Pleating invariants for punctured torus groups

Linda Keen^{a,1} and Caroline Series^{b}

aMathematics Department CUNY Lehman College Bronx, NY 10468, U.S.A.

bMathematics Institute Warwick University Coventry CV4 7AL, U.K.

Abstract

In this paper we give a complete description of the space QF of quasifuchsian punctured torus groups in terms of what we call pleating invariants. These are natural invariants of the boundary ∂C of the convex core of the associated hyperbolic 3-manifold M and give coordinates for the non-Fuchsian groups QF − F. The pleating invariants of a component of ∂C consist of the projective class of its bending measure, together with the lamination length of a fixed choice of transverse measure in this class. Our description complements that of Minsky in [35], in which he describes the space of all punctured torus groups in terms of ending invariants which characterize the asymptotic geometry of the ends of M.

Pleating invariants give a quasifuchsian analog of the Kerckhoff-Thurston description of Fuchsian space by critical lines and earthquake horocycles. The critical lines extend to pleating planes on which the pleating loci of ∂C are constant and the horocycles extend to BM-slices on which the pleating invariants of one component of ∂C are fixed.

We prove that the pleating planes corresponding to rational laminations are dense and that their boundaries can be found explicitly. This means, answering questions posed by Bers in the late 1960’s, that it is possible to compute an arbitrarily accurate picture of the shape of any embedding of QF into C^{2}.

  Email addresses: linda@lehman.cuny.edu, comet.lehman.cuny.edu/keenl

(Linda Keen), cms@maths.warwick.ac.uk (Caroline Series). 1 Research partially supported by NSF # 9622965, PSC-CUNY, EPSRC

Preprint submitted to Elsevier Science 6 June 2003

1 Introduction

In his recent paper [35], Minsky gave a full description of the space of punctured torus groups in terms of their ending invariants. These invariants are the conformal structures of the quotient surfaces of the regular set of the group acting on the Riemann sphere, or, if a component is degenerate, the corresponding ending lamination of Thurston.

In this paper we give an alternative description of quasifuchsian space QF in terms of what we call pleating invariants. These replace conformal structures at infinity by natural invariants of the geometry of the boundary of the convex core of the associated three manifold. These invariants again extend naturally to ending laminations for groups on the boundary of QF. Pleating invariants have considerable computational advantages: we show how they can be used to explicitly locate the group with given invariants, and to compute the shape and boundary of QF, for any embedding into C^{2}.

A punctured torus group < G; A, B > is a free marked two generator discrete subgroup of P SL(2, C) such that the commutator of the generators is parabolic. Such a group is the image of a faithful representation ρ of the fundamental group of a punctured torus T_{1} with presentation π_{1}(T_{1}) =< α, β >; the commutator of the generators represents a loop around the puncture and the ordered pair (A, B) = (ρ(α), ρ(β)) is the marking. The group G acts as a discrete group of isometries of hyperbolic space H^{3} and the quotient hyperbolic manifold M = H^{3} /G is a product T_{1} × (−1, 1).

A punctured torus group also acts as a group of conformal automorphisms of the Riemann sphere Ĉ and partitions it into two invariant subsets, the open (possibly empty) regular set Ω and the closed limit set Λ. The group G is quasifuchsian if Ω consists of two non-empty simply connected invariant components denoted Ω^{±}. The quotients Ω^{±}/G are punctured tori with conformal structures inherited from Ĉ.

Quasifuchsian space QF is the space of quasifuchsian marked punctured torus groups modulo conjugation in P SL(2, C); Fuchsian space F is the subset such that the components Ω^{±} are round disks.

The convex hull C of Λ in H^{3} is also invariant under G. The hyperbolic manifold C/G is called the convex core of G. If G is quasifuchsian, but not Fuchsian, ∂C/G consists of two components, ∂C^{±}/G. Each component is homeomorphic to T_{1} and admits an intrinsic hyperbolic structure making it a pleated surface in the sense of Thurston. Such a surface is a hyperbolic surface “bent” along a geodesic lamination called the pleating locus or bending lamination. The pleating locus carries a natural transverse measure, the bending measure pl^{±}(G).

For any measured geodesic lamination μ on a hyperbolic surface Σ, we denote the projective class of μ by [μ] and the underlying lamination by |μ|. Writing l_{μ} for the lamination length of μ, we note that if μ, μ′ are in the same projective class, so that μ = cμ′, c > 0, then l_{μ} = cl_{μ′}. We define the pleating invariants for G ∈ QF − F to be the projective class of the pair (μ^{±}, l_{μ^{±}}) for any choice of measured laminations μ^{±} in [pl^{±}].

We prove

Theorem 1 A non-Fuchsian quasifuchsian marked punctured torus group is determined by its pleating invariants, uniquely up to conjugacy in P SL(2, C).

The essential idea is to study the sets in QF on which some or all of the pleating invariants are constant; in particular, we study the set P_{μ,ν} ⊂ QF for which [pl^{+}] = [μ], [pl^{−}] = [ν]. Clearly P_{μ,ν} depends only on the projective classes [μ], [ν] of μ, ν. We prove that these sets are connected real two dimensional submanifolds of QF whose boundaries meet F and ∂QF in specific analytic curves; as the projective classes vary, the sets P_{μ,ν}, which for obvious reasons we call pleating planes, foliate QF − F. We are also able to describe exactly how the closure of P_{μ,ν} meets F.

The space QF has a natural C^{2}-holomorphic structure induced from P SL(2, C). Let U ⊂ QF. An R^{2}-locus in U is a set f^{−1}(R^{2}) ∩ U where f : U → C^{2} is a non-constant holomorphic function defined on U. A singularity is a point where Det(Jac_{f}(z)) = 0. For example, Fuchsian space is an R^{2}-locus in QF, (see section 7.1).

The starting point for our analysis of P_{μ,ν} is to prove that for μ ∈ ML, the length function l_{μ} on F extends to a holomorphic function λ_{μ}, called the complex length of μ, on QF, and that λ_{μ} is real valued at points where the projective class of pl^{±} is [μ]. Thus P_{μ,ν} is contained in the R^{2}-locus of the holomorphic function L_{μ,ν} = λ_{μ} × λ_{ν} from QF to C^{2}.

To describe P_{μ,ν} more precisely, we recall some facts about Fuchsian space F. Let μ be a measured geodesic lamination on a hyperbolic surface Σ. The distance t earthquake E_{μ}(t) along μ gives a one parameter family of deformations of F which generalize Fenchel-Nielsen twists along simple closed geodesics. For a point p ∈ F, we denote the earthquake path {E_{μ}(t)(p) : t ∈ R} through p by E_{μ}^{p}. The earthquake path is contained in F and meets ∂F, the Thurston boundary of F, in the point [μ]. Kerckhoff proved that for each measured lamination ν whose intersection i(μ, ν) with μ is non-zero, the length function l_{ν} has a unique minimum along E_{μ}^{p}.

In the special case of the punctured torus, it is an easy consequence of Kerckhoff’s results that for each c > 0, there is a unique earthquake path E_{μ,c} on which l_{μ} ≡ c. We denote the point at which l_{ν} is minimal on this path by p_{μ,ν,c}, and set f_{μ,ν}(c) = l_{ν}(p_{μ,ν,c}). For fixed μ, ν and variable c, the points p_{μ,ν,c} define an analytic path F_{μ,ν}, which we call a critical line; it meets ∂F in the points [μ], [ν]. The length functions l_{μ}, l_{ν} are monotonic on F_{μ,ν} and f_{μ,ν}(c) is continuous, decreasing monotonically from ∞ to 0 on its domain (0, ∞).

The following result completely describes the pleating planes P_{μ,ν}; in particular it shows that P_{μ,ν} can be viewed as an extension into QF of the critical line F_{μ,ν}.

Theorem 2 Let (μ, ν) be measured laminations on T_{1} with i(μ, ν) > 0. Then P_{μ,ν} is a non-empty connected non-singular component of the R^{2}-locus in QF − F of the function L_{μ,ν}. The restriction of L_{μ,ν} to P_{μ,ν} is a diffeomorphism to the open region under the graph of the function f_{μ,ν} in R^{+} × R^{+}.

The boundary of the closure of P_{μ,ν} in QF is the critical line F_{μ,ν} ⊂ F; it is mapped homeomorphically by L_{μ,ν} to the graph of f_{μ,ν}. The planes P_{ν,μ} and P_{μ,ν} are disjoint with common boundary F_{μ,ν} in QF. The set P_{μ,ν} ∪ P_{ν,μ} ∪F is an R^{2}-locus in QF and the union P_{μ,ν} ∪ P_{ν,μ} ∪F_{μ,ν} may be regarded as the extension of the μ, ν critical line to QF.

The three components of the boundary of the image of P_{μ,ν} in R^{+} × R^{+} correspond to three distinct parts of its closure in the set of algebraic limits of groups in QF. As above, the component corresponding to the graph of f_{μ,ν} represents groups on the critical line F_{μ,ν} ⊂ F. For limit groups corresponding to the axis λ_{μ} = 0 the component Ω^{+} has degenerated and the support |μ| of μ is an ending lamination; the bending measure of ∂C^{−}, however, is still in the projective class of ν. Likewise, for limit groups corresponding to the axis λ_{ν} = 0, the component Ω^{−} has degenerated and the ending lamination is |ν|. The boundary point (0, 0) represents a doubly degenerate group, unique by the results of [35] (or [17] in the rational case), with the two ending laminations |μ| and |ν|.

Theorems 1 and 2 together show that we have a nice coordinate system on QF − F: theorem 1 shows that the map to pleating invariants is injective and theorem 2 describes the image.

The measured lamination μ is called rational if its support is a simple closed geodesic. Such a geodesic can only belong to the pleating locus |pl^{±}| if its representatives V ∈ G are purely hyperbolic and hence have real trace. Given any embedding QF into C^{2}, the generators of G are holomorphic functions of the embedding parameters and Tr V is a polynomial in the entries of the generators. In particular, given any elements V, W ∈ G representing distinct simple closed curves |μ|, |ν| on T_{1}, one can compute the position of the critical line F_{μ,ν}. If both laminations μ, ν are rational, we call P_{μ,ν} a rational pleating plane. Theorem 2 implies

Theorem 3 Let μ, ν be rational laminations represented by non-conjugate elements V, W ∈ G. Then P_{μ,ν} and P_{ν,μ} are the unique components of the R^{2}-locus of the function Tr V × Tr W in QF − F whose closures meet F in F_{μ,ν}. On P_{μ,ν} ∪ P_{ν,μ} the function Tr V × Tr W is non-singular and the boundary of P_{μ,ν} ∪ P_{ν,μ} can be computed by solving Tr V = ±2 and Tr W = ±2 on this component.

We also prove

Theorem 4 The rational pleating planes are dense in QF.

In the late 1960’s, Bers asked whether it was possible to find the shape of quasifuchsian space by explicit computation; one would expect the punctured torus to be the easiest case. Partial results were obtained by a number of people, some using computational methods, among them [16,41,47], others developing new tools and techniques [13,30]. For the punctured torus, the above results give an effective means of finding the boundary of the image of any chosen embedding of QF into C^{2}, answering Bers’ question in full.

We also study the way in which the pleating planes fit together transversally to the real locus of L_{μ,ν}. This is done by fixing the pleating invariants of one side of ∂C; one can regard this as analogous to fixing the ending invariant on one side in QF, to obtain the classical Bers slice [1]. Thus for a fixed measured lamination μ and c > 0, we define the BM-slice BM_{μ,c}^{+} as the subset of QF on which [pl^{+}] = [μ] and λ_{μ} = c. The BM-slices are subsets of the quakebend planes Q_{μ,c} obtained by Thurston’s quakebend construction along the measured lamination μ (see [9] and section 7 below). These are extensions of the earthquake path E_{μ,c} into QF. Unlike the path E_{μ,c} which is completely contained in F, the quakebend plane Q_{μ,c} is not totally contained in QF. We prove

Theorem 5 Let μ be a measured lamination on T_{1} and let c > 0. Then the closures in QF of exactly two of the connected components of Q_{μ,c} ∩(QF −F) meet F. These components are the slices BM_{μ,c}^{±} and the closure of each slice meets F precisely in the earthquake path E_{μ,c}. Furthermore, each slice is simply connected and retracts onto E_{μ,c}.

