A areenough to determine P γ . Finally, since P γ ,γ is the union of the rays P ξ for ξ with support γ , γ , it must the connected component of the unique branchof V > ( γ ) ∩ V > ( γ ) which interpolates between P γ and P γ as τ i → ∞ .This means that in principle, given a means of computing sufficientlymany traces, P γ ,γ can be determined computationally without needing anyfurther tests to see if G is discrete . In particular, one can locate the one(real)-dimensional boundary of P γ ,γ on ∂ M , along which at least one ofthe elements γ , γ is parabolic. By [21], such geometrically finite groupcusp groups are dense in ∂ M . In the one dimensional case, this is exactlythe procedure which gives Figure 1. We hope to discuss the practical imple-mentation of this programme elsewhere.5.2. Examples.
Following [16], we will identify pleating rays and planesformed by various combinations of the curves corresponding to the elements W = S i , T, [ S i , T − ] , S S − T − and S − S T − in π (Σ). Let γ W be thecurve corresponding to element W and write P W for P γ W and Tr W forTr ρ ( W ) with ρ = ρ ( τ , τ ) defined as in Section 2.2.We compute: Tr T = 2 + τ τ , Tr [ S i , T − ] = 2 + 4 τ j , j (cid:54) = i (see Appendix1), and Tr S S − T − = ( τ − τ + 2), where Tr S S − T − can either becomputed directly or deduced from the fact that the right Dehn twist D σ i about S i induces the map τ i (cid:55)→ τ i + 2, see Section 2.2. Writing the canonicalcoordinates in the usual form i ( ξ ) = ( q ( ξ ) , p ( ξ ) , q ( ξ ) , p ( ξ )) , we find following the method outlined in Section 4: i ( T ) = (1 , , , , i ([ S , T − ]) = (0 , , , , i ([ S , T − ]) = (2 , , , , i ( S S − T − ) = (1 , − , , , i ( S − S T − ) = (1 , , , − . The curves S S − T − and S T − S − T are illustrated in Figure 6.The pleating planes discussed in Examples 1–3a below are shown in theleft frame of Figure 7 and those of Example 4 on the right. Example 1: The pleating ray P T . On P T , the surface Σ T := Σ \ γ T is flat and hence not only γ T , but also any curve contained in Σ T , has realtrace. Since γ T is disjoint from both γ [ S ,T − ] and γ [ S ,T − ] , it follows that2 + 4 τ j ∈ R , i = 1 , τ τ ∈ R . We deduce that on P T , (cid:60) τ = (cid:60) τ = 0. Since γ T is also disjoint from γ S S − T − , by the same reasoning, τ τ + 2 τ − τ ∈ R , from which we deduce (cid:61) τ = (cid:61) τ . (Disjointnessof curves can be easily checked by drawing disjoint representatives on the B B s S s S s T s T − s S − s S −
111 1 s S s S s T s T − s S − s S −
111 11 1
Figure 6.
The curves S S − T − (left) and S T − S − T (right).fundamental domain ∆, taking into account how the arcs link across theglued sides. In this instance, the curve T is an arc from s T to s T − which isclearly disjoint from both curves illustrated in Figure 6.)The conditions (cid:60) τ = (cid:60) τ = 0 , (cid:61) τ = (cid:61) τ define a line L in C . Moreover | Tr T | > L whenever (cid:61) τ >
2, and T is parabolic exactly at thepoint at which (cid:61) τ = 2. It follows from Theorem 3.4 and the discussionabove, that P T = L ∩ {(cid:61) τ > } . To compare with Theorem A, since p i ( T ) /q i ( T ) = 0 , i = 1 , q ( T ) /q ( T ) = 1, in this special case, P T isactually equal to the line (cid:60) τ i = 2 p i ( T ) /q i ( T ) , Arg τ i = π/ , (cid:61) τ / (cid:61) τ = q ( T ) /q ( T ) . The endpoint point E T = (2 i, i ) of P T represents the unique group G cusp ( T )for which S , S and T are all parabolic and the remaining part of ∂ C + /G is totally geodesic.In this very special case, it is also possible to verify directly, following themethods of [12], that groups on L have convex hull boundary which is bentexactly along the curve γ T , and that by symmetry the geodesic axis of γ T meets both the geodesic representatives of σ i = γ S i on ∂ C + /G orthogonally. Example 2: The pleating ray P S S − T − . The easy way to locate thisray is to note that S S − T − = D σ D − σ ( T ). Since D σ i is a symmetry of M ,it follows immediately that P S S − T − is the line (cid:60) τ = − , (cid:60) τ = 2 , (cid:61) τ = (cid:61) τ , (cid:61) τ >
2, in other words, the line L + 2( − ,
1) where L is as in Example1. Since in this case p /q = − , p /q = 1 and q /q = 1, this is again inaccordance with Theorem A. HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 19 , − − , P P P T P S S − T P S − S T E T E E S S − T τ = − ¯ τ < τ O = τ = τ (1 ,
4) (4 , , P P < τ = < τ = 0 Figure 7.
Pleating planes. Left: Examples 1 – 3a. Right:Example 4.The point E S S − T − = ( − i, i ) is the unique group for which S , S and S S − T − are all parabolic and the remaining part of ∂ C + /G istotally geodesic. Example 3: The pleating plane P = P T,S S − T − . By Theorem 3.4 andthe discussion above, P is a connected open subset of the plane Π defined by τ = − ¯ τ whose boundary contains the lines P T and P S S − T − . To computethe remaining boundary of P , note that the conditions | Tr T | , | Tr S S − T − | > | τ | > , | τ − | >
4. The equations | τ | = 2 , | τ − | = 2 definetwo circular arcs which meet at the point E at which (cid:60) τ = − , (cid:61) τ = √ T is parabolic along the arc of | τ | = 2 from E T to E and S S − T − is parabolic along the arc of | τ − | = 2 from E S S − T − to E . At E , both T and S S − T − are parabolic. Thus E represents a double cusp group inwhich T, S S − T − , S , S are all parabolic. (There is a unique such group,either by the ending lamination theorem, or more simply by [11].) It followsfrom the above discussion that P T,S S − T − = { ( τ , τ ) | τ = − ¯ τ , | τ | > , | τ − | > , − < (cid:60) τ < } , as illustrated in the left frame of Figure 7.The next example shows that distinct planes may be contained in thesame real trace variety in C . Example 3a: The pleating plane P = P T,S − S T − . Similarly to Ex-ample 2, we compute that P S − S T − is the line L + 2(1 , − P T,S − S T − is the region containedthe same plane Π, but bounded by the lines P T and P S − S T − and above the arcs | τ | = 2 and | τ + 2 | = 2. In other words P T,S − S T − = { ( τ , τ ) | τ = − ¯ τ , | τ | > , | τ + 2 | > , < (cid:60) τ < } . Thus the two pleating planes P and P are both contained same planeΠ ⊂ C . They meet along the line P T contained in the boundary of both,see the left frame of Figure 7.Our final example is of degenerate pleating planes. Example 4: The pleating planes P T, [ S i ,T − ] . In Example 3, the curves T and S S − T − are themselves are admissible, so that the correspondingpleating rays are non-empty. By contrast, since q = i ([ S , T − ] , S ) = 0,the curve [ S , T − ] by itself is not admissible so that P [ S ,T − ] = ∅ .From Tr T = 2+ τ τ , Tr [ S , T − ] = 2+4 τ we find P T, [ S i ,T − ] is containedin the plane Π (cid:48) = { ( τ , τ ) ∈ C : (cid:60) τ = (cid:60) τ = 0 } . To locate P T, [ S i ,T − ] , welocate its boundary curves in Π (cid:48) . Part of the boundary is the line P T = L of Example 1, along which (cid:61) τ = (cid:61) τ . In Π (cid:48) , the element [ S , T − ] isparabolic along the line τ = i and T is parabolic along the hyperbola {(cid:61) τ (cid:61) τ = 4 } . These two loci meet exactly once at the point (4 i, i ) whichtherefore represents the maximally pinched group for which S , S , T and[ S i , T − ] are all parabolic. We conclude that P T, [ S ,T − ] is the region P inFigure 7 bounded by the line { ( ti, ti ) : t ≥ } , the arc A of {(cid:61) τ (cid:61) τ = 4 } from (2 i, i ) to (4 i, i ) along which T is parabolic; and the line A = { ( ti, i ) : t ≥ } along which [ S , T ] is parabolic. Along A , the bending angle θ [ S ,T − ] increases from zero to π while θ T ≡ π ; along A , we have θ [ S ,T − ] ≡ π while θ T decreases from π to the unattainable value 0.The individual pleating ray P ξ with ξ = a T + a [ S i , T ], is asymptotic tothe line (cid:61) τ / (cid:61) τ = q ( ξ ) /q ( ξ ) = a / ( a + a ); the missing fourth bound-ary curve of P T, [ S ,T − ] would correspond to the non-existent ray P [ S ,T − ] asymptotic to the line (cid:61) τ / (cid:61) τ → q ([ S , T − ]) /q ([ S , T − ]) = 0.A similar argument shows that P T, [ S ,T − ] = { ( τ , τ ) ∈ Π (cid:48) : 1 ≤ (cid:61) τ ≤ (cid:61) τ , (cid:61) τ (cid:61) τ ≥ } , the region P in the figure. Thus P and P are contained in the same planein C . Also note that the curves γ S S − T − , γ S − S T − , γ [ S ,T − ] , γ [ S ,T ] are all disjoint from γ T and that the ray P T is the common boundary of allfour pleating planes P , . . . , P .6. Behaviour on a pleating variety
The heart of the proof of Theorem A is the geometry of ∂ C + /G for groups G = G ξ ( θ ) ∈ P ξ as θ →
0. Let σ + = σ + i denote the geodesic representativeof σ i on ∂ C + /G and let l + σ = l + σ i be its hyperbolic length in the hyperbolicstructure on ∂ C + /G . We show that l + σ → θ →
0, while σ + becomesasymptotically orthogonal to the bending lines. From this we deduce results HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 21 on the asymptotic behaviour of τ , τ . The main result of this section willbe: Theorem 6.1.
