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APPEARED IN BULLETIN OF THE

arXiv:math/9201263v1 [math.GT] 1 Jan 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 141-146PLEATING COORDINATES FOR THE TEICHM ¨ULLERSPACE OF A PUNCTURED TORUSLinda Keen and Caroline SeriesAbstract. We construct new coordinates for the Teichm¨uller space Teich of a punc-tured torus into R × R+.

The coordinates depend on the representation of Teich asa space of marked Kleinian groups Gµ that depend holomorphically on a parameterµ varying in a simply connected domain in C. They describe the geometry of thehyperbolic manifold H3/Gµ; they reflect exactly the visual patterns one sees in thelimit sets of the groups Gµ; and they are directly computable from the generators ofGµ.1. IntroductionIn this paper we announce results that will appear in [4], in which we constructa new embedding in R × R+ of the Teichm¨uller space T1,1 of a punctured torus.The pullbacks of the natural coordinates in R×R+ have simple geometric interpre-tations when T1,1 is realized as a subset of C using the Maskit embedding.

In theMaskit embedding, points of T1,1 correspond to marked Kleinian groups {Gµ} thatdepend holomorphically on a parameter µ that varies in a simply connected domainM in C. Figure 1 (see p. 142), drawn by David Wright, shows the domain M. The‘coordinate grid’ in M is the preimage under our embedding of the horizontal andvertical lines in R × R+.Our coordinates, which we call pleating coordinates, have the following virtues:they relate directly to the geometry of the hyperbolic manifold H3/Gµ; they reflectexactly the visual patterns one sees in the limit sets of the groups Gµ; and theyare directly computable from the generators of Gµ. The definition of the Maskitembedding prescribes that the regular set of each group Gµ should contain preciselyone invariant component Ω0(Gµ).

The Riemann surface Ω0(Gµ)/Gµ is a puncturedtorus, and this defines the correspondence between Gµ and a point in M. Ideally,one would like to relate the geometry of the surface Ω0(Gµ)/Gµ directly to theparameter µ. This seems to be a very hard problem.

However, it is possible todetermine the relationship of µ to the geometry of the hyperbolic manifold H3/Gµ.This is the basic idea of [4].The boundary of the convex hull of the limit set of a Kleinian group acting onH3 carries an intrinsic hyperbolic metric and is a union of pleated surfaces in thesense of Thurston (see [9,1]). (Roughly speaking, a pleated surface is a hyperbolicsurface in a hyperbolic 3-manifold that is bent or pleated along some geodesic1991 Mathematics Subject Classification.

Primary 30F40, 32G14.Research partially supported by NSF grant DMS-8902881Received by the editors March 29, 1991 and, in revised form, June 5, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2LINDA KEEN AND CAROLINE SERIESFigure 1. The Maskit embedding with pleating coordinates.

PLEATING COORDINATES FOR THE TEICHM¨ULLER SPACE3lamination called its pleating locus.) We use the term pleating coordinates becauseour embedding reflects the geometry of this pleating.Since there is a natural bijective correspondence between the connected compo-nents of the regular set of Gµ and those of its convex hull boundary, exactly oneof these boundary components, ∂C0 say, is invariant under the action of Gµ.

Thetwo quotients ∂C0/Gµ = bSµ and Ω0(Gµ)/Gµ are topologically, but not conformally,the same (see [1,3]). Thus the pleated surface bSµ is topologically a punctured torus(hereafter denoted by S).

The parameters we use in our embedding reflect thegeometry, not of Ω0(Gµ)/Gµ, but of bSµ. In the rest of this paper, we shall see howthe pleating coordinates describe bSµ in terms of its pleating locus pl(µ).2.

Pleating raysThe ‘vertical’ lines of the coordinate grid in Figure 1 are the locus of points inM along which pl(µ) is a particular fixed geodesic lamination on S. We call theselines pleating rays. They may be thought of as ‘internal rays’ in M since they playa role analogous to that of the ‘external rays’ of the Mandelbrot set in the study ofthe dynamics of quadratic polynomials.Define a projective measured lamination to be a geodesic lamination togetherwith a projective class of transverse measures.The set of projective measuredlaminations on S is naturally identified with bR = R ∪{∞} (see [8]).