Thus, just like the Bers slices, the BM-slices are complex planes in QF and like them, they foliate QF − F. We note that while the boundary of the pleating planes consists of smooth curves, the boundary of a BM-slice is typically a fractal-like curve. Pictures of such curves may be found in [18,38,47].

The basis of the proofs of the above results are two important theorems which control the local behavior of pleating invariants. We call these the limit pleating theorem and local pleating theorem respectively. Roughly, the limit pleating theorem states that if the pleating invariants of a sequence of groups in QF converge, then the groups converge to an algebraic limit; furthermore the limit group is in QF provided the limit pleating lengths are non-zero. It is closely related to Thurston’s double limit theorem [44], and also to the ‘Lemme de fermeture’ in [5].

The local pleating theorem makes essential use of the complex length function λ_{μ}. As mentioned above, if q ∈ QF, then λ_{pl^{+}}(q) ∈ R. In general, the converse of this result is false; however the local pleating theorem gives a partial result: if q ∈ P_{μ} so that λ_{μ}(q) ∈ R, then for q′ near q, the condition λ_{μ}(q′) ∈ R implies that q′ ∈ P_{μ}. (As discussed in the introduction of [20] this result does not hold for higher genus.)

The theory of quakebends as developed in [9] allows us to extend the earthquake paths E_{μ,c} into a family of holomorphic planes Q_{μ,c} in QF. We reduce the problem of studying the sets P_{μ,ν} by restricting to the subset P_{μ,ν,c} of P_{μ,ν} on which the value of λ_{μ} is fixed at c ∈ R^{+}. For reasons that will be clear below, we call such a set a pleating ray. In Q_{μ,c}, the complex length λ_{ν} restricts to a holomorphic function of one variable and it follows from the limit and local pleating theorems that P_{μ,ν,c} is both open and closed in the R-locus of λ_{ν} in Q_{μ,c} ∩ QF.

The fact that the pleating rays are non-empty and the discussion of how they meet Fuchsian space F results from the detailed study of the situation near F which was carried out in [20]. We also have detailed information from [39] about rays for which the laminations μ, ν are rational and correspond to a pair of generators of T_{1}. Combining this information allows us to prove

Theorem 6 Let μ, ν be measured laminations on T_{1} with i(μ, ν) > 0 and let c > 0. Then the set P_{μ,ν,c} ⊂ QF on which [pl^{+}] = [μ], [pl^{−}] = [ν] and l_{μ} = c, is a non-empty connected non-singular component of the R-locus of the restriction of λ_{ν} to Q_{μ,c}. This restriction is a diffeomorphism onto its image (0, f_{μ,ν}(c)) ⊂ R^{+}.

Theorem 2, and hence also theorem 1, are immediate consequences of this result. We also easily deduce theorem 5.

For groups on the boundary of QF, at least one of the components Ω^{±} degenerates and it is clear that our pleating invariants extend naturally to the corresponding ending laminations for which the length (and also the complex length) is always 0. It is also clear that these invariants should also characterize boundary groups; careful analysis requires the study of generalized Maskit slices in which the fixed ending lamination is irrational, see [31].

The reader is referred to [35] for a good outline of the history relating to the study of punctured torus groups.

Some of the ideas of this paper, in particular the relation of pleating planes to the Kerckhoff picture of F and the idea of looking at the BM-slices, grew out of discussions with John Parker, and we should like to thank him for his input into this work. We should also like to thank our referees for their detailed reading of earlier versions of this paper, in particular, for having signalled, in view of the examples in [24], a gap in our proof of theorem 15, as well as having suggested a more direct proof of lemma 41 and a simplification of the proof of theorem 23. We would also like to thank Yair Minsky for conversations which helped us precisely locate the above mentioned gap, and Francis Bonahon and Cyril Lecuire for very useful discussions about how to rectify it.

The paper is organized as follows. Section 2 contains background on the punctured torus, geodesic laminations and surfaces. Section 3 explains the picture of earthquake paths and critical lines in F and in section 4 we review results on pleated surfaces and the convex hull boundary. We prove the limit pleating theorem in section 5. In section 6 we show how to complexify the length functions and show that the complex length of the pleating locus is real. In section 7 we review results about quakebends and the convex hull boundary and then in section 8 prove the local pleating theorem. We also derive various important consequences of this result, including the proof of theorem 4. In section 9 we prove our main results, theorem 6 on pleating rays and theorem 2 on pleating planes. In section 10 we study BM-slices, proving theorem 5, and we conclude in section 11 with a discussion of rational pleating planes, computation, and some explicit examples. For readability, the proofs of three technical results are deferred to the appendix.

2 Background

2.1 Punctured torus groups and markings

Let T_{1} be a torus with one puncture and a fixed orientation. Any pair of simple closed loops on T_{1} that intersect exactly once are free generators of π_{1}(T_{1}). Let (α, β) be such an ordered pair of free generators, chosen so that their commutator αβα^{−1}β^{−1} represents a loop around the puncture that is positively oriented around the component of T_{1} not containing the puncture. The ordered pair (α, β) is called a marking.

A punctured torus group is a discrete subgroup G ⊂ P SL(2, C) that is the image of a faithful representation ρ of π_{1}(T_{1}) such that the image of the loop around the puncture is parabolic. If (α, β) is a marking of T_{1}, and if A = ρ(α), B = ρ(β), then the commutator K = ABA^{−1}B^{−1} is parabolic and the ordered pair (A, B) = (ρ(α), ρ(β)) is called a marking of G. If γ is any simple closed curve on T_{1}, then we can always choose a curve δ such that (γ, δ) is a marking of T_{1}. Setting ρ(γ) = V, ρ(δ) = W, then all possible markings (V, W′), W′ ∈ G of G are of the form (V, V^{m}W), m ∈ Z.

The group G is quasifuchsian if the regular set Ω consists of two non-empty simply connected invariant components Ω^{±}. The limit set Λ(G) is topologically a circle. Quasifuchsian space QF is the space of marked quasifuchsian punctured torus groups modulo conjugation in P SL(2, C); it has a holomorphic structure induced from the natural holomorphic structure of SL(2, C). Fuchsian space F is the subset such that the components Ω^{±} are round disks. It is canonically isomorphic to the Teichmüller space of marked conformal structures on T_{1}.

The quotients Ω^{±}/G are punctured tori with conformal structures, and hence also orientations, inherited from Ĉ; the orientations of Ω^{+}/G and T_{1} agree whereas those of Ω^{−}/G and T_{1} are opposite. This means Ω^{+}(G) is the component such that A^{−}, B^{+}, A^{+}, B^{−} occur in counterclockwise order around its boundary Λ(G), where, for a loxodromic g ∈ SL(2, C), g^{+} and g^{−} denote its attracting and repelling fixed points respectively. Thus an alternative way to choose a marking of G is to choose any pair of generators X, Y of G, and to specify the choice of Ω^{+} by choosing it to be the component such that the fixed points X^{−}, Y^{+}, X^{+}, Y^{−} run counterclockwise around its boundary.

A point q ∈ QF represents an equivalence class of marked groups in P SL(2, C). We choose once and for all a triple of distinct points in Ĉ and let G = G(q) denote the representative normalized by choosing A^{−}, A^{+}, K^{∞} to be this this fixed triple, where K^{∞} is the fixed point of the parabolic K. We will refer to this as the standard normalization. If it is clear from the context, for readability, we suppress the dependence on q.

Note that throughout this paper, QF and F refer to the special case of the once punctured torus T_{1} only.

2.2 Laminations

Let Σ be a hyperbolic surface. We denote by S the set of all simple closed geodesics on Σ. There is one such geodesic in each free homotopy class of simple closed non-boundary parallel loops, and the set S is independent of the hyperbolic structure on Σ.

Geodesic laminations were introduced by Thurston [45] as a generalization of simple closed geodesics. A geodesic lamination on Σ is a closed set that is a union of pairwise disjoint simple geodesics called its leaves. We denote by GL = GL(Σ) the set of of all geodesic laminations on Σ; GL(Σ) is also independent of the hyperbolic structure, see e.g. [7] section 4.1.4 and [19] section 3.7.

The Hausdorff topology on the set of closed subsets of Σ induces a topology on GL. Two laminations are close in this topology if any long segment of a leaf of either one is closely approximated by a long segment of a leaf of the other. See [7,9,40] for a complete discussion.

A measured lamination μ on Σ is a geodesic lamination, called the support of μ and denoted |μ|, together with a transverse measure, also denoted μ. We denote the set of all measured laminations on Σ by ML(Σ). The space ML is topologized by defining laminations to be close in ML if the measures they assign to any finite set of transversals are close, for details see [7] or [19]. Notice that the support of any measured lamination always avoids a definite neighborhood of each cusp. The relationship between the topologies on ML and GL is discussed in section 2.3 below.

Any element γ ∈ S carries a natural transverse measure δ_{γ} which assigns unit mass to each intersection with γ. We call a measured geodesic lamination on Σ rational if its support is a union of curves in S. The maximum number of disjoint loops in S on the punctured torus T_{1} is one, so that rational measured laminations are of the form μ = kδ_{γ}, k > 0. We denote the set of all rational measured laminations on Σ by ML_{Q}(Σ); the set ML_{Q} is dense in ML.

Two measured laminations μ, μ′ ∈ ML are projectively equivalent if |μ| = |μ′| and if there exists k > 0 such that for any arc σ transverse to the leaves of |μ|, μ′(σ) = kμ(σ). We write [μ] for the projective class of μ ∈ ML(Σ). We denote the set of projective equivalence classes on Σ by P ML(Σ). It is well known that P ML(T_{1}) is homeomorphic to S^{1} ≃ R ∪ {∞} (see for example [45]).

The length l_{γ} of a geodesic γ ∈ S generalizes to arbitrary laminations. Let ϕ represent a hyperbolic structure on Σ. For μ ∈ ML, the length l_{μ}(ϕ) is the total mass, on the surface with structure ϕ, of the measure that is the product of hyperbolic distance along the leaves of μ with the transverse measure μ. In particular, if μ ∈ ML_{Q}(Σ) with μ = δ_{γ}, then l_{γ} = ∫_{Σ} dδ_{γ}ds is just the hyperbolic length of γ.

Clearly, if μ′ = kμ then l_{kμ} = kl_{μ}. We define

[μ, l_{μ}] =^{def} {kμ, kl_{μ} ∈ ML × R^{+} : k > 0}

and call it the projective class of the pair (μ, l_{μ}).

The geometric intersection number i(γ, γ′) of two geodesics γ, γ′ ∈ S extends to a continuous function i(μ, ν) on ML(Σ) (see for example [21]). For Σ = T_{1}, i(μ, ν) > 0 is equivalent to [μ] ≠ [ν]. We also recall the well known fact that on T_{1}, measured laminations are uniquely ergodic; that is, if μ, μ′ ∈ ML(T_{1}) with |μ| = |μ′|, then [μ] = [μ′].

2.3 The convergence lemma

In general, laminations which are close in ML may not be close in the Hausdorff topology on GL. For example, one can put a transverse measure ν′ on a long closed geodesic γ′ spiralling in to a closed geodesic γ with transverse measure ν, such that ν, ν′ are close in ML but γ′ has arcs far from γ. A sequence of laminations may converge in ML to a measured lamination ν_{0} with support in one part of Σ, while simultaneously limiting on a closed curve with support disjoint from |ν_{0}|.

The following lemma gives conditions under which Hausdorff convergence is a consequence of convergence in ML. We note that the lemma depends crucially on the fact that on T_{1}, any irrational measured lamination is maximal. As stated, it is false for more general surfaces, and it is false if ν_{0} ∈ ML_{Q}(T_{1}).

Lemma 1 Suppose that ν_{0} ∈ ML(T_{1})−ML_{Q}(T_{1}), and that ν and ν_{0} are close in ML(T_{1}). Then |ν| and |ν_{0}| are close in the Hausdorff topology on GL(T_{1}).

This lemma is proved in appendix 12.1.

From now on, unless specifically stated, GL, ML, P ML will always refer to T_{1}.

3 Fuchsian space

Kerckhoff and Thurston used earthquake deformations to study the set of hyperbolic structures on a surface Σ. For T_{1} the description is especially simple. For an unpunctured torus, the Teichmüller space is a disk. Thinking of this disk as the hyperbolic plane D with boundary circle S^{1}, for each boundary point ξ there is a foliation of D by horocycles tangent to ∂D at ξ. Joining each pair of distinct boundary points ξ, η is a unique geodesic γ_{ξ,η} which, for fixed ξ and varying η, give another foliation of D. It follows from Kerckhoff’s results [21,23] and Thurston’s compactification of Teichmüller space [10] (see also [12]), that there is an analogous picture for F, the Teichmüller space of T_{1}. This picture is certainly well known and described for Teichmüller spaces of compact surfaces in [23]. As it is of central importance for us we explain it in detail here.