Fix ξ = (cid:80) , a i δ γ i as above. Let G = G ξ ( θ ) be the uniquegroup in M with β ( G ) = θξ . Then (cid:60) τ i = − p i ( ξ ) /q i ( ξ ) + O (1) and (cid:61) τ i = 4(1 + O ( cθ )) /θq i ( ξ ) where c is a constant depending q ( ξ ) , q ( ξ ) and O (1) denotes a universalbound independent of ξ , as θ → . Corollary 6.2.
Then | Arg τ i − π/ | ≤ c (cid:48) θ and (cid:12)(cid:12)(cid:12) q ( ξ ) q ( ξ ) − (cid:61) τ (cid:61) τ (cid:12)(cid:12)(cid:12) ≤ c (cid:48)(cid:48) θ where c (cid:48) , c (cid:48)(cid:48) > are constants depending on ξ , as θ → . Theorem 6.1 and the Corollary follow immediately from Propositions 6.6, 6.11and 6.14. At the end of the section, we also prove Theorem C.6.1.
A note on constants.
Throughout this section, we will make manyestimates of the form X ( σ i ) ≤ O ( θ e ), where X is some quantity whichdepends on the curve σ i , meaning that X ≤ cθ e as θ → c >
0, and e is an exponent (usually e = 1 , / ξ , so more precisely one has X ≤ c ( ξ ) θ e . Howeverit is easily seen by following through the arguments that the dependenceon ξ is always of the form X ( σ i ) ≤ cq i ( ξ ) e θ e , where now c is a universalconstant independent of ξ . The dependence of the constants on ξ is notimportant for our final arguments in Section 7, but we note it as it may beuseful elsewhere.In what follows, X ≥ − O ( θ ) Y means that X ≥ − cY θ for some constant c > Length estimates on ∂ C + . We begin by estimating the lengths l + σ i .We prove two main results, Propositions 6.4 and 6.6, which relate l + σ i to θ and τ respectively. The first shows in particular that l + σ i → θ → Lemma 6.3.
Let λ be a piecewise geodesic arc in H with endpoints P and P (cid:48) , and let ˆ λ be the H geodesic joining P to P (cid:48) . Suppose that for all X ∈ λ the angle between P X and λ is bounded in modulus by a ∈ (0 , π/ . Then l ˆ λ ≥ (cos a ) l λ for all X ∈ λ , where l λ and l ˆ λ are the lengths of λ and ˆ λ respectively.Proof. Join P to a variable point X on λ between P and P (cid:48) (see Figure 3in [28]). If P X has length x , the distance from P to X along λ is t , and theacute angle between P X and ˜ S + at X is ψ , then at every non-bend pointof λ , one has dx/dt = cos ψ , see [5] Lemma 4.2.12. (cid:3) Proposition 6.4.
Let ξ ∈ M L Q be admissible. As usual, let G ξ ( θ ) ∈ P ξ bethe unique group G ∈ M with β ( G ) = θξ . Then l + σ i ≤ i ( β ( G ξ ( θ )) , σ i )(1 + O ( θ )) = θi ( ξ, σ i )(1 + O ( θ )) as θ → .Proof. Here and in what follows, we work in the upper half space model of H and let ∂ C + denote the lift of ∂ C + /G to H . Using Lemma 2.3 if needed,normalise such that S i = S : z (cid:55)→ z + 2. Let ˜ S + = ˜ S + i denote the lift of σ + i to ∂ C + invariant under translation S . Then ˜ S + is made up a number of H -geodesic segments which meet at bending points where ˜ S + crosses a bendingline of ∂ C + . Choose a bending point P = P , let P , P , . . . , P k , P k +1 = P (cid:48) be in order the bending points along one period of ˜ S + between P and P k +1 = S ( P ), and let s i be the segment from P i − to P i , see Figure 8.Let φ i denote the exterior angle between the geodesic segments s i , s i +1 of˜ S + which meet at P i , measured so that φ i ≥ φ i = 0 meansthat s i , s i +1 are collinear. (This will be the case exactly when the bendingline containing P i is contained in the interior of a flat piece of ∂ C + .) If θ i isthe bending angle of ∂ C + at P i , then φ i ≤ θ i , see Appendix 3. Hence(2) k (cid:88) i =0 φ i ≤ i ( σ, β ( G ξ ( θ )) = θi ( σ, ξ ) . uφ w/ P v w/ P φ φ i φ k P i s i s i +1 E = z = 0 Figure 8.
The geodesic representative ˜ S + of σ on ∂ C + . HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 23
Now consider the triangle ∆ with vertices ∞ , P, P (cid:48) . Since P, P (cid:48) are atthe same Euclidean height above C in H , the angles in ∆ at P and P (cid:48) areequal, to w/ E be the (non-planar) polygon bounded by the H -geodesic P P (cid:48) and the segments s i of ˜ S + , and let u, v be the interior anglesin E at P, P (cid:48) respectively, see Figure 8. The Euclidean translation S carriesthe Euclidean configuration at P to that at P (cid:48) = S ( P ). It follows (usingthe triangle inequality in the spherical metric on the link of P ) that(3) φ + u + v + w ≥ π. Summing over the interior angles of E gives(4) k (cid:88) ( π − φ i ) + u + v < kπ. Combining (2), (4) and (3), we find(5) π − w < k (cid:88) φ i ≤ θi ( σ, ξ ) . Now the angles in ∆ at P and P (cid:48) are both w/
2, so by the angle ofparallelism formula, writing d = d H ( P, P (cid:48) ),sinh d/ w/ , so that from (5) we findsinh d/ ≤ θi ( σ, ξ )(1 + O ( θ )) / . Finally, we claim that l + σ ≤ d (1 + O ( θ ))for small θ , from which the result is immediate. This follows easily fromLemma 6.3. We have only to see that the angle ψ between the line P X from P to any point X on ˜ S + is bounded above by θi ( σ, ξ ). This follows sincealong the interior of any segment ψ decreases as x increases, and since atthe bend point P i it increases by at most θ i . This gives d ≥ l + σ (1 − O ( θ ))and the result follows. (cid:3) Corollary 6.5.
Let ξ ∈ M L Q be admissible and suppose that G ( τ , τ ) isthe unique group G ξ ( θ ) in M with β ( G ) = θξ . Then / (cid:61) τ i ≤ O ( θ ) , i = 1 , ,as θ → . Moreover the groups G ξ ( θ ) have no algebraic limit as θ → .Proof. As above, we work in the upper half space model and we assume G = G ξ ( θ ) normalised so that S = S i is the translation z (cid:55)→ z + 2. Define P, P (cid:48) as before. Let k be the Euclidean height k of P above C . Then1 /k = sinh d H ( P, P (cid:48) ) / d H ( P, P (cid:48) ) ≤ l σ + ≤ O ( θ ).It follows that 1 /k ≤ O ( θ ).Now P lies on a bending line ζ of ∂ C + which is contained in a supportplane to ∂ C + . This plane is a hemisphere H ⊂ H which meets ˆ C in a circle C . Since the convex core is contained entirely to one side of H , there are no limit points in one of the two discs in ˆ C bounded by C . Since the horizontallines (cid:61) z = 0 and (cid:61) z = (cid:61) τ i are contained in the limit set Λ, and since thehalf planes (cid:61) z < (cid:61) z > (cid:61) τ i are contained in Ω − , it follows that C iscontained in { z : 0 ≤ (cid:61) z ≤ (cid:61) τ i } and hence that its diameter is at most (cid:61) τ i .We deduce 2 / (cid:61) τ i ≤ /k so that 1 / (cid:61) τ i ≤ O ( θ ) as claimed.To prove that the algebraic limit of a sequence of groups does not exist,it is enough to show that the trace of some element becomes infinite. Theresult follows on recalling from Section 2.2 that Tr [ T, S − i ] = τ j + 2 , j (cid:54) = i mod 2. (cid:3) Now we establish a more precise link between l + σ i and (cid:61) τ i : Proposition 6.6.
Let ξ ∈ M L Q be admissible and suppose that G ( τ , τ ) isthe unique group G ξ ( θ ) ∈ M with β ( G ) = θξ . Then (cid:61) τ i (1 − O ( θ )) ≤ /l + σ i ≤ (cid:61) τ i (1 + O ( θ )) as θ → . Remark 6.7.
Kra [17] p. 568 gives the estimate (cid:61) τ i − ≤ π/l σ i (Ω + ) ≤ (cid:61) τ i for the hyperbolic length l σ i (Ω + ) of the geodesic representative of σ onΩ + /G . Combining this with Sullivan’s theorem [9], gives an alternativeproof of the final statement of Corollary 6.5. The discrepancy of the π/ G becomes asymptotically Fuchsian, thestructures on ∂ C + /G and Ω + /G would be asymptotically equal. For a quickconfirmation of Kra’s estimate, note that Ω + contains an S invariant stripof width (cid:61) τ −
1, so that there is an annular collar A of approximate modulus (cid:61) τ / S on Ω + . Kra’s estimate follows from the formula mod A = π/l σ (Ω + ) + O (1), see for example [23] p.255.We begin the proof of Proposition 6.6 with two lemmas. Lemma 6.8.