Because thepleating locus always carries a natural transverse measure, the bending measure (see[9,1]), we obtain for each µ a projective measured lamination on S, also denotedpl(µ). For each λ ∈R (λ ̸= ∞), there is a unique nonempty pleating rayPλ = {µ ∈M: pl(µ) = λ}.We show that the ray Pλ is asymptotic to the line ℜµ = 2λ as ℑµ →∞.As is well known [8], the simple closed curves on S correspond exactly to Q ∪{∞}.

More precisely, for each rational p/q, there is a unique free homotopy class[γp/q] ∈π1(S) for which the corresponding geodesic on the unpunctured torus is inthe (p, q)- homology class, and for which the geodesic γp/q on the punctured torusis simple. We call the sets Pp/q, rational pleating rays.The group Gµ is an embedding of π1(S) in SL(2, C), and the free homotopyclass of γp/q corresponds to a conjugacy class of Gµ under this embedding.

Choosegp/q(µ) in this conjugacy class.Theorem 1. The rational ray Pp/q is the unique branch of the locus {µ ∈C: Tr gp/q(µ) >2} that is asymptotically vertical as ℑµ →∞.

This branch contains no singularities,and it intersects ∂M in a unique point. At this point, Tr gp/q(µ) = 2.It is not hard to show that the rational ray Pp/q must be contained in thelocus where Tr gp/q(µ) is real.

The asymptotic behavior of the trace polynomialsis a straightforward consequence of the trace identities, as described in more detailbelow. What is much more interesting is that the pleating ray is precisely thatbranch of the real locus defined above.

The key point in proving this is to showthat along Pp/q, the invariant component Ω0(Gµ) is a circle chain, that is, a unionof overlapping circles that fit together in a manner reflecting the continued fractionexpansion of p/q. There are two main points: for sufficiently large c > 0, the rayPp/q intersects the line ℑµ = c in a unique point, and, circle chains persist under

4LINDA KEEN AND CAROLINE SERIEScontinuous deformations of the group Gµ along Pp/q. The endpoint of the pleatingray represents a cusp group in which the element gp/q(µ) has become parabolic.

In[2], we prove that there are no other cusp groups corresponding to gp/q(µ) in ∂M.The circle chain patterns are visually apparent in computer pictures of the limitsets of groups on the pleating ray and near to ∂M. When µ reaches the endpointof the pleating ray, the overlapping circles become tangent.

Such chains of tangentcircles were discovered by David Wright in the course of a computer investigationof ∂M, and our interest in them was the starting point of the present work.Wright obtained his striking pictures of ∂M by using an inductive procedurerelated to the continued fraction expansion of p/q to canonically choose a particularword Wp/q ∈Gµ in the conjugacy class of the image of [γ(p/q)]. He computedTr Wp/q as a polynomial in µ by means of the trace identities; and, by using hisenumeration scheme to give a systematic choice of initial point, he used Newton’smethod to find, for each p/q, a particular root of the equation Tr Wp/q(µ) = 2.These solutions form the boundary curve in Figure 1.3.

Pleating lengthAs µ moves down each rational ray, the length in H3 of the pleating locus pl(µ)provides a natural parameter. This parameter, however, is not continuous as µmoves across rays.

In fact, if µn ∈Ppn/qn converges to µ ∈Pλ, where λ is notrational, the hyperbolic lengths of the pleating loci pl(µn) always approach infinity.However, it is possible to define a global length parameter that is continuous asµ moves across rays by making a special choice of transverse measure, which wecall the pleating measure, for each projective measured lamination µ on S. Wedefine the pleating length PL(µ) of Gµ to be the length of pl(µ) with respect to thepleating measure of pl(µ). On a rational ray Pp/q, the pleating length of Gµ turnsout to be the hyperbolic length of γp/q(µ) divided by the intersection number ofγp/q with the fixed curve γ∞.

The pleating length gives a natural parameterizationof the pleating rays, and the horizontal lines of the coordinate grid in Figure 1 arelines of constant pleating length.To prove continuity properties of the pleating measure and pleating length, weuse the continuous dependence on µ of the hyperbolic structure of the convex hullboundary, the pleating locus, and the bending measure. These facts are also neededin the proof of Theorem 1.

We prove all of these results in a more general settingin [3].4. The coordinatesIt is apparent from Figure 1 that the partial foliation of M by the rational raysshould extend to a foliation by the real rays Pλ, λ ∈R.