Since the torus is homogeneous, F is holomorphically the same as the Teichmüller space of the unpunctured torus, namely D. The Thurston boundary of F is naturally identified with the circle S^{1}. The classical Fenchel-Nielsen coordinates for F are the length l_{α} of a generating curve α and a corresponding twist parameter t_{α}. In [21], [22], the Fenchel-Nielsen deformation defined by varying the twist parameter t_{α} is generalized to a map E_{μ}(t) : F → F defined relative to a measured lamination μ ∈ ML. The map E_{μ}(t) is called the time t earthquake along μ; when needed for clarity we write the parameter t as t_{μ}. The family E_{μ}(t), t ∈ R is a one parameter family of deformations of F; in particular E_{μ}(0) = id.

For p ∈ F, we define the earthquake path along μ through p by

E_{μ}^{p} = {E_{μ}(t)(p) ∈ F : t ∈ R}.

Clearly, E_{μ}^{p} is invariant under the earthquakes E_{μ}(t). In [22], Kerckhoff showed that E_{μ}^{p} is a real analytic path in F. Along E_{μ}^{p}, the length l_{μ} is constant. Thus for every p ∈ F, E_{μ}(t)(p) tends to the same point [μ] ∈ ∂F as t → ±∞.

In [21], Kerckhoff showed that if ν ∈ ML with i(μ, ν) > 0, then along an earthquake path E_{μ}^{p}, the length l_{ν} is a strictly convex real analytic function of t and l_{ν}(t) → ∞ as t → ±∞. Thus l_{ν} has a unique minimum on E_{μ}^{p}; at this point we say that l_{ν} is minimal with respect to E_{μ}^{p}. Wolpert showed in addition, that at the minimum, d^{2}l_{ν}/dt_{μ}^{2} > 0. (Actually Wolpert proved this only when μ, ν are rational; this case follows by inspection of his second derivative formula [46] Theorem 3.4, in which all terms are clearly positive. The case μ general but ν rational follows by inspection of [9] Theorem 3.10.1. For general ν, see Corollary 21 below.)

It follows from the anti-symmetry of the derivative formula

dl_{ν}/dt_{μ} = ∫_{T_{1}} cos θdμdν,

(where θ is the angle, measured counterclockwise, from a leaf of |μ| to a leaf of |ν| at each intersection point of the laminations |μ|, |ν|), that the minimum points for l_{ν} along E_{μ} and l_{μ} along E_{ν} coincide, and that at this minimum point p we have DE_{ν}(t_{ν})(p) = −DE_{μ}(t_{μ})(p).

The results which follow are simple consequences of Kerckhoff’s results applied to T_{1}.

Proposition 2 For any c ∈ R^{+} and μ ∈ ML, there is at most one earthquake path E_{μ}^{p} along which l_{μ} = c.

Proof: Suppose that there are two such paths, E_{1}, E_{2}. They are clearly disjoint, moreover since F ∪ ∂F is a closed disk and both E_{1} and E_{2} meet ∂F at the same point [μ], one path, E_{1} say, separates F ∪ ∂F so that one component of the complement contains both E_{2} and ∂F − {[μ]}. Choose ν ∈ ML with i(μ, ν) > 0 and let p be the minimum point for l_{ν} on E_{1}. Then E_{2} separates E_{1} from ∂F and hence p from [ν], so that E_{ν}^{p} must also cut E_{2} at a point p′. Since p is the unique minimum point for l_{μ} on E_{ν}^{p}, and since l_{μ}(p) = l_{μ}(p′) we have a contradiction. □

We denote the unique earthquake path on which l_{μ} = c by E_{μ,c}. It follows easily from proposition 3 below that E_{μ,c} ≠ ∅. Since for s > 0, E_{sμ}(t) = E_{μ}(st) and l_{sμ} = sl_{μ}, we have E_{sμ,sc} = E_{μ,c}. For ν ∉ [μ], we denote the minimum point for l_{ν} on E_{μ,c} by p(μ, ν, c). We define a function f_{μ,ν} : R^{+} → R^{+} by f_{μ,ν}(c) = l_{ν}(p(μ, ν, c)). Notice that from the definition, f_{μ,ν} = f_{ν,μ}^{−1}.

For each pair μ, ν ∈ ML × ML, μ ∉ [ν], set

F_{μ,ν} = {p ∈ F|dl_{ν}/dt_{μ}(p) = 0}

Note that F_{μ,ν} depends only on [μ], [ν], and that by the antisymmetry of the derivative, F_{μ,ν} = F_{ν,μ}. We call F_{μ,ν} the μ, ν-critical line. This is justified by the following proposition.

Proposition 3 For each pair μ, ν, i(μ, ν) > 0, the locus F_{μ,ν} is a real analytic path in F with endpoints at [μ] and [ν] in ∂F. Both l_{μ} and l_{ν} are strictly monotonic on F_{μ,ν} and vary from 0 to ∞ in opposite directions.

Proof: By Wolpert’s result, d^{2}l_{ν}/dt_{μ}^{2} > 0 at every point of F_{μ,ν}. Therefore F_{μ,ν} is a disjoint union of real analytic arcs.

We claim the function l_{μ} is strictly monotonic on each component of F_{μ,ν}. If not, there is an earthquake path E_{μ,c} that meets F_{μ,ν} in two distinct points. Both these points are critical for l_{ν} on E_{μ,c} which is impossible.

Since l_{μ} is real analytic, its restriction to F_{μ,ν} is open and proper and hence its range must be (0, ∞). Clearly, as l_{μ}(p) → 0 along F_{μ,ν}, we have p → [μ] ∈ ∂F. Thus each component of F_{μ,ν} is an embedded arc with endpoints [μ] and [ν] in ∂F.

If F_{μ,ν} had two components, then, for some c > 0, we could find a path E_{μ,c} intersecting both components of F_{μ,ν}. Thus l_{ν} would be minimal at two points on E_{μ,c} which is impossible.

By the anti-symmetry in the formulas, we see that l_{ν} also varies monotonically from 0 to ∞ along F_{μ,ν} but in the opposite direction. □

Corollary 4 For any c ∈ R^{+} and μ ∈ ML there is a unique earthquake path E_{μ}^{p} along which l_{μ} = c.

Remark 5 In [23], Kerckhoff proves that given (μ, ν) ∈ ML with i(μ, ν) > 0 and such that μ, ν fill up the surface (that is, the complement of their union consists of pieces which are either simply connected or a neighborhood of the puncture), then for each t ∈ (0, 1) there is a unique p ∈ F at which the function tl_{μ}(p) + (1 − t)l_{ν}(p) attains minimum. As t varies keeping μ, ν fixed, the set of these minima is a line. For the punctured torus, any pair (μ, ν) ∈ ML with i(μ, ν) > 0 fills up the surface. While not strictly needed for our development, the following lemma confirms that for the punctured torus, Kerckhoff’s line of minima is identical with our critical line, see also [23] theorem 3.4.

Lemma 6 Suppose that i(μ, ν) > 0. Then p ∈ F_{μ,ν} if and only if p is the global minimum for some function tl_{μ}(p) + (1 − t)l_{ν}(p) for some t ∈ (0, 1).

Proof: At a minimum of tl_{μ}(p) + (1 − t)l_{ν}(p), since l_{μ} is constant along the earthquake path E_{μ}(p), we find dl_{ν}/dt_{μ}(p) = dl_{μ}/dt_{ν}(p) = 0 so that p ∈ F_{μ,ν}. Conversely, if dl_{ν}/dt_{μ}(p) = 0, the earthquake paths E_{μ}(p) and E_{ν}(p) must be tangent at p because p is the unique minimum of l_{ν} on E_{μ}(p). Thus E′_{μ}(p) = −kE′_{ν}(p) for some k ≠ 0, where′ denotes the tangent vector to the corresponding earthquake path. From the derivative formula dl_{ν}/dt_{μ} = −dl_{μ}/dt_{ν} it follows that k > 0. We get dl_{η}/dt_{μ}(p) = −kdl_{η}/dt_{ν}(p) for any η ∈ ML, which, using the derivative formula again, gives dl_{μ}/dt_{η}(p) = −kdl_{ν}/dt_{η}(p). Since the tangent vectors E′_{η}(p), η ∈ ML certainly span the tangent space to F at p, we must be at a critical point of l_{μ} + kl_{ν}. □

Using the identification of the critical line F_{μ,ν} with the Kerckhoff line of minima, the following proposition follows immediately from [23] theorem 2.1. Here is another proof.

Proposition 7 Fix [μ] ∈ P ML. Then the arcs F_{μ,ν}, [ν] ∈ P ML − {[μ]} are pairwise disjoint and foliate F.

Proof: Given p ∈ F, following Kerckhoff we define β = β_{p} : ML → T_{p}F to be the map which takes μ ∈ ML to DE_{μ}(t_{μ})(p)|_{t_{μ}=0}, the derivative with respect to t_{μ} of the earthquake path E_{μ}(t_{μ})(p) through p evaluated at p. By [23], Theorem 3.5 the map β is a homeomorphism. Clearly, β induces a homeomorphism between P ML and the set of rays through the origin in T_{p}F.

Suppose [μ], [ν], [ν′] ∈ P ML are distinct, and suppose that p ∈ F_{μ,ν} ∩ F_{μ,ν′}.

Pick representatives μ, ν, ν′ of [μ], [ν], [ν′] and let c = l_{μ}(p), d = f_{μ,ν}(c), d′ = f_{μ,ν′}(c). The earthquake paths E_{ν,d} and E_{ν′,d′} both go through p and, because l_{μ} is minimal at p with respect to both E_{ν} and E_{ν′}, from the derivative formula we see that DE_{ν}(t_{ν})(p)|_{t_{ν}=0} = DE_{ν′}(t_{ν′})(p)|_{t_{ν′}=0}. By the injectivity of β_{p} on P ML, [ν] = [ν′].

Now let p ∈ F. By the surjectivity of β, there is some ν ∈ ML such that DE_{ν}(t_{ν})(p)|_{t_{ν}=0} = −DE_{μ}(t_{μ})(p)|_{t_{μ}=0}. Therefore the earthquake paths E_{ν,l_{ν}}(p) and E_{μ,l_{μ}}(p) are tangent at p. Since earthquake paths can intersect in at most two points it follows that l_{ν} is minimal at p with respect to E_{μ}, so that p ∈ F_{μ,ν}.

These two facts show that the sets F_{μ,ν} foliate F. □

We shall also need

Corollary 8 For fixed μ ∈ ML, c ∈ R^{+}, the map ψ : P ML − {[μ]} → E_{μ,c}, ψ([ν]) = p(μ, ν, c), is a homeomorphism.

Proof: Proposition 7 shows that ψ is well defined and a bijection. It is also clear, thinking of P ML−{[μ]} and E_{μ,c} as intervals, that ψ is monotonic. The result follows. □

Corollary 4 implies that for μ ∈ ML, the paths E_{μ,c}, c ∈ R^{+} are pairwise disjoint and foliate F. This is the analogue of the foliation of the hyperbolic disk D by horocycles tangent at to a point on the boundary. Likewise, the critical lines F_{μ,ν} are the analogue of the geodesics in D joining a pair of distinct points in S^{1}. For fixed [μ] the foliation by leaves F_{μ,ν}, [ν] ≠ [μ] is clearly transverse to that by the earthquake paths E_{μ,c}.

This is the picture that we shall extend to QF below.