Let ξ ∈ M L Q be admissible. Normalize G ξ ( θ ) so that S i istranslation z (cid:55)→ z + 2 . Let γ be a bending line of the pleating lamination ξ which intersects σ i . Then there is a lift ˜ γ of γ with endpoints γ ± such that |(cid:60) ( γ + − γ − ) | ≤ and (cid:61) τ i − < |(cid:61) ( γ + − γ − ) | < (cid:61) τ i .Proof. We continue with the set up of Corollary 6.5. We can choose alift ˜ γ of γ to be identified with the bending line ζ in that proof. Theendpoints γ ± of ˜ γ lie in C ∩ Λ, where C is a circle contained in the strip B between the lines (cid:61) z = 0 and (cid:61) z = (cid:61) τ i and as usual Λ is the limit setof G . By Proposition 2.1, Λ is contained in the union of the two strips0 ≤ (cid:61) z ≤ / (cid:61) τ i − / ≤ (cid:61) z ≤ (cid:61) τ i . (This proposition holds onthe hypothesis that (cid:61) τ i >
1, which holds for small θ by Corollary 6.5.)Moreover there are no points of Λ in the intersection of the two open disks D and S ( D ) bounded by C and S ( C ) and inside B . We deduce that ζ either has Euclidean height at most O (1), or has endpoints one of whichis in the rectangle |(cid:60) z | < , ≤ (cid:61) z ≤ / HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 25 |(cid:60) z | < , (cid:61) τ i − / ≤ (cid:61) z ≤ (cid:61) τ i . Since P ∈ ˜ γ and since the Euclidean heightof P is large by Corollary 6.5, the result follows. (cid:3) Continuing with the assumption that ξ ∈ M L Q is admissible and that G ξ ( θ ) is normalized so that S i is translation z (cid:55)→ z + 2, we call any lift of abending line satisfying the conditions of Lemma 6.8, good . Combining withCorollary 6.5 we obtain easily: Corollary 6.9.
Let ˜ γ be a good lift of a bending line γ and set γ + − γ − =2 re iα , where without loss of generality we take (cid:61) ( γ + − γ − ) > . Then r = (1 + O ( θ )) (cid:61) τ i / and | π/ − α | = O ( θ ) as θ → . Lemma 6.10.
Let ˜ γ be a good lift of a bending line of the pleating laminationwhich intersects ˜ S + i . Then the complex distance D between ˜ γ and S i (˜ γ ) isgiven by − r e iα = sinh D/ . Proof.
Recall that the complex distance between two axes is d + iψ where d is the real perpendicular distance and ψ is the rotation of one axis relativeto the other along their common perpendicular, see e.g. [27] for details.We use the cross ratio formula for complex distance. Let z , z and w , w be endpoints of two oriented geodesics in H at complex distance D . Wecan conjugate so that z , z move to 1 , − w , w move to e D , − e D respectively. Then[ z , w , w , z ] := z − w z − w · w − z w − z = coth D/ . Applying this formula to the four endpoints of ˜ γ , S i (˜ γ ) which are at points γ + , γ − and γ + + 2 , γ − + 2 respectively givescoth D/ γ + , γ + + 2 , γ − , γ − + 2]which simplifies to the claimed result. (cid:3) Proof of Proposition 6.6.
Pick a good lift ˜ γ of a bending line, and let P be the point where ˜ γ meets ˜ S + . Let Q be the highest point of ˜ γ and let Q (cid:48) = S ( Q ), so that the Euclidean height of both Q, Q (cid:48) above C is r . Writing d ∂ C for the induced hyperbolic metric on ∂ C + , we have l σ + ≤ d ∂ C ( Q, Q (cid:48) ) ≤ d H ( Q, Q (cid:48) )(1 + O ( θ ))where the first inequality follows since the curve on ∂ C + from Q to Q (cid:48) is inthe homotopy class of ˜ S + , and the final one follows by Lemma 6.3.Now sinh d H ( Q, Q (cid:48) ) / /r and (cid:61) τ − ≤ r ≤ (cid:61) τ by Lemma 6.8. Thus d H ( Q, Q (cid:48) ) / ≤ /r ≤ / ( (cid:61) τ − from which we deduce l σ + ≤ O ( θ )) (cid:61) τ − (cid:61) τ (cid:16) O (1 / (cid:61) τ ) (cid:17)(cid:16) O ( θ ) (cid:17) = 4 (cid:61) τ (cid:16) O ( θ ) (cid:17) by Corollary 6.5. Hence(6) (cid:61) τ / ≤ /l σ + (1 + O ( θ )) . To find an upper bound for 1 /l σ + , we use Lemma 6.10. Writing D = d + iψ , note that since d ≤ l σ + it is enough to find an upper bound on 1 /d .Comparing real and imaginary parts in the formula of Lemma 6.10 givessinh d/ ψ/ ± sin α/r and cosh d/ ψ/ ± cos α/r. Since d ≤ l σ + ≤ O ( θ ) we find2 d ≤ (1 + O ( θ ))sinh d/ r (1 + O ( θ )) | sin α | | cos ψ/ | . By Corollary 6.9, we have 1 / | sin α | = 1 + O ( θ ). Thus2 /d ≤ r (1 + O ( θ ))from which(7) 2 /l σ + ≤ (cid:61) τ (1 + O ( θ )) / . Inequalities (6) and (7) together complete the proof. (cid:3)
Asymptotic orthogonality.
Propositions 6.4 and 6.6 are not enoughto give the detailed asymptotics of Theorem 6.1. We also need the followingmore refined comparison:
Proposition 6.11.
Along the pleating variety P ξ , we have θi ( ξ, σ )(1 − O ( θ )) ≤ l + σ ≤ θi ( ξ, σ )(1 + O ( θ )) as θ → . This result is a direct consequence of the fact that asymptotically, ˜ S + becomes orthogonal to the bending lines, see Proposition 6.13. The intuitionfor this is the following. Suppose that ˜ S + were actually perpendicular toall bending lines. Then each good lift of a bending line cut by ˜ S + wouldhave Euclidean height between (cid:61) τ / − (cid:61) τ /
2, and in the proof ofProposition 6.4, Equations (2) and (3) would be equalities. Since E hasarea O ( θ ), in this situation (5) becomes an equality up to O ( θ ). Thissituation actually pertains in the case in which there is unique bending linein the class of T (in which case E = ∅ ), see Example 1 in Section 5. Lemma 6.12.
Let ˜ γ be a good lift of a bending line and as above, let P, P (cid:48) bethe points at which ˜ S + meets ˜ γ and S (˜ γ ) respectively. If K is the Euclideancentre on C of the semicircle ˜ γ , then ∠ P KQ = O ( √ θ ) . HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 27
Proof.
As in Corollary 6.5, let k denote the Euclidean height of P . Wehave sinh d H ( P, P (cid:48) ) / /k and 1 /k < O ( θ ). Hence d H ( P, P (cid:48) ) = 2(1 + O ( θ )) /k . We will estimate k/r = cos ∠ P KQ .By Lemma 6.3: d H ( P, P (cid:48) ) ≤ l σ + ≤ (1 + O ( θ )) d H ( P, P (cid:48) )and hence l σ + = 2(1 + O ( θ )) /k. On the other hand, by Proposition 6.6 and Lemma 6.8,2 /l σ + = r (cid:0) O ( θ ) (cid:1) . Combining these two equations gives | k/r − | = O ( θ )so that ∠ P KQ ≤ O ( √ θ ) as claimed. (cid:3) Now we prove our result on asymptotic orthogonality.
Proposition 6.13.
Along the pleating variety P ξ , the curve ˜ S + is asymp-totically orthogonal to the bending lines as θ → . More precisely, supposethat ˜ S + meets an (oriented) bending line ˜ γ at a point P so that the acuteangle between ˜ S + and ˜ γ is ψ ( P ) . Then | ψ ( P ) − π/ | ≤ O ( √ θ ) as θ → .Proof. As usual, we may suppose that ˜ γ is a good lift of a bending line,and we let P (cid:48) = S ( P ). Let κ denote the geodesic arc in H from P to P (cid:48) . Let u , v , w be forward pointing unit vectors at P along ˜ S + , κ , and ˜ γ respectively, where κ, ˜ S + are given the same orientation and the orientationof ˜ γ is from γ − to γ + , where (cid:61) γ − < (cid:61) γ + . Let Ψ( x , y ) denote the anglebetween the vectors x and y at P , so that ψ ( P ) = Ψ( u , w ).From Equation (4) in the proof of Proposition 6.4, we have Ψ( u , v ) ≤ θi ( σ, ξ ). Thus it will suffice to show that | Ψ( v , w ) − π/ | = O ( √ θ ). Weprove this by a finding a sequence of small rotations which, when applied to v and w , results in a pair of perpendicular vectors.Let Γ be the vertical plane containing ˜ γ and let Z be the footpoint of theperpendicular from P to C . As in Lemma 6.12, let K be the footpoint ofthe perpendicular from the highest point Q on ˜ γ to C . Let Θ be rotationby an angle ∠ P KQ about a line perpendicular to Γ through P , so thatΘ ( w ) is horizontal (i.e. parallel to the base plane C ). By Lemma 6.12, ∠ P KQ = O ( √ θ ), so that Θ ( w ) = w + O ( √ θ ).The vectors w and Θ ( w ) are in the plane Γ containing γ , which is at angle α to the real axis and hence to the vertical plane Π containing P and P (cid:48) .These two planes intersect in the line ZP . Thus if Θ denotes anticlockwiserotation by π/ − α about ZP , then Θ (Θ ( w )) is orthogonal to Π at P .Since by Lemma 6.8, π/ − α = O ( θ ), we have w = Θ (Θ ( w )) + O ( √ θ ).Finally, let ζ be the angle between P Z and the arc κ at P , so thatsinh d H ( P, P (cid:48) ) / ζ . Since d H ( P, P (cid:48) ) ≤ l σ + this gives π/ − ζ = O ( θ ). If Θ is rotation by an angle π/ − ζ about a line perpendicular to Π through P , then Θ ( v ) is horizontal and lies in Π and Θ ( v ) = v + O ( θ ).By construction Θ (Θ ( w )) is orthogonal to Θ ( v ) , so that putting thesethree estimates together we find | Ψ( v , w ) − π/ | = O ( √ θ ) as claimed. (cid:3) Finally, we can prove Proposition 6.11.