To show that it does, wecharacterize the irrational pleating rays as the real loci of a family of holomorphicfunctions. The complex translation length of a loxodromic element g ∈SL(2, C) isdefined as 2 arccosh(Tr g)/2 (see [9]).

It follows from Theorem 1 that on the rationalray Pp/q, the polynomial Tr gp/q(µ) is real valued and, hence, that the complextranslation length is real. Since Pp/q is connected, we can choose a well-definedbranch of the complex translation length of gp/q(µ) by specifying that it be real onPp/q.

We show that the family of functions {Lp/q = 1/q arccosh(Tr gp/q(µ))}p/q∈Qis normal in M and that on Pp/q, the function Lp/q(µ) coincides with the pleating

PLEATING COORDINATES FOR THE TEICHM¨ULLER SPACE5length PL(µ). Taking limits in O(M), the space of analytic functions on M withthe topology of uniform convergence on compact subsets, we proveTheorem 2.

The family {Lp/q} extends to a family {Lλ}λ∈R of complex analyticfunctions defined on M, such that the function from M to R, given by µ 7→PL(µ),and the function from R to O(M), given by λ 7→Lλ, are both continuous and suchthat the function Lλ is real valued on Pλ.That the real rays are a codimension one foliation of M follows fromTheorem 3. The real pleating ray Pλ is a connected component of the real locus ofLλ in M. This component contains no singularities and is asymptotic to ℜµ = 2λas ℑµ →∞.Our main theorem isTheorem 4.

The map from M to R × R+ defined by µ 7→(pl(µ), PL(µ)) is ahomeomorphism onto its image.The fact that the map described in Theorem 4 is surjective will be proved else-where.In a future paper, we expect to use the methods described here to give a completedescription of ∂M and of the approach to ∂M along the internal rays. In particular,we hope to give proofs of McMullen’s theorems [6,7], conjectured by Bers, that thecusp groups are dense in ∂M and that ∂M is a Jordan curve.

Although our workhere relates to the punctured torus, most of the techniques we have developedapply more generally. We plan to extend our analysis to any union of surfacesof finite topological type.

David Wright has already produced computer picturesof the analogous coordinatization for the (one complex dimensional) Riley slice ofSchottky space, and the discussion in [4] goes over to that situation (see [5]).AcknowledgmentsWe wish to express our thanks to a number of people. David Wright introducedus to this problem and has generously allowed us to use his computer pictures.Curt McMullen has graciously shared his ideas and work with us.

We also want tothank Jonathan Brezin, David Epstein, Michael Handel, Steve Kerckhoff, PaddyPatterson, and Bill Thurston for many helpful conversations throughout the courseof this work. Finally, we would like to acknowledge the support of the NSF, theSERC in the United Kingdom, the Danish Technical University, and the IMS atSUNY at Stonybrook.References1.

D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan,and measured pleated surfaces, Analytical and Geometric Aspects of Hyperbolic Space (D.B.

A. Epstein, ed. ), London Math.

Soc. Lecture Note Ser., vol.

111, Cambridge Univ.Press, Cambridge and New York, 1987, pp. 112–253.2.

L. Keen, B. Maskit, and C. Series, Geometric finiteness and uniqueness for Kleimangroups with circle packing limit sets, IMS SUNY preprint, 1991.3. L. Keen and C. Series, Continuity of convex hull boundaries, IMS SUNY, preprint, 1990/16.4., Pleating coordinates for the Maskit embedding of the Teichm¨uller space of punc-tured tori, IMS SUNY, 1991/2.5., The Riley of Shottky space, Warwick Univ., preprint, 1991.6.

C. T. McMullen, Cusps are dense, Ann. of Math.

(2) 133 (1991), 217–247.

6LINDA KEEN AND CAROLINE SERIES7., personal communication.8. C. Series, The geometry of Markoffnumbers, Math.

Intelligencer 7 (1985), 20–29.9. W. P. Thurston, Geometry and topology of three manifolds, Lecture notes, Princeton Univ.,NJ, 1979.Mathematics Department, CUNY, Lehman College, Bronx, New York 10468Mathematics Institute, Warwick University, Coventry CV4 7AL, United Kingdom

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