4 Hyperbolic 3-manifolds

4.1 The pleating locus

Let q ∈ QF and let G = G(q) be a group representing q with the standard normalization of section 2.1. The group G acts as a discrete group of isometries of hyperbolic space H^{3} and the quotient hyperbolic manifold M = H^{3} /G is a product T_{1} × (−1, 1). If G is quasifuchsian, but not Fuchsian, the boundary ∂C of the hyperbolic convex hull C of Λ in H^{3} has two components ∂C^{±} each of which is also G-invariant. Each quotient ∂C^{±}/G is homeomorphic to T_{1}, see for example [19] proposition 3.1. The metric induced on the components ∂C^{±} from H^{3} makes them pleated surfaces. This means, see for example [9], that there are surjective isometric maps ψ^{±} : D → ∂C^{±} such that for each point z in D there is at least one geodesic segment through z that is mapped to a geodesic segment in ∂C^{±}. The group G acts as a discrete group of isometries on each component ∂C^{±}. Since ∂C^{±}/G are both homeomorphic to T_{1}, these two groups of isometries are both isomorphic to π_{1}(T_{1}) and inherit a marking in the obvious way. (The marking on ∂C^{−}/G has its orientation reversed.) The isometries ψ^{±} induce isomorphisms to marked Fuchsian punctured torus groups F^{±} = F^{±}(q) acting on D, which we may again take to have the standard normalization. We refer to both the marked groups F^{±}(q) and the quotients D/F^{±}(q) as the flat structures of either the surfaces ∂C^{±}/G(q) or of their universal covers ∂C^{±}(q).

The bending laminations of ∂C^{±}/G carry natural transverse measures, the bending measures pl(q)^{±}, see [9,19]. The underlying laminations |pl(q)^{±}| are the pleating loci of G. If G ∈ QF is a Fuchsian group acting on the hyperbolic disk D ⊂ H^{3}, then C = D is degenerate and we regard ∂C and ∂C/G as 2-sided surfaces, each side of which is a pleated surface with empty pleating locus (and zero measure).

The following proposition follows immediately from [20] proposition 3.3 and corollary 3.4.

Proposition 9 Suppose that q ∈ QF − F. Then the projective class of the bending measure cannot be the same on both sides of the convex core; that is, [pl^{+}(q)] ≠ [pl^{−}(q)].

Remark 10 The work in [20] depends heavily on the λ-lemma and the theory of holomorphic motions which is usually stated in the context of one complex variable. In the present case we shall be studying families of groups parameterized by a two dimensional complex manifold; in fact the theory of holomorphic motions extends to motions over any complex manifold, see [33].

In [19] we prove:

Theorem 11 The map QF → F which sends q ↦ F^{±}(q), and the map QF − F → ML which sends q ↦ pl^{±}(q), are continuous.

4.2 Pleating Varieties

Given μ ∈ ML we set

P_{μ}^{±} = {q ∈ QF : [pl^{±}(q)] = [μ]} and P_{μ} = P_{μ}^{+} ∪ P_{μ}^{−} .

We call these sets the μ-pleating varieties.

Given the ordered pair (μ, ν) ∈ ML × ML, we set

P_{μ,ν} = {q ∈ QF : [pl^{+}(q)] = [μ], [pl^{−}(q)] = [ν]}.

We call this set the μ, ν-pleating plane. Note that two these definitions depend only on the projective classes [μ], [ν].

Finally, given the ordered pair μ, ν ∈ ML × ML, and c > 0 we set

P_{μ,ν,c} = {q ∈ P_{μ,ν} : l_{μ}(q) = c}.

We call this set a pleating ray. Note that for s ∈ R^{+}, P_{μ,ν,c} = P_{sμ,ν,sc}. Thus P_{μ,ν,c} depends on the projective class of the pair (μ, c), (recall section 2.2), and on the projective class [ν].

Theorems 2 and 6, to be proven in section 9 below, will justify the terminology rays and planes.

Proposition 9 implies P_{μ,μ} = ∅. It is also clear that P_{μ,ν} ∩ P_{μ′,ν′} = ∅ unless [μ] = [μ′], [ν] = [ν′]. In particular P_{μ,ν} ≠ P_{ν,μ} whenever i(μ, ν) > 0.

Remark 12 Whether a group is in P_{μ,ν} or in P_{ν,μ} depends on our conventions in labelling the sides ∂C^{±} of ∂C. This is based on the labelling of the components of the regular set Ω^{±}. The point here is that two groups which differ only in the labelling of their + side and their − side are not the same as marked groups in QF.

The main result of [20] is that the pleating varieties are non-empty. Precisely, we prove

Theorem 13 Let μ, ν ∈ ML, [μ] ≠ [ν]. Then P_{μ,ν} ≠ ∅.

We shall need to study the ideas in the proof of this result in some detail; see 7.2 below.

4.3 Lamination length in M = H^{3} /G.

For the proof of theorem 15 below, we need also to discuss briefly the length l_{μ}(M) of a measured lamination μ ∈ ML in the hyperbolic 3-manifold M = H^{3} /G. First, suppose that μ = δ_{γ} where γ ∈ S is represented by an element V ∈ G. The multiplier λ_{V} is related to its trace by the formula Tr V = 2 cosh λ_{V} /2. The translation length of V, ℜλ_{V}, is the minimum distance that V moves a point in H^{3}. Equivalently it is the length of the geodesic representative of γ in M, so that l_{δ_{γ}}(M) = ℜλ_{V}.

In [45], p.9.21 and [3], p.117, it is shown that this definition can be extended by linearity and continuity to define the lamination length l_{μ}(M) for an arbitrary μ ∈ ML. In the proof of theorem 15 below, we shall need to make crucial use of the fact that one can extend this definition continuously to the algebraic closure of QF.

Suppose G is a (discrete) punctured torus group associated to the faithful representation ρ : π_{1}(T_{1}) → G ⊂ P SL(2, C). This representation marks the associated hyperbolic 3-manifold M = H^{3}/G. One says that a lamination |μ| on T_{1} is realized in M relative to the marking ρ, if there is a Fuchsian group Γ, a homeomorphism h : T_{1} → S = H^{2}/Γ, and a pleated surface f : S → M with pleating locus containing |μ|, such that fh induces ρ.

Let AH(T_{1}) denote the set of Kleinian once punctured torus groups as defined in section 2.1, modulo conjugation in P SL(2, C). By abuse of notation, we also denote by AH(T_{1}) the set of hyperbolic 3-manifolds {M = H^{3}/H : [H] ∈ AH(T_{1})}, where [H] is the conjugacy class of H in P SL(2, C).

Clearly, whether or not a lamination is realized is a conjugacy invariant. Simple closed curves are always realized in any hyperbolic 3-manifold M ∈ AH(T_{1}) unless they are represeneted by parabolics. The closed geodesics are dense in the set of realizable laminations, [7] theorem 5.3.11. Since length is a conjugacy invariant, the above definition of lamination length l_{μ}(M) extends by continuity to any M ∈ AH(T_{1}) containing a realization of |μ|. If |μ| is connected and not realized in M, set l_{μ}(M) = 0. (If the closure \overline{|μ|} of |μ| is not connected one has to be more careful with this definition since some components of μ may be realized and others not; for example on a general surface, |μ| might consist of disjoint loops some but not all of whose components are accidentally parabolic. In this case only the accidental parabolics are not realized and l_{μ} must be defined by summing over the connected components of \overline{|μ|}. In the case of a punctured torus \overline{|μ|} is always connected (since μ is measured) and this difficulty does not occur.)

In the next section, we shall make important use of the following result.

Proposition 14 The function L : AH(T_{1}) × ML → R, L(H, μ) = l_{μ}(H^{3}/H) is continuous.

Proof: This result was asserted by Thurston in [44]; detailed proofs appear in [36] Lemma 4.2 and [2] Theorem 5.1. We remark that the proof in [36] seems to have overlooked the above mentioned difficulties when |μ| is not connected. See [2] section 7 for a discussion of the general case. □

Note that if a lamination μ ∈ ML is realized in M ∈ AH(T_{1}), then the length of μ in M is equal to the hyperbolic length of μ on the surface Σ, where ψ : Σ → M is the pleated surface map realizing |μ|, and so is strictly positive.

In general, the lamination lengths l_{μ}(∂C) on ∂C and l_{μ}(M) in M are not the same, and we shall take care to indicate which length we mean. In the special case in which q ∈ P_{μ}^{+}, however, the lengths l_{μ}(∂C^{+}) and l_{μ}(M) coincide, and may be safely denoted by l_{μ} = l_{μ}(q). This is the situation we are discussing in theorem 15 below.

In section 6, we shall show how to extend the holomorphic multiplier λ_{V} to a holomorphic function called the complex length λ_{μ} of μ on QF. Again by linearity and continuity, we have l_{μ}(M) = ℜλ_{μ}. We also prove in section 6 that q ∈ P_{μ}^{+} implies λ_{μ} ∈ R. Combining these observations gives that q ∈ P_{μ}^{+} implies λ_{μ} = l_{μ}(∂C^{+}) = l_{μ}(M).

5 The limit pleating theorem

Classically, the ending invariants of a quasifuchsian group are the marked conformal structures ω^{±}(q) of the tori Ω^{±}(q)/G(q) and so are points in the Teichmüller space Teich. Suppose we have a sequence q_{n} ∈ QF with ω^{±}(q_{n}) → ω^{±} ∈ Teich. It then follows from Bers’ simultaneous uniformization theorem that the groups G(q_{n}) have an algebraic limit in QF. If both of the sequences ω^{±}(q_{n}) converge to distinct points in the Thurston boundary of Teich, then Thurston’s double limit theorem [44] again asserts the existence of an algebraic limit G_{∞}; the intermediate situation works in a similar way and is discussed in [35].

We need an analogous result which asserts the existence of a limit group when our pleating invariants converge. We also need to understand the behavior of the pleating invariants when an algebraic limit exists. The results we need are collected in the following limit pleating theorem, which will be a key factor in the proof of our main results in section 9.

Theorem 15 Limit Pleating Theorem. Let μ, ν ∈ ML, [μ] ≠ [ν] and suppose that {q_{n}} ∈ P_{μ,ν}. Then

(1) if l_{μ}(q_{n}) → c ≥ 0 and l_{ν}(q_{n}) → d ≥ 0, then there is a subsequence of the groups {G(q_{n})} with an algebraic limit G_{∞};

(2) if the sequence {G(q_{n})} has algebraic limit G_{∞}, then the sequences {l_{μ}(q_{n})} and {l_{ν}(q_{n})} have finite limits c ≥ 0, d ≥ 0 respectively. The group G_{∞} represents a point in QF if and only if c > 0 and d > 0.

We remark that in the case of a more general surface, the second statement as it stands is false, as is seen by taking |μ| to be a multiple loop such that one, but not all of its components, becomes accidentally parabolic. It works in our case because any measured lamination on T_{1} is automatically connected. The result is closely related to, but not the same as, the ‘Lemme de fermeture’ in [5], which concerns the existence of the limit groups under hypotheses on the limits of bending measures as opposed to lengths.

The first statement, the existence of the algebraic limit, follows from a deep estimate of Thurston’s about lengths of geodesics in hyperbolic 3-manifolds, [44], theorem 3.3 (Efficiency of pleated surfaces). The same estimate is fundamental in Thurston’s proof of the double limit theorem in [44]. A detailed discussion and proof of Thurston’s estimate is to be found in [6], where a limit theorem similar to our first statement in the context of Schottky groups is proved.

To prove the second statement we use continuity of lamination length described in section 4.3 above. This allows us to deduce that the laminations μ, ν must be realized in the algebraic limit. We complete the proof by showing that the pleated surfaces which realize μ and ν are in fact components of the convex hull boundary of the algebraic limit. This idea is in essence the same as that used in [5], and we would like to thank F. Bonahon for suggesting this approach.

The statement, and the theorem on continuity of lamination length, conceals much subtlety. The hypothesis that G_{n} ∈ P_{μ,ν} is crucial; examples like the one described in [24] show that it is not enough just to require that some fixed curve on ∂C^{+} have bounded length. Again, if one takes a varying sequence μ_{n} → μ as in [5], then it is essential to add the hypothesis that the laminations converge in the Hausdorff topology as well as in measure, otherwise examples similar to the one in [24] again show that the convergence may not be strong.

Proof: First we suppose that l_{μ}(q_{n}) → c ≥ 0 and l_{ν}(q_{n}) → d ≥ 0, and show that there is some subsequence of {q_{n}}, along which an algebraic limit exists. Choose and fix an ideal triangulation λ on T_{1}; specifically, take λ as the lines from the cusp to itself in the homotopy classes of the curves α, β and αβ, where < π_{1}(T_{1}); α, β > corresponds to < G; A, B >.