Proof of Proposition 6.11.
In view of Proposition 6.4, it will be enoughto find a bound l σ + ≥ θi ( ξ, σ )(1 − O ( θ )) as θ → P and P (cid:48) . This plane intersects ∂ C + in a path which is a union of H -geodesic segments similar to but notthe same as the geodesic path s , s , . . . , s k of Proposition 6.4. Denote thebending points along this path P = ˆ P , ˆ P , . . . , ˆ P k +1 where ˆ P k +1 = S ( P ) = P (cid:48) , and let ˆ s i be the segment from ˆ P i − to ˆ P i .Let ˆ φ i denote the exterior angle between the segments ˆ s i , ˆ s i +1 which meetat ˆ P i , measured in the same way as φ in the proof of Proposition 6.4, andlet θ i be the bending angle between the support planes of ∂ C + which meetat ˆ P i . We will prove below that(8) ˆ φ i /θ i = 1 + O ( θ ) . Denote by l (ˆ s i ) the hyperbolic length of ˆ s i . We observe: l + σ ≥ d H ( P, P (cid:48) ) ≥ (1 − O ( θ )) (cid:88) l (ˆ s i )where the last inequality follows by Lemma 6.3 since from (8) ˆ φ i = O ( θ )for all i . To estimate l (ˆ s i ), join each point ˆ P i to ∞ in the plane Π and let y i = ∠ ˆ P i − ˆ P i ∞ and x i = ∠ ∞ ˆ P i ˆ P i +1 , so that (since all angles are measuredin the plane Π), x i + y i + ˆ φ i = π . The formula for the length of the finiteside of triangle with angles x, y, l (ˆ s i ) = cos x i + cos y i +1 sin x i + sin y i +1 . We also prove below that(9) | π/ − x i | = O ( θ ) and | π/ − y i | = O ( θ ) . This gives l (ˆ s i ) = ( π − x i − y i +1 )(1 + O ( θ ))from which k (cid:88) l (ˆ s i ) = ( k (cid:88) ˆ φ i )(1 + O ( θ )) = ( k (cid:88) θ i )(1 + O ( θ ))by (8). Hence l + σ ≥ θi ( ξ, σ )(1 − O ( θ ))as θ →
0. This completes the proof, modulo the proofs of (8) and (9).
Proof of (8). This follows from Appendix A.4 in [9]. Suppose that twoplanes J , J meet at an angle ζ along a line L , so that ζ is the angle HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 29 between the lines of intersection of J , J with the plane H orthogonal to L . Suppose that H (cid:48) is another plane slightly skewed to H , and let ζ ∗ bethe angle between the lines of intersection of J , J with H (cid:48) . Setting thingsup so that H has unit normal (0 , ,
1) and so that the bisector of the planesorthogonal to L is the vector (1 , , ζ ∗ /ζ in terms of the unit normal ( x , x , x ) to H (cid:48) for small x , x . Infact if x i = O ( (cid:15) ) , i = 1 ,
2, then we find easily either from the formula fortan ζ ∗ / / tan ζ/ x i on p. 246, or from Theorem A.4.2 on p.247, that ζ ∗ /ζ = 1 + O ( (cid:15) ).To apply the theorem in our case, we want to find the angle between thesupport planes which meet along the bending line at P i , measured in theplane Π which is slightly skewed to the plane orthogonal to the lift ˜ γ i of γ through ˆ P i . In the above set up, the unit vector along ˜ γ at ˆ P i is e = (0 , , γ is e = (1 , , e (cid:48) , e (cid:48) be unit vectors at ˆ P i orthogonal to Π, and in Π pointing verti-cally upwards, respectively. It will be sufficient to show that e (cid:48) = e + O ( √ θ )and e (cid:48) = e + O ( √ θ ). Now if ˆ P i were replaced by the similar configura-tion at the point P i , then the first result would follow from Corollary 6.9and Lemma 6.12, while the second would follow from the proof of Proposi-tion 6.4, since as illustrated in Figure 8, e (cid:48) bisects the angle between s i and s i +1 up to O ( θ ).In the move from P i to ˆ P i , the estimates for e (cid:48) , e (cid:48) will change by termson the order of ∠ P i K i ˆ P i , the angular distance from P i to ˆ P i along ˜ γ i . (Here K i is the centre of the Euclidean semi-circle ˜ γ i .) We estimate this as follows.Let d i be the perpendicular distance between the plane Π through P parallelto the real axis, and the parallel plane Π i through P i . Since P is joined to P i by segments s , s , . . . s i − along ˜ S + , and since it follows from (4) inProposition 6.4 that each segment s j makes an angle at most O ( θ ) with theplane P j − , we find d i ≤ i − (cid:88) l ( s j ) O ( θ ) = O ( θ ) . Let Z i , ˆ Z i denote the footpoints in C of the vertical lines through P i , ˆ P i .The estimate on d i combined with Corollary 6.9 gives | Z i − ˆ Z i | = O ( θ ). Itfollows that the error in replacing P i by ˆ P i is of a lower order than thosealready obtained, and we conclude that e (cid:48) = e + O ( √ θ ) and e (cid:48) = e + O ( √ θ ) as claimed. Proof of (9). Let L be a line from ∞ to a variable point X on somesegment ˆ s i , and let x = x ( X ) > ∠ ˆ P i − X ∞ and ∠ ∞ X ˆ P i . We want to show that | π/ − x | = O ( θ ).Since P and S ( P ) are at the same Euclidean height, there is certainlysome ˆ s i containing a point X at which the tangent to ˆ s i is horizontal,so x ( X ) = π/
2. Moving away from this point in either direction, x ( X ) decreases according to the formula sinh t = cot x ( X ( t )), where the point X ( t ) is at distance t from X along ˆ s i . The change in angle at a bendpoint is ˆ φ i = θ i (1 + O ( θ )) by (8), and on the adjacent segment the argumentproceeds as before. Since (using Lemma 6.3 again) (cid:80) i l (ˆ s i ) ≤ O ( θ ), theresult follows. (cid:3) Twisting.
To complete the proof of Theorem 6.1, it remains to bound (cid:60) τ i . We have: Proposition 6.14.
Let ξ ∈ M L Q be admissible and suppose that γ ∈ S is contained in the support of ξ . Then if ( τ , τ ) ∈ P ξ , we have (cid:60) τ i = − p i ( γ ) /q i ( γ ) + O (1) and hence in particular | Arg τ i − π/ | = O ( θ ) as θ → . Since this result holds for any γ contained in the support of ξ we have: Corollary 6.15.
Suppose that γ, γ (cid:48) ∈ S are supported on a common ad-missible lamination ξ . Then | p i ( γ ) /q i ( γ ) − p i ( γ (cid:48) ) /q i ( γ (cid:48) ) | ≤ and hence (cid:60) τ i = − p i ( ξ ) /q i ( ξ ) + O (1) . The condition that γ, γ (cid:48) are supported on a common admissible lamina-tion is equivalent to the condition that together with σ , σ they fill up Σ.The value 10 is obtained by following through the constants in the argumentbelow; it could certainly be improved by more careful inspection.We prove this result using the concept of the twist of one geodesic aroundanother following Minsky [23]. Suppose given a hyperbolic metric h on thesurface Σ. The twist tw β ( γ, h) of a curve γ about another curve β is definedas follows. Let p be an intersection point of γ with β . Let P be a lift to H of p and let ˜ γ, ˜ β be the lifts of γ, β through P . Orient ˜ γ, ˜ β with positiveendpoints Z, W respectively on ∂ H so that the anticlockwise arc from Z to W does not contain the other two endpoints. Let R be the footpoint of theperpendicular from Z to ˜ β . Let t be the oriented distance P R , where t > R follows P in the positive direction along ˜ β and t ≤ t/l β ( h ) is independent up to an additiveerror of 1 of the choices made, including the choice of p . Finally, definetw β ( γ, h) = inf t / l β ( h ), where we take the infimum over all possible choicesof lifts as above.Note that the twist is independent of the orientation of β, γ but dependson the choice of hyperbolic metric h ∈ T , where T is the Teichm¨uller spaceof Σ. However: Lemma 6.16. ( [23] Lemma 3.5, see also [7]
Sec. 4.3.) For any two γ , γ ∈S , the relative twist tw α ( γ , h) − tw α ( γ , h) is independent of h ∈ T , up toa bounded additive error of . We define the signed relative twist of γ , γ with respective to β to be i β ( γ , γ ) = inf h ∈T tw β ( γ , h) − tw β ( γ , h). Here is a useful way of computingit: HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 31
Lemma 6.17.
Let γ , γ ∈ S and let ˜ γ , ˜ γ be lifts of γ , γ which cut thefixed axis ˜ β corresponding to β , and let b ∈ Γ be the primitive element whoseaxis is ˜ β and whose attracting fixed point is the positive endpoint of B , where Γ is the Fuchsian group uniformising h . Then tw β ( γ , h) − tw β ( γ , h) is equalin magnitude to the number of times the images b n (˜ γ ) , n ∈ Z intersect ˜ γ ,up to a bounded additive error of . The sign is negative if b (˜ γ ) follows ˜ γ in the positive direction along ˜ γ and positive otherwise.Proof. This is clear from the definition in a metric in which ˜ γ is orthogonalto B . Whether or not two axes intersect, depends only on the relativeposition of their endpoints round the boundary at infinity ∂ H = S . Sincea quasiconformal deformation of H induces a homeomorphism of ∂ H , wededuce that the relative positions of endpoints of axes are independent ofthe metric h , from which the result follows. (cid:3) We shall prove Proposition 6.14 by computing i σ i ( γ, T ) in two differentways, where as usual i σ i ( γ, T ) means i σ i ( γ, γ T ) where γ T ∈ S is the curvecorresponding to the generator T ∈ π (Σ). We have: Lemma 6.18.