Let M_{n} = H^{3}/G_{n} and realize λ as the pleating locus of a pleated surface S_{n} in M_{n}. The lamination λ has no closed leaves and its complement is a pair of ideal triangles. Pick ξ ∈ ML. When an oriented arc on a leaf |ξ| cuts two consecutive sides of one of these complementary triangles T, the two sides meet in an ideal vertex which is either to its left or its right. The arc of leaf containing an intersection point P of |ξ| and λ goes from one triangle T_{1} to another T_{2}. Following Thurston, [44], we call P a boundary intersection if the right-left location of the ideal vertex switches as we cross from T_{1} to T_{2}, and we define the alternation number a(ξ, λ) as the total ξ-measure of the set of boundary intersection points. Recall from section 4.3 that l_{ξ}(S_{n}) denotes the length of the lamination ξ measured in the flat structure of S_{n} and l_{ξ}(M_{n}) denotes the length of the lamination ξ in M_{n}. Then by [44] theorem 3.3, there exists a constant C > 0, depending only on a fixed choice of structure for T_{1}, such that

l_{ξ}(S_{n}) ≤ l_{ξ}(M_{n}) + Ca(ξ, λ).

(We remark that since a(ξ, λ) ≤ i(ξ, λ) the usual intersection number would be just as good a bound in the present case.) Applying this inequality in our case to the pleating laminations |μ| and |ν| we find,

l_{μ}(S_{n}) ≤ l_{μ}(q_{n}) + Ca(μ, λ), l_{ν}(S_{n}) ≤ l_{ν}(q_{n}) + Ca(ν, λ).

It follows that the sequences {l_{μ}(S_{n})} and {l_{ν}(S_{n})} are bounded.

Since [μ] ≠ [ν], the laminations |μ|, |ν| fill up T_{1} and we conclude from [44] proposition 2.4 that the hyperbolic structures of the surfaces S_{n} lie in a bounded subset of F and thus that the lengths l_{α}(S_{n}) and l_{β}(S_{n}) of the geodesic representatives of the marking curves α and β on S_{n} are bounded. From the discussion in section 4.3, we conclude that, since l_{α}(M_{n}) ≤ l_{α}(S_{n}) and l_{β}(M_{n}) ≤ l_{β}(S_{n}), the sequences {| Tr A_{n}|}, {| Tr B_{n}|} are also bounded. Therefore we can find a convergent subsequence along which Tr A_{n} and Tr B_{n} converge and thus, (because from the Markov identity Tr A and Tr B determine at most two normalized punctured torus groups up to conjugation) we conclude that a subsequence of {G_{n}} has an algebraic limit G_{∞}. This proves statement 1.

Now suppose that G_{∞} is the algebraic limit of a sequence G_{n} = G(q_{n}) ∈ P_{μ,ν}. By the continuity of lamination length on AH(T_{1}), the sequences {l_{μ}(q_{n})}, {l_{ν}(q_{n})} converge to {l_{μ}(G_{∞})}, {l_{ν}(G_{∞})}, in particular the limits exist. We have to prove that G_{∞} ∈ QF if and only if both limits are non-zero. We note immediately that if G_{∞} ∈ QF, then, using our assumption that q_{n} ∈ P_{μ,ν}, we have {l_{μ}(q_{n})} → c > 0 and {l_{ν}(q_{n})} → d > 0 by the continuity theorem 11. This can also be seen from the fact that all laminations, in particular μ and ν, are realized in G_{∞}, see [45], [7] theorem 5.3.11.

Suppose that one of the laminations μ or ν, for definiteness say μ, is not realized in G_{∞}. Since |μ| is connected, l_{μ}(G_{∞}) = 0 and by the continuity of lamination length on AH(T_{1}) we deduce that c = 0. Thus we need only prove that if μ, ν are both realized in G_{∞}, and if c > 0, d > 0, then G_{∞} ∈ QF.

Our strategy is to show that the lifts of the pleated surfaces which realize |μ| and |ν| are in fact invariant components of ∂C(G_{∞}) which face simply connected invariant components of the regular set Ω(G_{∞}). The key point is to show that if |μ| is realized in the algebraic limit M_{∞} = H^{3}/G_{∞}, then the lift of any leaf of |μ| to H^{3} is the limit of corresponding lifts of leaves of |μ| in their realizations in M_{n}.

To prove this involves comparing geodesics in the universal covers of M_{∞} and M_{n}. We need to use the fact that geometric structures vary continuously with the holonomy. The clearest statement of what we need is in [15], Lemma 14.28. Our normalizations have been fixed in such a way that ρ_{n}(g) → ρ_{∞}(g) for each g ∈ π_{1}(T_{1}). Let M_{∞}^{1} be the manifold with boundary obtained by removing from M_{∞} a horoball neighborhood of the cusp corresponding to ρ_{∞}(K), where K corresponds to a loop around the cusp of T_{1}. A relative compact core M_{c} for M_{∞}^{1} is a compact submanifold M_{c} ⊂ M_{∞}^{1} such that the induced map on fundamental groups is an isomorphism, and such that M_{c} ∩∂M_{∞}^{1} is a compact subsurface of ∂M_{∞}^{1}. The assertion of Kapovich’s lemma is that under the hypothesis that ρ_{n}(g) → ρ_{∞}(g), if M_{c} is any relative compact core for M_{∞}^{1}, then there exists a sequence of smooth maps j̃_{n} : M̃_{c} → H^{3} which intertwine the actions of ρ_{∞} and ρ_{n} and which converge C^{1} to the identity, uniformly on compact subsets of M̃_{c}. Here M̃_{c} ⊂ H^{3} is the universal cover of M_{c}. For the proof see [15] theorem 7.2 or [7] theorem 1.7.1. Following [7], one can actually make j̃_{n} be C^{r} close to the identity for any r. It follows that the map j_{n} : M_{c} → M_{n} induced by j̃_{n} is close to a local isometry, see the similar assertion in [32] section 3.1.

Now we have to be careful about our set-up of pleated surfaces. Let Γ_{0} be a fixed Fuchsian group acting on D. Identify T_{1} with D/Γ_{0}, choosing a fixed isomorphism of π_{1}(T_{1}) with Γ_{0}. The action of G_{n} = ρ_{n}(Γ_{0}) on ∂C_{n}^{+} pulls back to the action of a Fuchsian group Γ_{n} on D. This induces a pleated surface map f_{n} : D → H^{3} with image ∂C_{n}^{+}, intertwining the action of Γ_{n} on D and G_{n} on ∂C_{n}^{+}. Let h_{n} : D → D denote a homeomorphism which intertwines the actions of Γ_{0} and Γ_{n}, so that f_{n}h_{n} : D → H^{3} induces the representation ρ_{n} : Γ_{0} → G_{n}. Since |μ| is realized in M_{∞}, there is also a Fuchsian group Γ_{∞}, and a pleated surface f : D → H^{3} intertwining the actions of Γ_{∞} and G_{∞} with pleating locus containing |μ|, together with a homeomorphism h : D → D intertwining the actions of Γ_{0} and Γ_{∞} such that fh induces ρ_{∞} : Γ_{0} → G_{∞}. Thus all the maps f_{n}h_{n} and fh have the same domain D and range H^{3} and intertwine the groups Γ_{0} and G_{n}, and Γ_{0} and G_{∞}, respectively.

Now let l be a lift to D of some leaf of μ in T_{1} = D/Γ_{0}, that is, in the structure induced by Γ_{0}. The corresponding leaves for the structures induced by Γ_{n}, Γ_{∞} are the geodesics l_{n}, l_{∞} which have the same endpoints on ∂D as h_{n}(l), h(l) respectively. From the definition of pleated surfaces, under f_{n} and f these leaves are mapped to geodesics in H^{3}. To make precise the statement that leaves of |μ| in M_{∞} = H^{3}/G_{∞} are close to leaves of the corresponding realizations in M_{n}, we shall prove that f_{n}(l_{n}) → f(l_{∞}).

Let S_{0}^{1} be the surface with boundary obtained by removing from D/Γ_{0} a horocycle neighborhood of the cusp. The pleating locus of the convex hull boundary cannot contain any leaf going out to the cusp (because otherwise the developing image of the boundary in a horoball neighborhood of the cusp would not be embedded), so we may suppose that the support of μ on D/Γ_{0} is contained in S_{0}^{1}. Now M_{∞} is geometrically tame and homeomorphic to S × R, see [3], or [15] theorem 14.17. It follows that M_{∞} may be exhausted by relative compact cores; in particular we may assume that M_{c} contains the image under the map induced by fh of S_{0}^{1}. Choose a map j̃_{n} as in Kapovich’s lemma above. As already remarked, since j̃_{n} is C^{∞} close to the identity on compact subsets of M̃_{c}, the induced map j_{n} : M_{c} → M_{n} is close to a local isometry. The same is therefore true of j̃_{n} on the whole of M̃_{c}. Now the image of a geodesic under a map which is close to a local isometry is clearly a quasi-geodesic with small constants, and hence close to its geodesic representative. Hence j̃_{n}f(l_{∞}) has definite endpoints on ∂H^{3} and is arbitrarily close (depending on n) to the geodesic A_{n} with the same endpoints.

Since j̃_{n} → id uniformly on compact sets in M̃_{c} and since f(l_{∞}) ⊂ M̃_{c}, we see that j̃_{n}f(l_{∞}) converges to f(l_{∞}). (Here we use the fact that both curves are quasigeodesic, so it suffices to prove convergence on compact subsets of H^{3}.) Thus to complete the proof, it will suffice to show that f_{n}(l_{n}) = A_{n}. We shall do this by showing that the curves f_{n}(l_{n}) and j̃_{n}f(l_{∞}) are a bounded distance apart.

First let us show that f_{n}h_{n}(l) and j̃_{n}fh(l) are a bounded distance apart. Pick a fundamental domain for the action of Γ_{0} on D, and let R be the closure of the intersection of this region with the lift S̃_{0}^{1} of S_{0}^{1} to D. Since R is compact, there exists K > 0 such that d_{H^{3}}(j̃_{n}fh(x), f_{n}h_{n}(x)) ≤ K for all x ∈ R. Now by construction, both maps j̃_{n}fh, f_{n}h_{n} are equivariant, meaning that j̃_{n}fh(γx) = ρ_{n}(γ)j̃_{n}fh(x) for all x ∈ D and γ ∈ Γ_{0}, and similarly for f_{n}h_{n}. Thus d_{H^{3}}(j̃_{n}fh(x), f_{n}h_{n}(x)) ≤ K for all x ∈ S̃_{0}^{1}. Parameterize the leaf l as t ↦ l(t) for t ∈ R. Since l projects to S_{0}^{1}, it follows that d_{H^{3}}(j̃_{n}fh(l(t)), f_{n}h_{n}(l(t))) ≤ K for all t ∈ R, in other words the two curves are a bounded distance apart over the whole of their lengths.

Since S_{0}^{1} is compact, by equivariance the restriction of h_{n} to S̃_{0}^{1} is Lipschitz with constant depending on n. Hence h_{n}(l) is a quasigeodesic, and thus lies at a bounded distance from its geodesic representative l_{n}. Now the restriction of f_{n} to the compact set h_{n}(S̃_{0}^{1}) is also Lipschitz, so that f_{n}(l_{n}) and f_{n}(h_{n}(l)) are also a bounded distance apart (again with constant depending on n). Similarly, so are j̃_{n}f(l_{∞}) and j̃_{n}fh(l). We conclude that f_{n}(l_{n}) is a bounded distance from A_{n} (with bound depending on n). However f_{n}(l_{n}) is a geodesic, and geodesics which are a bounded distance apart over the whole of their lengths coincide. Thus f_{n}(l_{n}) = A_{n} as claimed.

We now use this fact to prove that the image of the pleated surface f : D → H^{3} is a component of the convex hull boundary of G_{∞}. The projection of the pleating locus of f to D/Γ_{∞} is a geodesic lamination which contains |μ|. If this pleating locus is not maximal, then by area considerations we can make it maximal by adding at most three extra leaves. (If μ is irrational, there may be extra leaves running from the cusp and spiralling into the two boundary leaves; if μ is rational there may be a further leaf running from the cusp to itself, or a leaf spiralling around μ at both ends.) Call this maximal lamination μ̂ and for simplicity, use the same symbols μ, μ̂ to denote the lifts to D. Notice that since the pleating locus of f_{n} actually equals |μ| (since the pleating locus of the convex hull boundary cannot contain any leaf going out to the cusp or spiralling onto a closed geodesic) the additional leaves of μ̂−|μ| are necessarily mapped to geodesics by f_{n}. Moreover any additional endpoints of these leaves are cusps and hence their lifts move continuously as n → ∞.