Suppose that γ ∈ S has canonical coordinates i ( γ ) = ( q ( γ ) , p ( γ ) , q ( γ ) , p ( γ )) . Then i σ i ( γ, T ) = − p i ( γ ) /q i ( γ ) + O (1) .Proof. We work in the fundamental domain ∆ of Σ , and label the sidesas in Figure 2. We also suppose that p ≥
0. Let ˜ A be a horizontal stripjoining s S to s S − , shown shaded in the figure. This projects to an annularneighbourhood A of σ on Σ. We may take our lift ˜ σ of σ to be the centreline of ˜ A and the lift ˜ γ T of the curve γ T to be the arc joining the midpoints of s T and s T − . This intersects ˜ A in a single arc λ which joins the boundaries ∂ − ˜ A, ∂ + ˜ A of ˜ A . By the previous lemma, to compute i σ ( γ, T ) we have toexamine how many images S n (˜ γ T ) cut a fixed lift ˜ γ of γ , equivalently howmany images S n (˜ γ ) cut ˜ γ T .The lift ˜ γ appears as a collection of disjoint arcs joining sides of ∆, thenumbers of arcs joining particular pairs of sides being determined by thecanonical coordinates i ( γ ). It is not hard to see that the magnitude of therelative intersection number i σ i ( γ, T ) is, up to an additive error of 1, thenumber of times that a connected component κ of γ ∩ A cuts the projection π ( λ ) of λ to Σ. For convenience, replace λ by L = s S ∩ ˜ A . This changesthe intersection number by at most 2.Denote the lift of κ to ∆ by ˜ κ . Clearly ˜ κ contains at most two ‘cornerarcs’ (see Section 4) of ˜ γ ∩ ∆, which can be ignored in our count. If p ≤ q then ˜ γ contains no horizontal strands and i ( κ, L ) ≤ p > q . In this case ˜ γ ∩ ∆ contains m > A joining L to S ( L ). Thus after entering A across ∂ − A ,the component κ travels around A cutting π ( L ) either m or m + 1 timesbefore exiting across ∂ + A . (This is the well known combinatorics of simple curves crossing a cylinder.) The total number of such connected componentsis i ( σ , γ ) = q . On the other hand, the total number of strands of ˜ γ ∩ ∆which meet L is by definition p . Thus mq ≤ p < ( m + 1) q + q and so m = [ p /q ] + O (1). Finally, we check from the definition that in the obviousmetric h on ∆ in which ˜ σ is orthogonal to ˜ γ T , we have i σ ( γ, h ) < i σ ( γ T , h ) = 0. Thus i σ i ( γ, T ) = − p i ( γ ) /q i ( γ ) + O (1) as claimed.The arguments for p < S are similar. (cid:3) Proof of Proposition 6.14.
From Lemma 6.18 we have i σ i ( γ, T ) = − p i ( γ ) /q i ( γ ) + O (1). On the other hand we can also compute i σ i ( γ, T ) asfollows. As usual, after normalizing suitably let ˜ S + = ˜ S + i be the lift of σ i to ∂ C + which is invariant under S i : z (cid:55)→ z + 2. If ˜ γ is a good lift of γ , then ˜ γ certainly intersects ˜ S + . Now referring to Figure 3, let B , B be the circleswith equal diameters 2 / (cid:61) τ i tangent to R at 0, and R + τ i at τ i , respectively.It follows from the usual ping-pong theorem methods, that there is a lift ˜ T of the axis of T to ∂ C + which has one endpoint inside B and one inside B .This lift also clearly cuts ˜ S + . By Lemma 6.17, i σ i ( γ, T ) is up to sign thenumber of images S ni (˜ γ ) of ˜ γ which cut ˜ T . Since ˜ γ is a good lift , orientingas in Lemma 6.17, we see that i σ i ( γ, T ) = [ (cid:60) τ i /
2] + O (1). We deduce that (cid:60) τ i = − p i ( γ ) /q i ( γ ) + O (1) . and the result follows. (cid:3) Proof of Theorem C . As usual let ξ ∈ M L Q be admissible and let G ξ ( θ )be the unique group for which β ( G ) = θξ . Let h ( θ ) denote the hyperbolicstructure of ∂ C + /G ξ ( θ ). Since l + σ i → , i = 1 ,
2, the limit of the structures h ( θ ) in P M L is in the linear span of δ σ , δ σ . We want to prove that thelimit is the barycentre δ σ + δ σ .Let δ, δ (cid:48) ∈ S . Since σ , σ are a maximal set of simple curves on Σ,the thin part of h ( θ ) is contained in collars A i around σ i of approximatewidth log(1 /l + σ i ) and the lengths of δ, δ (cid:48) outside the collars A i are bounded(with a bound depending only on the combinatorics of δ, δ (cid:48) and hence thecanonical coordinates i ( δ ) , i ( δ (cid:48) )). By Proposition 6.14 the twisting around A i is bounded. We deduce that for any curve transverse to σ i we have(10) l + δ = (cid:88) i =1 , q i ( δ ) log(1 /l + σ i ) + O (1) , see for example [23] Lemma 7.2. By Theorem 6.1 we have l + σ /l + σ → q ( ξ ) /q ( ξ ), and since ξ is admissible, q ( ξ ) , q ( ξ ) >
0. Thus log l + σ / log l + σ →
1. Hence l + δ /l + δ (cid:48) → (cid:88) i =1 , q i ( δ ) / (cid:88) i =1 , q i ( δ (cid:48) ) = i ( δ, σ + σ ) /i ( δ (cid:48) , σ + σ ) . The result follows from the definition of convergence to a point in
P M L . (cid:3) HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 33
Remark 6.19.
The above length estimate above coincides with that comingfrom the top terms Theorem 4.1. Namely from that formula we have(11) log Tr δ = (cid:88) i =1 , q i ( δ ) log( τ i ) + O (1) . Since l + σ i is small, any transverse curve has definite length. Hence by forexample Proposition 5.1 of [28], l + δ is close to the hyperbolic length of thegeodesic representative of δ in H /G and thus to log Tr δ . Since by Propo-sition 6.6, l + σ i (cid:61) τ i = O (1), the formula (11) is compatible with (10).7. Asymptotic directions
In this section we prove our main results, Theorems A and B.Throughout this section, to simplify notation, X = O ( θ ) will mean X ≤ cθ where the constant c depends on the lamination ξ and a small numberof related curves chosen during the proofs. With a bit more effort, thedependence could be controlled more carefully, but this is not needed for ourresults here. We write Tr γ to mean Tr ρ ( W ) where W is a word representing γ ∈ π ( S ).Suppose that γ is a bending line of ∂ C + /G for a group G ( τ , τ ) ∈ P ξ . Thetop terms Theorem 4.1, together with the condition Tr γ ∈ R of Lemma 3.3,gives asymptotic conditions for ( τ , τ ) ∈ P ξ , in terms of the canonical co-ordinates i ( γ ) of γ . For τ , τ ∈ C set τ i = x i + iy i , ρ = (cid:112) y + y , and η i = y i /ρ . Define E γ ( τ , τ ) = ( q x + 2 p ) η + ( q x + 2 p ) η , where as usual i ( γ ) = ( q , p , q , p ) and y i > , i = 1 , Proposition 7.1.
Suppose that ξ ∈ M L Q is an admissible lamination, that G ( τ , τ ) ∈ P ξ has bending measure β ( G ) = θξ , and that γ is a bending lineof ξ . Then E γ ( τ , τ ) = O ( θ ) as θ → .Proof. Suppose first that q i = q i ( γ ) > , i = 1 , a i = − p i ( γ ) /q i ( γ ).By Theorem 4.1 we have(12) Tr γ = ± | q − q | (cid:0) τ − a (cid:1) q (cid:0) τ − a (cid:1) q + R (cid:0) q + q − (cid:1) where R ( q + q −
2) is a polynomial of degree at most q i in τ i but with totaldegree in τ and τ at most q + q − x i − a i = O (1) and by Theorem 6.1:(13) (cid:12)(cid:12) q ( ξ ) /q ( ξ ) − η /η (cid:12)(cid:12) = O ( θ ) . (Notice that the terms in (13) involve q i ( ξ ) as opposed to q i = q i ( γ ) in (12).)Hence arranging the terms of (12) in order of decreasing powers of ρ , and using Equation (1) in Section 4, we get ± Tr γ −| q − q | = ρ q + q η q η q + iρ q + q − η q − η q − ( q η ( x − a )+ q η ( x − a )) + O ( ρ q + q − ) . By Lemma 3.3, Tr γ ∈ R . We deduce η q − η q − E γ ( τ , τ ) + O (1 /ρ ) = 0from which using (13), q η ( x − a ) + q η ( x − a )) = O (1 /ρ ) . Since 1 /ρ = O ( θ ) by Corollary 6.5, this proves the result.We still have to deal with the case that, say, q ( γ ) = 0. Then Tr γ is apolynomial in τ only, of the form(14) Tr ρ ( γ ) = ± q (cid:0) τ − a (cid:1) q + R (cid:0) q − (cid:1) . The result then follows easily by similar reasoning to the above. (cid:3)
Solving the asymptotic equations.
Suppose we want to locate thepleating ray P γ of γ ∈ S . If G ∈ P γ , then ∂ C + /G \ γ is flat, so that notonly γ , but also any curve δ ∈ wh ( γ ), is a bending line of ξ , where wh ( γ )(the wheel of γ ) denotes the set of all curves δ ∈ S disjoint from γ . (Byconvention, γ / ∈ wh ( γ ).) Thus τ , τ are constrained by the equations (cid:61) Tr γ = (cid:61) Tr δ = 0and hence, by the above proposition, E γ ( τ , τ ) + O ( θ ) = 0 , and E δ ( τ , τ ) + O ( θ ) = 0for all δ ∈ wh ( γ ). Our proof of Theorem A amounts to solving these equa-tions for τ , τ .In order to do this, note that for any curve ω ∈ S : E ω ( τ , τ ) = i ( ω ) · u where i ( ω ) = ( q ( ω ) , p ( ω ) , q ( ω ) , p ( ω )) and(15) u = ( x η , η , x η , η )with x i = (cid:60) τ i , η i = (cid:61) τ i /ρ as above. Effectively what we will do is use linearalgebra to solve the equations i ( δ ) · u = 0 for all δ ∈ wh ( γ ). This is donewith the aid of Thurston’s symplectic form Ω Th introduced in Section 4.2.This induces a map ξ → ξ ∗ of R such thatΩ Th ( i ( γ ) , i ( δ )) = i ( γ ) · i ( δ ) ∗ where · is the usual inner product on R : if i ( γ ) = ( q , p , q , p ), then i ( γ ) ∗ = ( − p , q , − p , q ). By Proposition 4.2, i ( γ ) ∗ is orthogonal not onlyto i ( γ ), but also to all curves in wh ( γ ). We need: Lemma 7.2.
HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 35 (i)
Suppose that γ, δ ∈ S and that δ ∈ wh ( γ ) . Then γ and δ are sup-ported on a common canonical train track and i ( γ ) , i ( δ ) are indepen-dent vectors in M L . (ii) Given γ ∈ S , we can find δ, δ (cid:48) ∈ wh ( γ ) such that i ( γ ) , i ( δ ) , i ( δ (cid:48) ) aresupported on a common canonical train track and span a subspace ofdimension in M L .Proof. ( i ) That γ, δ are supported on a common canonical train track followsimmediately since they are disjoint. If i ( γ ) , i ( δ ) are dependent, since theylie on the same track all their coefficients are integers which pairwise havethe same sign. Thus we must have n i ( γ ) = m i ( δ ) for some n, m ∈ N . Sinceboth are connected simple curves, n = m = 1.( ii ) The surface Σ γ := Σ \ γ is either a one holed torus and a sphere withtwo punctures and a hole; or a sphere with two holes and two punctures.In either case, P M L (Σ γ ) is a topological circle in which the set of rationallaminations supported on a simple curve is dense. Let π denote the mapwhich associates to a curve ω ∈ S the measured lamination δ ω ∈ M L (Σ),and likewise define the quotient map ¯ π ( ω ) = [ δ ω ] ∈ P M L . Then π, ¯ π areinjective and ¯ π ( whγ ) is a dense subset of an embedded circle K in P M L (Σ).If (ii) is false, then in particular π ( wh ( γ )) ⊂ M L is contained in a 2-planewhose image in
P M L is an affine line L . (Since the canonical coordinatesare global coordinates for M L , there is no need to consider the subdivisionof
M L into cells.) Since
P M L (Σ γ ) injects into L , it is open and closedin K . Thus K = L , which is impossible. Thus π ( wh ( γ )) spans at least a3-dimensional subspace in M L .Suppose that π ( γ ) is in the linear span of π ( δ ) , π ( δ (cid:48) ) ∈ π ( wh ( γ )) ⊂ M L .Then by the above we can find δ (cid:48)(cid:48) ∈ wh ( γ ) with π ( δ (cid:48)(cid:48) ) not contained in thelinear span of π ( γ ) , π ( δ ) and π ( δ (cid:48) ) which proves the result. (cid:3) Theorem 7.3 (Theorem A) . Suppose that ξ = (cid:80) , a i γ i is admissible andnot exceptional. Let i ( ξ ) = ( q , p , q , p ) and set tan ψ ( ξ ) = q /q . Let L ξ : [0 , ∞ ) → C be the line t (cid:55)→ ( w ( t ) , w ( t )) where w ( t ) = − p /q + t cos ψ, w ( t ) = − p /q + t sin ψ. Let ( τ ( θ ) , τ ( θ )) ∈ C be the point corresponding to the group G ξ ( θ ) with β ( G ) = θξ , so that the pleating ray P ξ is the image of the map p ξ : θ → ( τ ( θ ) , τ ( θ )) for a suitable range of θ > . Then P ξ approaches L ξ as θ → in the sense that if t ( θ ) = 4 Q/θq q with Q = (cid:112) ( q + q ) then |(cid:60) τ i ( θ ) − (cid:60) w i ( t ( θ )) | = O ( θ ) and |(cid:61) τ i ( θ ) − (cid:61) w i ( t ( θ )) | = O (1) , i = 1 , . Proof.
As above, write τ i ( θ ) = τ i = x i + iy i , ρ = y + y and η i = y i /ρ , wherethe dependence on θ is understood. By Theorem 6.1, we have y i − θq i = O (1). On the other hand, with t = t ( θ ) as in the statement of the theorem,we find (cid:61) w ( t ) = tq /Q = 4 /θq and similarly (cid:61) w ( t ) = 4 /θq . Thus |(cid:61) τ i ( θ ) − (cid:61) w i ( t ( θ )) | = O (1) , i = 1 , θ → x i = (cid:60) τ i ( θ ) is more subtle. Consider firstthe special case ξ = γ ∈ S . By Lemma 7.2 we can choose δ, δ (cid:48) ∈ wh ( γ ) suchthat i ( γ ) , i ( δ ) , i ( δ (cid:48) ) span a subspace of dimension 3 in M L . If ( τ , τ ) ∈ P γ ,the conditions (cid:61) Tr γ = (cid:61) Tr δ = (cid:61) Tr δ (cid:48) = 0must be satisfied. By Proposition 7.1, we can write this as E ζ ( τ , τ ) + O ( θ ) = 0for ζ = γ, δ, δ (cid:48) . With τ i = x i + iρη i as above, we can as in (15) regard theseas equations in R for u = u ( θ ) = ( x η , − η , x η , − η ):(16) i ( ζ ) · u = O ( θ )for ζ = γ, δ, δ (cid:48) .Now we use Thurston’s symplectic form. By Proposition 4.2 we haveΩ Th ( i ( γ ) , i ( ζ )) = 0 for all ζ ∈ wh ( γ ). Hence i ( ζ ) · i ( γ ) ∗ = 0for ζ = γ, δ, δ (cid:48) . Since i ( γ ) , i ( δ ) , i ( δ (cid:48) ) are independent, it follows that we canwrite(17) u ( θ ) = λ ( θ ) i ( γ ) ∗ + µ ( θ ) v ( θ )where v = v ( θ ) is in the linear span of i ( γ ) , i ( δ ) , i ( δ (cid:48) ) and || v || = 1. We findusing (16) that u · v = O ( θ ) (where the constants depend on i ( γ ) , i ( δ ) , i ( δ (cid:48) )).Then v · i ( γ ∗ ) = 0 gives µ ( θ ) = O ( θ ), with the same proviso on the constants.Equating the two sides of (17) gives x η = − λp ( γ ) + O ( θ ) , η = λq ( γ ) + O ( θ ) ,x η = − λp ( γ ) + O ( θ ) , η = λq ( γ ) + O ( θ ) . (18)Since || u || = ( x + 4) η + ( x + 4) η ≥ | λ ( θ ) | = || u ( θ ) − O ( θ ) |||| i ( γ ) ∗ || ≥ c for some constant c >
0. It follows easily that | x i + 2 p i /q i | = O ( θ ), provingTheorem A in the special case ξ = γ .Now we turn to the case of a general admissible lamination ξ = a γ γ + a δ δ ∈ M L Q . In this case, if ( τ , τ ) ∈ P ξ then γ and δ are both bending lines of G ( τ , τ ). It follows as above that i ( γ ) · u = O ( θ ) and i ( δ ) · u = O ( θ ) . By Lemma 7.2, i ( γ ) and i ( δ ) are independent and by Proposition 4.2, i ( γ ) ∗ and i ( δ ) ∗ are orthogonal to both. Thus(19) u = λ i ( γ ) ∗ + µ i ( δ ) ∗ + ν w HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 37 where w is in the span of i ( γ ) , i ( δ ), and || w || = 1. Thus u · w = O ( θ ) andsince w · i ( γ ) ∗ = w · i ( δ ) ∗ = 0 we find ν = ν || w || = u · w = O ( θ ).Now from Theorem 6.1,(20) (cid:12)(cid:12)(cid:12) y y − a γ q ( γ ) + a δ q ( δ ) a γ q ( γ ) + a δ q ( δ ) (cid:12)(cid:12)(cid:12) = O ( θ ) . On the other hand setting z i = λq i ( γ ) + µq i ( δ ), we find from (19) that | z | + | z | ≥ c > c depending only on γ, δ . It followsfrom (19) and (20) that | z /z − y /y | = O ( θ ). Since by hypothesis weare not in the exceptional case q ( γ ) /q ( γ ) = q ( δ ) /q ( δ ), we deduce that (cid:12)(cid:12) λ/µ − a γ /a δ (cid:12)(cid:12) = O ( θ ). Substituting back in (19) we see u = κi ( ξ ) ∗ + O ( θ )for some κ >
0, and a little bit of algebra completes the proof. (cid:3)
The following lemma proves the second part of the statement of Theo-rem B.
Lemma 7.4.