We call any ideal triangle in H^{3} formed by the lifts of the images of the boundary leaves of a complementary region of μ̂ under a pleated map a plaque. The vertices of such a triangle are either the endpoints of leaves of the lamination or parabolic fixed points. For clarity, denote the images in H^{3} of μ̂ under the pleated surface maps f_{n}, f by μ̂_{n}, μ̂_{∞} respectively. We have just shown that any plaque of μ̂_{∞} is arbitrarily closely approximated in H^{3} by a plaque of μ̂_{n} for all sufficiently large n. Notice also that any plaque of μ̂_{n} is contained in a support plane for ∂C_{n}^{+}.

Denote the image of f by Π^{+}. We want to show that Π^{+} is a component of ∂C(G_{∞}). Let X be a plane containing a plaque of |μ_{∞}| and let X_{n} be a sequence of planes containing approximating plaques for ∂C^{+}(G_{n}). We claim that all of Π^{+} lies on the same side of X so that X is a support plane for Π^{+}. If not, we can find points y, y′ ∈ Π^{+} on opposite sides of X so that the geodesic joining y to y′ crosses X transversally. By choosing n sufficiently large, we can find y_{n}, y′_{n} near to y, y′ in ∂C_{n}^{+}, and a support plane X_{n} to ∂C_{n}^{+} close to X, such that the geodesic from y_{n} to y′_{n} crosses X_{n}, which is impossible.

Denote by H_{X} the closed half space bounded by X containing Π^{+} and set K = ∩_{X}H_{X} where X runs through all planes containing plaques of Π^{+}. By the above, Π^{+} ⊂ K so K ≠ ∅. By its construction, K is convex and closed. Moreover K is G_{∞} invariant since the same is true of Π^{+}. Since C(G_{∞}) is the smallest closed convex G_{∞} invariant set in H^{3}, we conclude C(G_{∞}) ⊂ K.

We claim Π^{+} ⊂ ∂C(G_{∞}). Let P be a plaque of Π^{+}. Clearly P ⊂ C(G_{∞}) and so P ⊂ K. Since P is by definition contained in a support plane for K, we conclude P ⊂ ∂C(G_{∞}). Since Π^{+} is the closure in H^{3} of the union of its plaques, the claim follows.

We prove in lemma 16 below that Π^{+} is embedded in H^{3}. (This rules out the possibility that, for example, |μ| is rational and the bending angle is π.)

Thus Π^{+} is isometric to a complete hyperbolic surface and hence is both open and closed in ∂C(G_{∞}). Since Π^{+} is connected, it must be a component of ∂C(G_{∞}). As such, it faces a component Ω^{+} of Ω(G_{∞}). Moreover since Π^{+} is simply connected, so is Ω^{+}. (This also follows from the fact that the limit representation ρ : π(T_{1}) → G_{∞} is faithful.) Also G_{∞} invariance of Ω^{+} follows from that of Π^{+}.

Now there is a similar image Π^{−} for the pleated surface map which realizes |ν|, from which we deduce the existence of another simply connected invariant component Ω^{−} of Ω(G_{∞}). We conclude, see for example [28] lemma 3.2, that G_{∞} ∈ QF. □

Lemma 16 With the notation and conditions above, the image Π^{+} of the pleated surface map f is embedded in H^{3}.

Proof: If Π^{+} is not embedded then f(x) = f(y) for some distinct points x, y ∈ D; these cannot be in the same plaque since f is an isometry on plaques. We begin by reducing to the case in which x and y are both contained in leaves of μ̂. If not, suppose that x is in a complementary region of μ̂, and let P_{x} be the image plaque containing f(x). Now y is either in a distinct complementary region with image plaque P_{y}, or on a leaf with image a geodesic L. If P_{y} or L cuts P_{x} transversally, then the same is true for all nearby pleated surfaces f_{n}, since the endpoints which determine plaques and leaves move continuously. This is impossible since f_{n}(D) = ∂C_{n}^{+} is embedded. Thus P_{y} (or L) and P_{x} are in a common plane. In the first case there is some point on boundary leaves of both P_{y} and P_{x}, and in the second L meets some boundary point of P_{x}.

Now let l_{x}, l_{y} be leaves of the lift of μ̂ to D through x, y respectively. We claim that the image leaves f(l_{x}) and f(l_{y}) meet at a non-zero angle in H^{3}. This follows from [43] Theorem 5.6. A pleated surface map k from a surface S into a 3-manifold M induces an obvious map K from its pleating locus to the projective unit tangent bundle ℙM of M. Thurston’s result states that if k is weakly doubly incompressible, then K is an embedding. In our situation, the pleated surface map f̄ induced on the quotient D/Γ_{∞} is weakly doubly incompressible since the induced map Γ_{∞} → G_{∞} is an isomorphism. Thus the conclusion of the theorem implies immediately that the image geodesics f(l_{x}) and f(l_{y}) are distinct. (Actually Thurston’s proof simplifies slightly in our situation, see remark 17 below.)

Consider the plane P containing these two leaves. It meets Ĉ in a circle C. Notice that any circle through the endpoints of f(l_{x}) other than C separates the endpoints of f(l_{y}). Now for any nearby group G_{n}, there are leaves f_{n}(l_{x}), f_{n}(l_{y}) near f(l_{x}), f(l_{y}). Any support plane to ∂C_{n}^{+} through either of these leaves meets Ĉ in a circle which cannot separate the other pair of endpoints. One deduces easily that any pair of support planes for ∂C_{n}^{+} must meet Ĉ in circles both of which are close to C, and that Λ(G_{n}) is contained in the thin ring or crescent between them. Now every support plane of ∂C_{n}^{−} is disjoint from every support plane of ∂C_{n}^{+}; moreover one of the two disks defined by the circle it bounds on Ĉ contains no limit points.

It follows that every support plane of ∂C_{n}^{−} must have very small diameter, and hence that the distance of any such support plane to f(x) tends to ∞ with n. On the other hand, any support plane for ∂C_{n}^{−} contains points close to some plaque of the pleated surface which realizes |ν| in M_{∞}. Pick a point z ∈ H^{3} on a lift of a leaf of |ν|, at distance D say from f(x). Since z is on a plaque of |ν| it is close to a support plane of ∂C_{n}^{−}. This shows there are points in ∂C_{n}^{−} which stay at bounded distance, with bound close to D, from f(x). This contradiction completes the proof. □

Remark 17 We explain the simplification of Thurston’s result alluded to above. Notice that μ (the lamination on the quotient D/Γ_{∞}) is recurrent. Thus following the first part of the proof of [43] Theorem 5.5, one can use a shadowing argument to show that the canonical lift F̄ of f̄ induces a local embedding of |μ| into the projective unit tangent bundle ℙM_{∞}. The observation that the injectivity radius of M_{∞} is bounded below in a neighborhood of f̄(|μ|) follows since |μ| is compact. Since Margulis tubes in M_{∞} are separated by a definite distance, only a finite number can intersect the image f̄(|μ|).

Still following Thurston, one then argues that F̄ extends to a covering map on a neighborhood of |μ|. (Notice that the inequality in [43] concerning the degree of the covering should be reversed.) We now need to see that the covering has degree 1. If μ is a closed geodesic, this follows immediately since f induces an isomorphism of fundamental groups. Otherwise, the complement of |μ| is a punctured bigon B with two boundary leaves. Thus any non-trivial covering must have degree two and identify the two boundary leaves. Choose a loop β in B which is very close to the boundary leaves but homotopic to a loop round the puncture. Since the covering is degree 2, f(β) is the square of the generator of the maximal parabolic subgroup in G_{∞}. This is impossible since f induces an isomorphism.

6 Complex length

In this section we introduce the complex length of a measured lamination. Just as lamination length as defined in section 2.2 is a real analytic function on F, the complex lamination length is a holomorphic function on QF. The relationship of this holomorphic function to pleating varieties, in particular theorem 23, is a central tool in everything which follows. Complex lamination length has also been introduced using somewhat different techniques by Bonahon [4].

6.1 Complex length of a loxodromic

Let M ∈ P SL(2, C). Its complex translation length λ_{M} ∈ C/2πiZ is given by the equation

± Tr M = 2 cosh λ_{M}/2 (1)

where Tr M is the trace of M and we choose the sign so that ℜλ_{M} ≥ 0.

Complex length is invariant under conjugation by Möbius transformations and has the following geometric interpretation, provided M is not parabolic. Let x ∈ Ax M and let v̄ be a vector normal to Ax M at x. Then ℜλ_{M} is the hyperbolic distance between x and M(x) and ℑλ_{M} is the angle mod 2π between M(v̄) and the parallel transport of v̄ to M(x), measured facing the attracting fixed point M^{+} of M. In particular, if M is loxodromic then ℜλ_{M} > 0 and if M is purely hyperbolic then in addition ℑλ_{M} ∈ 2πZ; equivalently Tr M ∈ R, | Tr M| > 2. (We refer to [39] for a detailed discussion of the sign ambiguity in equation 1; note that in our notation here λ_{M} is twice the multiplier denoted by λ_{M} in [39].)

Let q ∈ QF, let γ ∈ S and denote the element representing γ in the group G(q) by W (q). Because the trace is a conjugation invariant, the complex translation length λ_{W} (q) depends only on q and is independent of the normalization of G(q). We want to define the complex length λ_{γ}(q) = λ_{W} (q) as a holomorphic function on QF with values in C, not C/2πiZ. To do this, we choose the branch that is real valued on F. Since λ_{γ} ≠ 0 on QF this choice uniquely determines a holomorphic function λ_{γ} : QF → C. From now on, the term “complex length” will always refer to this branch.

We define the complex length of the rational lamination μ = cδ_{γ} ∈ ML_{Q}, c > 0, as λ_{μ}(q) = cλ_{γ}(q).

To define the complex length λ_{μ}(q) for arbitrary μ ∈ ML and q ∈ QF, we would like to choose μ_{n} ∈ ML_{Q}, μ_{n} → μ and set

λ_{μ}(q) = lim_{n→∞} λ_{μ_{n}}(q).

To justify this, we need to show these limits exist and are independent of the sequence {μ_{n}}.

We do this using the following theorem which summarizes the results of [22], lemma 2.4 and [21], theorem 1. In the statement, l_{μ} denotes lamination length defined in section 2.2.

Theorem 18 The function (cδ_{γ}, p) ↦ cl_{δ_{γ}}(p) from ML_{Q} × F to R^{+} extends to a continuous function (μ, p) ↦ l_{μ}(p) from ML × F to R^{+}. If μ_{n} ∈ ML_{Q}, μ_{n} → μ then l_{μ_{n}}(p) → l_{μ}(p) uniformly on compact subsets of F, and the limit function l_{μ}(p) is non-constant.

We also need an elementary lemma about holomorphic functions.

Lemma 19 If f : QF → C is holomorphic and if f ≡ c on F for some constant c, then f ≡ c on QF.

Proof: Because F is the R^{2}-locus of the complex Fenchel-Nielsen coordinates (λ, τ) in QF, see [27], and section 7 below, the conclusion follows directly from the Cauchy-Riemann equations applied to each variable separately. □

Theorem 20 The function (μ, q) ↦ λ_{μ}(q) from ML_{Q} × QF to C extends to a continuous function from ML × QF to C, also denoted λ_{μ}(q). The function q ↦ λ_{μ}(q) is holomorphic and non-constant for all μ and the family {λ_{μ}} is bounded and equicontinuous on compact subsets of QF.

Proof: By construction, the functions {λ_{μ}}, μ ∈ ML_{Q}, omit the half plane ℜz < 0 and thus form a normal family on compact subsets of QF. It follows that if μ_{n} → μ, μ_{n} ∈ ML_{Q}, then suitable subsequences of {λ_{μ_{n}}} converge to limit functions that are holomorphic.

We note that on F, if μ ∈ ML_{Q}, then λ_{μ} is real and coincides with l_{μ}. By theorem 18, if μ_{n} → μ, μ_{n} ∈ ML_{Q}, then {l_{μ_{n}}} is uniformly convergent on compact subsets of F; further, the limit function l_{μ} is finite, non-constant and independent of the choice of the sequence {μ_{n}}. The result now follows from lemma 19. □

Corollary 21 Let μ, ν ∈ ML. Then the zero of dl_{ν}/dl_{μ} at the minimum of l_{ν} along the earthquake path E_{μ}^{p} is simple.