Suppose that γ ∈ S is admissible. Then there exists γ ∈ wh ( γ ) such that the pair γ , γ is not exceptional.Proof. Write X = ( X , X , X , X ) ∈ R . Then δ ∈ wh ( γ ) implies that i ( δ )is in the codimension one hyperplane H defined by i ( γ ) ∗ · X = 0. If thepair γ , δ is exceptional then i ( δ ) is also in the codimension one hyperplane K defined by q ( γ ) X − q ( γ ) X = 0. Note that i ( γ ) is also in H ∩ K .Now since q ( γ ) , q ( γ ) >
0, the normal vectors( − p ( γ ) , q ( γ ) , − p ( γ ) , q ( γ )) and ( q ( γ ) , , q ( γ ) , H and K respectively are not collinear. Thus H ∩ K is two dimensional.However by Lemma 7.2, we can find δ, δ (cid:48) ∈ wh ( γ ) such that i ( δ ) , i ( δ (cid:48) ) , i ( γ )are independent, so at least one of the pairs i ( γ ) , i ( δ ) and i ( γ ) , i ( δ (cid:48) ) mustbe non-exceptional as claimed. (cid:3) Proof of Theorem B.
We need to show that if ξ = (cid:80) , a i δ γ i is admis-sible and if the pair γ , γ is not exceptional, then the equationsTr γ , Tr γ ∈ R have a unique solution as τ i → ∞ in the direction specified by Theorem A.First consider the equations(21) τ q τ q , τ q (cid:48) τ q (cid:48) ∈ R , as τ , τ → ∞ , where q i = q i ( γ ) and q (cid:48) i = q i ( γ ). Setting z i = 1 /τ i we define G : C → C by G ( z , z ) = ( z q z q , z q (cid:48) z q (cid:48) ), so that (21) is equivalent to theequation G ( z , z ) ∈ R as z , z →
0. Writing z i = (cid:15) i e iθ i , i = 1 , e i ( q θ + q θ ) , e i ( q (cid:48) θ + q (cid:48) θ ) ∈ R , so that writing A = (cid:18) q q q (cid:48) q (cid:48) (cid:19) , we must have A ( θ , θ ) T = π ( n, m ) T for n, m ∈ Z . Since A is invertible by hypothesis, these equations have a discrete set of solutions for ( θ , θ ). Since q + q , q (cid:48) + q (cid:48) ∈ Z , one of these solutionsis θ i = π/ , i = 1 ,
2. It follows that G is a branched covering C → C in aneighbourhood of 0, and G − ( R ) is a discrete set of 2-planes meeting onlyat 0.Now consider our actual equations (cid:61) Tr γ = (cid:61) Tr δ = 0. As above set z i = 1 /τ i and define H : C → C by H ( z , z ) = (1 / Tr γ, / Tr δ ). ByTheorem 4.1 we haveTr γ = ( τ + 2 p /q ) q ( τ + 2 p /q ) q (1 + R ) , Tr δ = ( τ + 2 p (cid:48) /q (cid:48) ) q (cid:48) ( τ + 2 p (cid:48) /q (cid:48) ) q (cid:48) (1 + R ) . (23)where p i = p i ( γ ) , q i = q i ( γ ) and p (cid:48) i = p i ( δ ) , q (cid:48) i = q i ( δ ) and R , R denotepolynomials of total order at most q + q − τ , τ . Hence we can expand H ( z , z ) as a Taylor expansion about 0 to obtain H ( z , z ) =( z q z q (1 + p z + p z + ˆ R ) , z q (cid:48) z q (cid:48) (1 + p (cid:48) z + p (cid:48) z + ˆ R ))(24)where ˆ R , ˆ R denote terms of total order at least 2 in z , z and p i = p i ( γ ) , p (cid:48) i = p i ( δ ).There is clearly a neighbourhood U of 0 such that H is locally injectiveon U \ { } , and moreover in which the homotopy G + tH, t ∈ [0 ,
1] between G and H is regular at every point. It follows that H is also a branchedcovering of C near zero, of the same order as G , and that there is a naturalbijective correspondence between the branches of G − ( R ) and H − ( R ).To complete the proof, we need to show that for ( t , t ) ∈ R sufficientlynear 0, the point H − ( t , t ) is arbitrarily close to the point G − ( t , t ) onthe corresponding sheet of G − ( R ).In a neighbourhood of 0, we view (24) as a perturbation of (21) and usethe ideas of Appendix B in [22]. If g : C n → C n is an analytic function withan isolated zero at Z ∈ C n , define the multiplicity of g to be the degreeof the mapping g/ || g || : S δ → S , where S δ is the sphere radius δ centre Z and S is the unit sphere. The same proof as Lemma B.1 of [22] proves‘Rouch´e’s principle’ that if r : C n → C n with r (0) = 0 and if || r || < || g || on S δ , then the degrees of ( g + r ) / || g + r || and g/ || g || on S δ are equal.Now take Z ∈ U to be an isolated solution of the equation G ( z , z ) =( t , t ) ∈ R . Choose δ > S δ ( Z ) ⊂ U and so that G ( z , z ) = ( t , t ) has no other solutions in S δ ( Z ). We can also choose U small enough so that || ( H − G ) || < || G || on U . It follows from the aboveRouch´e’s principle that H and G have the same degree on S δ . Then LemmaB.2 of [22] shows that H has exactly one zero inside S δ as required.In particular, there is a unique branch of H − ( R ) close to the branch z i = (cid:15) i i, i = 1 , G ( z , z ) ∈ R . If (cid:15) /(cid:15) is bounded away from 0 and ∞ and we set (cid:15) = (cid:112) (cid:15) + (cid:15) , we can clearly write points on this branch inthe form z i = (cid:15) i ie iα i , i = 1 , α i = O ( (cid:15) ) as (cid:15) →
0. The arguments of
HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 39
Theorem A are then sufficient to show the solution to the equations in thattheorem is unique. (cid:3)
The exceptional case.
Recall that a pair of curves γ , γ is said to beexceptional if q ( γ ) q ( γ ) = q ( γ ) q ( γ ). As an example, the coordinates i ( γ ) = (2 , , ,
1) and i ( γ ) = (2 , − , , i ( γ ) = ( a + b, , a + b, a ) and i ( γ ) = ( a + b, − , a + b, a + 1), can be easily be checked torepresent exceptional pairs of disjoint connected curves. Theorem 7.5.
Suppose that ξ = (cid:80) , a i γ i is an admissible lamination suchthat the pair γ , γ is exceptional. For s ∈ [0 , let ξ ( s ) = (cid:80) , sa γ +(1 − s ) a γ . Let i ( ξ ( s )) = ( q ( s ) , p ( s ) , q ( s ) , p ( s )) and set tan ψ ( s ) = q ( s ) /q ( s ) . Let L γ ,γ : [0 , × [0 , ∞ ) → C be the map ( s, t ) (cid:55)→ ( w ( s, t ) , w ( s, t )) where w ( s, t ) = 2 p ( s ) /q ( s ) + t cos ψ ( s ) , w ( s, t ) = 2 p ( s ) /q ( s ) + t sin ψ ( s ) . Let ( τ ( s, θ ) , τ ( s, θ )) ∈ C be the point corresponding to the group G ξ ( s ) ( θ ) with β ( G ) = θξ ( s ) , so that the (closure of the) pleating plane P γ ,γ is theimage of the map p γ ,γ : ( s, θ ) → ( τ ( θ ) , τ ( θ )) for s ∈ [0 , and a suitablerange of θ > . Then P γ ,γ approaches L γ ,γ as θ → in the sensethat if t ( s, θ ) = 4 Q/θq ( s ) q ( s ) with Q = (cid:112) ( q ( s ) + q ( s )) , then for allsufficiently small θ there exists a continuous function f θ : [0 , → [0 , suchthat f θ (0) = 0 , f θ (1) = 1 and |(cid:60) τ i ( s, θ ) − (cid:60) w i ( f θ ( s ) , t ( s, θ )) | = O ( θ ) and |(cid:61) τ i ( s, θ ) − (cid:61) w i ( f θ ( s ) , t ( s, θ )) | = O (1)(25) for i = 1 , . Moreover every point on L γ ,γ is close to a point on P γ ,γ , inthe sense that for all sufficiently small θ , for each s there exists s (cid:48) ∈ [0 , such that |(cid:60) τ i ( s (cid:48) , θ ) − (cid:60) w i ( s, t ( s (cid:48) , θ )) | = O ( θ ) and |(cid:61) τ i ( s (cid:48) , θ ) − (cid:61) w i ( s, t ( s (cid:48) , θ )) | = O (1)(26) Remark 7.6.
The difference between this statement and that of Theo-rem 7.3 is that in that theorem, P ξ ( s ) is close to a point on L ξ ( s ) , while herewe can only assert closeness of points on the pleating ray P ξ ( s ) to a pointon some line L ξ ( f θ ( s )) where possibly f θ ( s ) (cid:54) = s . Proof.
We proceed as in the proof of Theorem 7.3 above, up to (19) whichsays that u = u ( θ ) is close to a convex combination of i ( γ ) ∗ and i ( γ ) ∗ .Define f θ ( s ) by letting ξ ( f θ ( s ) ) be the orthogonal projection of u ( s, θ ) onto theplane spanned by i ( γ ) ∗ , i ( δ ) ∗ . Then (25) follows as before, while continuityof the path s (cid:55)→ f θ ( s ) is clear. Equation (26) follows choosing s (cid:48) such that s = f θ ( s (cid:48) ). (cid:3) Remark 7.7.
One might expect to be able to prove Theorem A in theexceptional case by a limiting argument with laminations ξ n → ξ . However the interaction of the double limits as n → ∞ and θ → Remark 7.8.