Proof: From the discussion preceding Proposition 2, we have only to consider the case in which ν is not rational. By Theorem 20, l_{ν} extends to a holomorphic function λ_{ν} on QF which is locally uniformly approximated by the complex lengths λ_{ν_{n}} of a sequence of rational laminations ν_{n} which converge to ν. Now apply Hurwitz’ theorem. □

For μ ∈ ML, we call λ_{μ} the complex length of μ. Throughout this paper, the complex length functions are a fundamental tool. We remark that (1) Suppose q ∈ QF and let F^{±}(q) ∈ F denote the flat structures (see

section 4.1) on the convex core boundary ∂C^{±}(q)/G(q). If μ ∈ [pl^{±}(q)], then l_{μ}(F^{±}(q)) = ℜλ_{μ}(q), see Proposition 22 below.

(2) For q ∈ QF, μ ∈ ML, ℜλ_{μ} coincides with the lamination length l_{μ}(M(q)) in the 3-manifold M(q) = H^{3} /G(q) as discussed in section 4.3 above.

For μ ∈ ML_{Q}, ℜλ_{μ}(q) = l_{μ}(M(q)), so by continuity, both statements hold for all μ ∈ ML.

6.2 Complex length and Pleating varieties

The first step in proving our main theorems is to show that for any μ ∈ ML, the complex length λ_{μ} is real valued on P_{μ}.

First consider the case μ ∈ ML_{Q}. We have

Proposition 22 Suppose μ ∈ ML_{Q}. Let q ∈ QF and suppose pl^{+}(q) = μ. Then λ_{μ}(q) ∈ R.

Proof: This is just a reformulation of the easy observation, proved in [18], lemma 4.6, that if a geodesic γ is contained in |pl^{±}(G)|, then any representative in G is purely hyperbolic. □

We now extend proposition 22 to arbitrary laminations.

Theorem 23 Suppose μ ∈ ML. Let q ∈ QF and suppose pl^{+}(q) = μ. Then λ_{μ}(q) ∈ R.

Proof: For [μ] ∈ ML_{Q} this is proposition 22, so suppose [μ] ∉ ML_{Q}.

The map QF −F → ML×F that takes q ∈ QF −F to (pl^{+}(q), F^{+}(q)) where F^{+}(q) is the flat structure of ∂C^{+}/G(q) is continuous by theorem 11. The map is also injective because the hyperbolic structure F^{+} together with the bending data pl^{+} determine the group G = G(q), see [9] Chapter 3 especially Lemma 3.7.1. Let U ⊂ QF −F be an open ball containing q; if [pl^{+}(q′)] were constant on U, a four dimensional neighborhood would have a three dimensional image, violating the invariance of domain for a continuous injective map.

By the continuity theorem 11, since P ML is one dimensional, we may find a sequence q_{n} → q in U such that pl^{+}(q_{n}) = μ_{n} with μ_{n} ∈ ML_{Q}. By proposition 22, λ_{μ_{n}}(q_{n}) ∈ R. By the continuity theorem again, μ_{n} → μ and hence λ_{μ_{n}} → λ_{μ} uniformly on compact subsets of QF. Thus taking a diagonal limit we have λ_{μ_{n}}(q_{n}) → λ_{μ}(q) and λ_{μ}(q) ∈ R. □

7 Twists and Quakebends

In this section we briefly discuss complex Fenchel-Nielsen coordinates and quakebends, and the connection with the convex hull boundary ∂C. This circle of ideas is at the heart of the proof of the local pleating theorem 26 in section 8; some of the ideas are also needed in section 9, where we work in quakebend planes as defined in section 7.3 below.

7.1 Complex Fenchel Nielsen coordinates

Complex Fenchel Nielsen parameters were introduced in [27,42] (see also [20]) as a generalization to QF of the classical Fenchel Nielsen coordinates for Fuchsian groups. Here we briefly summarize the main points as applied to T_{1}.

Let ⟨G; A, B⟩ be a marked quasifuchsian punctured torus group constructed from a pair of marked generators α, β of π_{1}(T_{1}) as described in 2.1. Complex Fenchel Nielsen coordinates (λ_{A}, τ_{A,B}) for ⟨G; A, B⟩ are obtained as follows. The parameter λ_{A} ∈ C/2πiZ is the complex translation length of the generator A = ρ(α), or equivalently the complex length λ_{α}. The twist parameter τ_{A,B} ∈ C/2πiZ measures the complex shear when the axis Ax B^{−1}AB is identified with the axis Ax A by B. More precisely, if the common perpendicular δ to Ax B^{−1}AB and Ax A meets these axes in points Y, X respectively, then ℜτ_{A,B} is the signed distance from X to B(Y) and ℑτ_{A,B} is the angle between δ and the parallel translate of B(δ) along Ax A to X, measured facing towards the attracting fixed point of A. On the critical line F_{α,β}, τ_{A,B} ≡ 0 mod 2πi and Ax A, Ax B intersect orthogonally. Thus a point on this line corresponds to a rectangular torus with generators (A, B). The conventions for measuring the signed distance and the angle are explained in more detail in [20] but are not important here.

As shown in [11,20,27], given the parameters λ_{A}, τ_{A,B}, and a fixed a normalization, one can explicitly write down the matrix generators for a marked two generator group G(λ_{A}, τ_{A,B}) ⊂ P SL(2, C) in which the commutator [A, B] is parabolic. This group may or may not be discrete. The matrix coefficients of G depend holomorphically on the parameters. The construction thus defines a holomorphic embedding of QF into a subset of C/2πiZ × C/2πiZ, in which Fuchsian space F is identified with the image of R^{2}.

We want to lift this to an embedding into C^{2}. In section 6 we discussed how to lift the length function λ_{A} on QF to a holomorphic function on C. We can similarly lift the twist parameter τ_{A,B} by specifying that it be real valued on F.

On F, the real valued parameters λ_{A}, τ_{A,B} reduce to the classical Fenchel Nielsen parameters l_{A}, t_{A,B} defined by the above construction with λ_{A} the hyperbolic translation length l_{A} of A and τ_{A,B} the twist parameter t_{A,B}.

Clearly, the complex Fenchel Nielsen construction can be made relative to any marking V, W of G. As described in detail in section 5 of [20], for fixed λ ∈ R^{+} and τ ∈ C, the complex Fenchel Nielsen construction relative to V, W determines a map D → H^{3}. This map is the composition of the earthquake E_{γ}(ℜτ) along the geodesic γ represented by V with a map ψ : D → H^{3} which is an isometry restricted to each conjugate of Ax A and also to each component of the complement of these axes in D. The earthquake E_{γ}(ℜτ) : D → D intertwines the action of the rectangular torus group G(λ, 0) with the group G(λ, ℜτ). The map ψ is a pleated surface map with with pleating locus γ and angle ℑτ between the outward normals to adjacent flat planes. It conjugates the actions of G(λ, ℜτ) on D and G(λ, τ) on its image in H^{3}. We set D_{γ}(λ, τ) = ψ(D). We note for future use that the bending measure of a transversal σ is i(γ, σ)ℑτ.

7.2 Quakebends

Quakebends are a complex version of earthquakes. The construction was introduced by Thurston and is explained in detail in [9] and also summarized in [20]. An alternative discussion can be found in [31].

Let p ∈ F and let G_{0} = G(p) act on the disk D ⊂ H^{3}. For μ ∈ ML and τ ∈ C, the quakebend construction defines an isomorphism Q_{μ}(τ) from G_{0} to its image Q_{μ}(τ)(G_{0}) = G_{μ}^{p}(τ), together with a pleated surface ψ_{μ}^{p}(τ) : D → H^{3} conjugating the actions of E_{μ}(ℜτ)(G_{0}) = G_{μ}^{p}(ℜτ) on D and G_{μ}^{p}(τ) on the image D_{μ}^{p}(τ) = ψ_{μ}^{p}(τ)(D). If ℑτ ≠ 0, then D_{μ}^{p}(τ) has pleating locus |μ|. When τ = 0, ψ_{μ}^{p}(τ) = id and G_{μ}^{p}(τ) = G_{0}. When ℑτ = 0 and ℜτ = t, Q_{μ}(τ) coincides with the earthquake E_{μ}(t), D_{μ}^{p}(τ) = D and G_{μ}^{p}(t) is discrete and Fuchsian for all t ∈ R. If ℜτ = 0, we call the quakebend a pure bend.

If the lamination μ is rational, μ = kδ_{γ}, an earthquake along μ reduces to a Fenchel Nielsen twist. In terms of Fenchel Nielsen coordinates (l_{V} , t_{V,W}) relative to a marking (V, W), where V ∈ G represents the geodesic γ, this is given by the formula (λ_{V} , τ_{V,W}) ↦ G(λ_{V} , τ_{V,W} + kt). Likewise a quakebend along kδ_{γ} is the complex Fenchel Nielsen twist given by the formula Q_{μ}(τ) : G(λ_{V} , τ_{V,W}) ↦ G(λ_{V} , τ_{V,W} + kτ). In particular, if the base point p ∈ F is the rectangular group G(λ, 0) relative to its marking (V, W), the image pleated surface D_{μ}^{p}(τ) is exactly D_{γ}(λ, τ) as described in 7.1 above. We shall make frequent use of this observation below. Note that the bending measure of a transversal σ to D_{μ}^{p}(τ) is always i(σ, μ)ℑτ.

So far, we have only discussed quakebends when the basepoint p is in F. Examining [9], however, it is clear that one can make the same construction starting from a basepoint q ∈ P_{μ}^{+}. More precisely, let pl^{+}(q) be the bending measure on ∂C^{+}(q), so that (by the unique ergodicity of measured laminations on a punctured torus) pl^{+}(q) = kμ for some k > 0. Let the flat structure of ∂C^{+}(q) be represented by the Fuchsian group F^{+}(q) acting in D. One can define the quakebend Q_{μ}^{q}(τ) as the group obtained by the quakebend Q_{μ}^{q}(τ+ik) acting on F^{+}(q); in other words compose an earthquake along μ by ℜτ with a pure bend by ℑτ + k. In this case, we should consider the time zero pleated surface D_{μ}^{q}(0) to be the surface ∂C^{+}. (See also [26], [31] for other versions of this construction.)

We shall not need to discuss here the problems associated with defining a quakebend from an arbitrary basepoint in QF.

7.3 Quakebend planes

In what follows, we shall often want to regard the quakebend parameter τ as a holomophic function on the space of representations ρ : π_{1}(T_{1}) → P SL(2, C), modulo conjugation in P SL(2, C). When the basepoint is Fuchsian, this is justified by the following proposition, which is [9], Lemma 3.8.1.

Proposition 24 Let p ∈ F, τ ∈ C, μ ∈ ML, and let G_{μ}^{p}(τ) = Q_{μ}(τ)(G(p)). Then the matrix coefficients of the elements of G_{μ}^{p}(τ) are holomorphic functions of τ.

It is clear that the Epstein-Marden proof still works when the basepoint q is in P_{μ}^{+}.

This result enables us to introduce quakebend planes, which are the device used in section 9 to reduce the investigation of pleating varieties to a tractable problem in one complex dimension.

For q ∈ P_{μ}^{+} ∪F, we set Q_{μ}^{q} = {G_{μ}^{q}(τ) : τ ∈ C}; we call Q_{μ}^{q} the μ-quakebend plane based at q and sometimes write Q_{μ}^{q}(τ) for G_{μ}^{q}(τ). By proposition 25 below, a neighborhood of q in Q_{μ}^{q} is contained in P_{μ} — but we emphasize once again that that in general the whole of Q_{μ}^{q} is not contained in QF (see proposition 35 below and [31]).

In the rational case μ ∈ ML_{Q}, Q_{μ}^{q} has a very easy description in terms of complex Fenchel Nielsen coordinates. Suppose that μ = δ_{γ}, γ ∈ S and that (γ, γ′) are a pair of marked generators for π_{1}(T_{1}). Let (λ_{V} , τ_{V,W}) ⊂ C^{2} be complex Fenchel Nielsen coordinates relative to corresponding marked generators (V, W) of G. Let c = λ_{μ}(q). Then it is clear from the discussion above that Q_{μ}^{q} is just the slice {(c, τ)} ⊂ C^{2}. We denote this slice by Q_{γ,c}. Clearly, Q_{γ,c} meets F along the earthquake path E_{γ,c}.