Suppose that the pair γ , γ is exceptional. Then we claimthat it is not possible to deduce from the top terms Theorem 4.1 alone thatthe equations (cid:61) Tr γ i = 0 , i = 1 , τ i = π/ , (cid:61) τ / (cid:61) τ = q ( ξ ) /q ( ξ ). To study this question,as above we replace the equations by equations (cid:61) f ( z , z ) = (cid:61) f ( z , z ) = 0in a neighbourhood of 0, where f i : C → C are analytic functions withlowest order terms z q ( γ )1 z q ( γ )2 and z q ( γ )1 z q ( γ )2 respectively.To show that many different behaviours are possible, consider the func-tions f ( z , z ) = z z , f ( z , z ) = z z (1 − z + z + z z ) , f ( z , z ) = z z (1 − z − z + z z ) and f ( z , z ) = z z (1 − z + z + z ). We look forsolutions to each of the three pairs of equations (cid:61) f = 0 , (cid:61) f i = 0 , i = 1 , , z = (cid:15)ie iα , z = (cid:15)ie − iα where (cid:15) → α = O ( (cid:15) ).As is easily verified, the equations (cid:61) f = 0 , (cid:61) f = 0 are satisfied forarbitrary choices of α . The equations (cid:61) f = 0 , (cid:61) f = 0 have no solutions ofthe required form near (0 , (cid:61) f = 0 , (cid:61) f = 0 havea unique suitable solution for each (cid:15) >
0, namely α = 0. Appendix 1
The following proofs are taken from [16].
Proof of Proposition 2.1
Let S = T S T − , so that S is parabolicwith fixed point T (0) = τ . Let J j = (cid:104) S j (cid:105) for j = 1 , ,
3. We constructfundamental domains D j for the S j as follows.Referring to Figure 3 in Section 2.2, let l consist of the vertical line below − i/
2, the vertical line above τ − − i/ − i/ τ − − i/
2. This last line segment has positive slope because (cid:61) τ >
1, from which it follows that l and S ( l ) do not intersect and hencethat the strip D between l and S ( l ) is a fundamental domain for J . Afundamental domain for J is D = (cid:8) z ∈ ˆ C : | z + 1 / | > / , | z − / | > / (cid:9) , and a fundamental domain for J is D = (cid:8) z ∈ ˆ C : | z − τ + 1 / | > / , | z − τ − / | > / (cid:9) . The hypothesis (cid:61) τ > D j is the whole of ˆ C . Moreover the boundaries of the D j only intersectat parabolic fixed points. Therefore by a simple application of the firstKlein-Maskit combination theorem (see [20], p.149 or [1], p.103) we see that F = (cid:104) S , S , S (cid:105) is discrete with fundamental domain D = D ∩ D ∩ D .Now let J = (cid:104) T (cid:105) . We will construct a fundamental domain D for J .Let B be the disk centred at i/ (cid:61) τ with radius 1 / (cid:61) τ and let B be thedisk centred at τ − i/ (cid:61) τ with radius 1 / (cid:61) τ . One checks that T takes B to the complement of B . (Note that T ( z ) = τ + τ +1 /z and consider the HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 41 action of z (cid:55)→ τ +1 /z on B .) Thus the domain D exterior to both disksis a fundamental domain for J . Since (cid:61) τ > (cid:61) τ (cid:61) τ >
4, it is easyto see that B and B are contained in the strip D , B is contained in D and B is contained in D . Moreover S j ( B j ) = B j for j = 2 ,
3. Thus B j isprecisely invariant with respect to J j for j = 2 ,
3; that is, W ( B j ) = B j for W ∈ J j and W ( B j ) ∩ B j = ∅ for W ∈ F − J j . Therefore the hypotheses ofthe second combination theorem are satisfied and so G = (cid:104) F , T (cid:105) is discretewith domain D = D ∩ D ∩ D ∩ D .Now we verify that G ∈ M . By construction, D consists of three compo-nents. The first, in the lower half plane H − , is a component of a fundamentaldomain for the Fuchsian subgroup F = (cid:104) S , S (cid:105) and H − /F is a triply punc-tured sphere. Similarly the second, in the half plane H τ above the horizontalthrough τ , is a component of a fundamental domain for the Fuchsian sub-group F = (cid:104) S , S (cid:105) . Again H τ /F is a triply punctured sphere. Becausethe components of D in H − and H τ are disjoint, F and F are not conjugatein G .Let D ∗ denote the third component D . It is contained in the strip betweenthe horizontal lines through 0 and τ . It has eight sides, one pair of sidescontained in the boundaries of each of D , D , D , D and identified by S , S , S and T respectively. Performing these identifications we obtain a toruswith two punctures corresponding to S S − and S S − . Developing D ∗ by G we see that it corresponds to a simply connected G -invariant componentof the regular set of G , and we conclude that G ∈ M .Finally, since translates of D by S cover the strip 1 / < (cid:61) z < (cid:61) τ − / H − , H τ are in Ω( G ), the limit set Λ is contained in the two strips0 ≤ (cid:61) z ≤ / (cid:61) τ − / ≤ (cid:61) z ≤ (cid:61) τ as claimed. (cid:3) Proof of Proposition 2.2
This is based on a similar result for the Maskitspace of the once punctured torus due to David Wright [29]. Let W ∈ G − {(cid:104) S , S (cid:105) ∪ (cid:104) S , S (cid:105)} and let H − be the lower half plane. Then W ( H − )is a disk contained in the strip { < (cid:61) z < (cid:61) τ } . Let W = (cid:18) a bc d (cid:19) with ad − bc = 1 and suppose that the circle C = W ( R ∪ ∞ ) has radius r and centre z , so that (cid:61) z ≥ r >
0. Using the fact that the points T − ( z ) , T − ( ∞ ) are inverse points with respect to R (see also [25] p.91),we find that r = i/ ( c ¯ d − d ¯ c ) and z = ( a ¯ d − b ¯ c ) / ( c ¯ d − d ¯ c ). The inequality (cid:61) z > (cid:61) c ¯ d > (cid:61) z ≥ r simplifies to (cid:60) ( b ¯ c − a ¯ d ) ≥ T we see that (cid:61) τ (cid:61) τ ≥
1. Applying it to[ S , T − ] = (cid:18) − τ + 4 τ τ τ τ (cid:19) we see that (cid:61) τ ≥ /
2, and similarly applying it to [
T, S − ] we have (cid:61) τ ≥ / (cid:3) Appendix 2
To shed more light on the definition of canonical coordinates, we recallthe familiar situation for Σ , , see [2]. Consider first a Euclidean torus Σ , viewed as the quotient of C by translations S : z → z + 1 and T : z → z + i .The unit square 0 ≤ (cid:60) z, (cid:61) z ≤ with oppositesides identified by S and T . Label each side by the translation which carriesit to a paired side, so that the bottom side (cid:61) z = 0 is labelled s T since T carries it to the top side (cid:61) z = 1; similarly the left side (cid:60) z = 0 is labelled s S since S carries it to the right side (cid:60) z = 1. Any closed geodesic γ onΣ , lifts to a line in C . The lifts of γ cut ∆ in a number of pairwisedisjoint parallel arcs running from one side of ∆ to another: if the linehas slope p/q there are q ≥ s T and s T − , and | p | strands which intersect s S and s S − . By collapsing all the strands which joina fixed pair of sides into one arc we obtain one of the four configurationsshown in Figure 9, which we can view as four train tracks on Σ , . Note thatboth the supporting train track and the weight on each branch is completelydetermined by the signed slope p/q . For example, if p/q < − q ≥
0, there is one branch from s S to s S − of weight | p | − q and there arebranches joining s S − to s T − and s T to s S each of weight q , shown in thelower left hand quadrant of Figure 9.We obtain the canonical coordinates for M L (Σ , ) by viewing ∆ as aschematic representation of a fundamental domain for a hyperbolic oncepunctured torus Σ , , with the puncture at the vertices. A simple closedcurve on Σ , is also a simple closed curve on Σ , , and every simple closedcurve on Σ , can be represented in this way [2]. (The proof is to eliminatethe possibility of non zero weights on all four corner arcs and then use theswitch conditions as in Section 4.) Thus any such curve is supported on oneof the four train tracks in Figure 9 and the spaces of weights on these fourtracks are a cell decomposition for M L (Σ , ) in the usual way.Let γ p/q ∈ S (Σ , ) be the curve associated to the line of slope p/q in C . The canonical coordinates i ( γ p/q ) = ( q, p ) ∈ Z + × Z of γ p/q are the signed weights obtained as above from the original line in C . We alwaystake q = i ( S, γ ) ≥
0. We take p > s S to s T − and take p < s S − to s T − . (If p = 0 the diagonal arcs have zero weight and we areon the boundary of two cells.) It is easy to see from the above discussionthat i ( γ p/q ) determines both a train track track on Σ , and the weights onthat track, thus giving global coordinates for the homotopy classes of simpleclosed curves on M L (Σ , ). Note that the coordinates ( | p | , q ) of γ p/q are not in general equal to the weights on the branches. In fact, up to the choiceof a base point for the twist, p/q is the Dehn-Thurston twist coordinate of γ . These coordinates extend naturally by linearity and continuity to globalcoordinates for M L (Σ , ). HE MASKIT EMBEDDING OF THE TWICE PUNCTURED TORUS 43 p < p > q > pp > qq > − p − p > qq + p − p − p q − pp pp − qq q − p − qqq Figure 9.
Canonical train tracks for the once puncturedtorus. In all charts, the total weight on the vertical sides is | p | and on the horizontal sides is q . Appendix 3
We give the (presumably well known) formula for the bending angle φ between two consecutive segments of a geodesic s on ∂ C + which crosses abending line L of ∂ C + making an angle ψ with L . Let the bending anglebetween the two planes Π , Π which meet along L be θ . Let s i ⊂ Π i bethe two segments of s which meet at P ∈ L . Measure φ so that φ = θ when ψ = π/
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2, let Y ∈ Π
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