More generally, if μ ∈ ML and p, p′ ∈ E_{μ,c}, it is clear that Q_{μ}^{p} = Q_{μ}^{p′}; we denote this plane by Q_{μ,c}. Clearly, Q_{μ,c} meets F along the earthquake path E_{μ,c}. In general, however, if q, q′ ∈ P_{μ} and λ_{μ}(q) = λ_{μ}(q′), then it is not immediately clear whether or not Q_{μ}^{q} = Q_{μ}^{q′}. It is a consequence of our main results that λ_{μ}(q) = λ_{μ}(q′) always implies Q_{μ}^{q} = Q_{μ}^{q′}; this is proved in corollary 45 below.

As explained above, for a basepoint q ∈ P_{μ} ∪F, the quakebend plane Q_{μ}^{q} is not, in general, contained in QF. We note that in the special case p ∈ F, since QF is an open neighborhood of F (in the space of representations into P SL(2, C) modulo conjugation), it follows that for small τ, G_{μ}^{p}(τ) is quasifuchsian. The following stronger result shows that, as one would naively expect, as one quakebends along μ away from a basepoint q ∈ P_{μ}^{+} ∪F (for which ∂C^{+} = D_{μ}^{q}(0)), the pleated surface D_{μ}^{q}(τ) remains equal to ∂C^{+} for all small τ.

Proposition 25 Given μ ∈ ML and q ∈ P_{μ}^{+} ∪F, there exists ϵ > 0, depending on μ and q, such that if |τ| < ϵ, then G_{μ}^{q}(τ) ∈ QF and D_{μ}^{q}(τ) is a component of ∂C(G_{μ}^{q}(τ)).

Proof: This is proved in [20], prop 8.10 for the case in which the basepoint q is in F. It is clear that the same proof works in our more general case. □

We note that if G_{μ}^{q}(τ) ∈ QF and D_{μ}^{q}(τ) = ∂C^{+}(G_{μ}^{q}(τ)), then the flat structure of ∂C^{+}(G_{μ}^{q}(τ)) is represented by the Fuchsian group E_{(ℜτ)μ}(F^{+}(q)) obtained by earthquaking a distance ℜτ along the pull-back of μ to D. This observation will be important in section 8 below.

8 The local pleating theorem

In this section we prove the local pleating theorem 26. We derive various consequences including the density theorem 4 of the introduction and a detailed description of how pleating varieties meet F. The statement of the theorem is as follows.

Theorem 26 Local Pleating Theorem Suppose that ν ∈ ML and q_{0} ∈ P_{ν} ∪F. Then there exists a neighborhood U of q_{0} in QF such that if q ∈ U and λ_{ν}(q) ∈ R^{+}, then q ∈ P_{ν} ∪F.

Our starting point for proving this theorem is proposition 7.6 of [20], part of whose content can be stated in the following way. We write G_{γ}^{q_{0}}(τ) for G_{δ_{γ}}^{q_{0}}(τ) = Q_{δ_{γ}}(τ)(G(q_{0})).

Proposition 27 Suppose that γ ∈ S and q_{0} ∈ F. Then there exists η > 0 such that if |τ| < η, then G_{γ}^{q_{0}}(τ) ∈ P_{γ} ∪F.

This proposition can be regarded as the special case of 26 in which ν = kδ_{γ} ∈ ML_{Q}, the basepoint q_{0} is Fuchsian and we restrict the discussion to the quakebend plane Q_{γ}^{q_{0}} through q_{0}. Notice that in this plane, λ_{γ}(G_{γ}^{q_{0}}(τ)) is fixed and hence automatically real.

We begin by reviewing the argument in [20]. Suppose γ ∈ S, let V ∈ G represent γ and choose W ∈ G such that (V, W) is a marking. Let (λ_{V} , τ_{V,W}) be complex Fenchel Nielsen coordinates for QF relative to (V, W); thus we regard (λ_{V} , τ_{V,W}) as holomorphic functions on QF. As described in section 7.1, whenever λ_{V} = λ_{V} (q) ∈ R^{+}, the complex Fenchel Nielsen construction determines a pleated surface map D → H^{3} with pleating locus γ. To indicate more clearly the relevant variables, we shall write P_{γ}(q) for the image D_{γ}(λ_{V} , τ_{V,W}) ⊂ H^{3}.

If q_{0} ∈ F, then ℑτ_{V,W} (q_{0}) = 0, hence for q near q_{0}, ℑτ_{V,W} (q) is small. In [20], we argued that for ℑτ sufficiently small, P_{γ}(q) = D_{γ}(λ_{V} , τ_{V,W}) is embedded and bounds a convex half space in H^{3}. It follows by proposition 7.2 of [20], that P_{γ}(q) is a component of ∂C(q).

There are two problems in applying this argument in the present circumstances. First, we wish to include the case q_{0} ∉ F, and thus can no longer assume that ℑτ_{V,W} is small. Second, we want to prove theorem 26 for an irrational lamination ν by taking a limit of rational laminations. Since the constant ϵ of proposition 25 depends on γ and is not uniform, (in fact ϵ ∼ 2 exp (−l_{γ}/2)), the limiting process fails, indicating that we need to scale the approximating laminations properly. To resolve these problems, we digress to study the geometry of the pleated surfaces P_{γ}(q) more carefully.

Fix q_{0} ∈ QF, γ ∈ S and a marking (V, W) as above. Suppose that q ∈ QF and that λ_{V} (q) ∈ R^{+}. Let ϕ_{γ}(q) be the normalized Fuchsian group with (real) Fenchel Nielsen coordinates (λ_{V} (q), ℜτ_{V,W} (q)). The surface P_{γ}(q) is the image of the pleated surface map D → H^{3} defined by a pure bend along δ_{γ} by iℑτ_{V,W} (q). We refer to ϕ_{γ}(q) as the flat structure of P_{γ}(q).

We can associate a transverse measure b_{ν}(q) to P_{γ}(q) in an obvious way: for any arc σ on P_{γ}(q) transverse to its pleating locus γ, set b_{γ}(q)(σ) = i(σ, γ)ℑτ_{V,W} (q). Thus we can also write P_{γ}(q) = Q_{b_{γ}(q)}^{p}(i), where p is the image of ϕ_{γ}(q) in F.

We remark that we are not making the assumptions that P_{γ}(q) is a component of ∂C(q), or that ϕ_{γ}(q) is one of the flat structures F^{±}(q) of ∂C(q) (see section 4.1); in fact, this is exactly what we must prove. In particular, we cannot assume that b_{γ}(q) is the bending measure pl^{±}(q). The following result, however, gives information about ϕ_{γ}(q) and b_{γ}(q) for q near q_{0} ∈ P_{ν_{0}} for irrational ν_{0}.

Proposition 28 Given ν_{0} ∈ ML − ML_{Q}, and q_{0} ∈ P_{ν_{0}}^{+} ∪F, let F^{+}(q_{0}) ∈ F and pl^{+}(q_{0}) be the flat structure and bending measure of ∂C^{+}(q_{0}) respectively. (If q_{0} ∈ F, then pl^{+}(q_{0}) = 0 and F^{+}(q_{0}) is the Fuchsian group representing q_{0}.) Then, given neighborhoods V of F^{+}(q_{0}) in F and W of pl^{+}(q_{0}) in ML, there exist neighborhoods U of q_{0} in QF and X of [ν_{0}] in P ML such that if q ∈ U, [δ_{γ}] ∈ X ∩ P ML_{Q} and λ_{γ}(q) ∈ R^{+}, then the flat structure ϕ_{γ}(q) of P_{γ}(q) is in V and the transverse measure b_{γ}(q) is in W.

The idea of the proof of this proposition is that by the convergence lemma 1, for ν_{0} ∈ ML − ML_{Q}, nearby rational laminations are close in the Hausdorff topology, so that the bending loci and hence the structures of the associated pleated surfaces are also close. The details are a technical modification of the arguments in [19] and are given in appendix 12.2. (We remark that the result is still true for ν_{0} ∈ ML_{Q}, however the details of the proof differ since the convergence lemma does not apply. We omit this case since it is not needed here.)

The plan of the proof of theorem 26 is the following. The hard case to handle is ν ∉ ML_{Q}. We shall show in theorem 31 below, that if q_{0} ∈ P_{ν}, then for q in a neighborhood of q_{0}, if [δ_{γ}] is sufficiently close to [ν] in P ML, the condition λ_{γ}(q) ∈ R^{+} implies that P_{γ}(q) is a also a component of ∂C. Theorem 26 then follows by an easy limiting argument using the continuity theorem 11.

We prove theorem 31 using an extension of proposition 25, which we state as proposition 29. Stated roughly it says that if p ∈ F and the pleated surface D_{μ}^{p}(τ) associated to the quakebend Q_{μ}^{p}(τ) is a component of ∂C, then the same is true of any surface D_{μ′}^{p′}(τ′) obtained by quakebending a nearby amount τ′ from a nearby point p′ ∈ F along a nearby lamination μ′. Now, a component of ∂C can be obtained from the Fuchsian group representing its flat structure by a pure bend along the pleating lamination |μ|. Proposition 28 allows us to apply proposition 29 to P_{γ}(q) for [δ_{γ}] close to [ν] and q close to q_{0}, thus proving theorem 31.

Proposition 29 Let p_{0} ∈ F be represented by G_{0} = G(p_{0}) and suppose that τ_{0} ∈ C is such that q_{0} = Q_{μ_{0}}^{p_{0}}(τ_{0}) ∈ P_{μ_{0}}^{+}. Then there exist neighborhoods X, Y and Z of μ_{0}, p_{0} and τ_{0} in ML, F and C respectively, such that if μ ∈ X, p ∈ Y and τ ∈ Z, then q = Q_{μ}^{p}(τ) ∈ QF and D_{μ}^{p}(τ) is a component of ∂C(q).

The proof of this result is identical with the version in [20] once we note that the constants involved depend continuously on G and μ. This follows from the following variant of lemma 8.2 of [20].

Lemma 30 Let X and Y be compact sets in ML and F respectively. Then there exist constants d > 0 and K > 0 such that if μ ∈ X and G ∈ Y, and if σ is any geodesic segment on D/G of length less than d, then μ(σ) < K.

We can now prove theorem 31, which is important in its own right.

Theorem 31 Suppose ν_{0} ∈ ML − ML_{Q} and q_{0} ∈ P_{ν_{0}}^{+} ∪F. Then there are neighborhoods U of q_{0} in QF and X of [ν_{0}] in P ML such that if [δ_{γ}] ∈ P ML_{Q} ∩ X, q ∈ U and if λ_{γ}(q) ∈ R^{+}, then P_{γ}(q) is a component of ∂C(q).

Proof: By proposition 28, there are neighborhoods X of [ν_{0}] in P ML and U of q_{0} in QF such that for q ∈ U and [δ_{γ}] ∈ X, the flat structures F^{+}(q_{0}) of ∂C^{+}(q_{0}) and ϕ(q) of P_{γ}(q) are close in F, and the transverse measures pl^{+}(q_{0}) and b_{γ}(q) are close in ML.

As remarked earlier, ∂C^{+}(q_{0}) is just the pleated surface obtained from F^{+}(q_{0}) under a pure bend by i along the measured lamination pl^{+}(q_{0}) while P_{γ}(q) is obtained from ϕ_{γ}(q) by a pure bend by i along b_{γ}(q). The result now follows from proposition 29. □

We now prove theorem 26.

Proof: Suppose first that ν ∈ ML_{Q}. In this case the result follows as in the discussion following proposition 27 above, using proposition 29 as a substitute for the condition ℑτ near 0 when the base point q_{0} is not Fuchsian.

Suppose therefore that ν ∉ ML_{Q}, and pick ν_{n} ∈ ML_{Q}, ν_{n} → ν. Find neighborhoods U of q_{0} in QF and X of [ν] in P ML satisfying the conclusion of theorem 31.

Assume q ∈ U and λ_{ν}(q) ∈ R^{+}. Since λ_{ν_{n}} → λ_{ν} uniformly on U, and since λ_{ν} is non-constant on U, by Hurwitz’s theorem we can find q_{n} ∈ U, q_{n} → q, such that λ_{ν_{n}}(q_{n}) = λ_{ν}(q), and in particular such that λ_{ν_{n}}(q_{n}) ∈ R^{+}. Applying theorem 31, we see that for sufficiently large n, P_{|ν_{n}|}(q_{n}) is one of the components of ∂C^{+}(q_{n}) so that q_{n} ∈ P_{ν_{n}}^{+}. Hence, by the continuity theorem 11, we get q ∈ P_{ν}^{+}. This completes the proof of theorem 26. □

Corollary 32 Suppose μ, ν ∈ ML, [μ] ≠ [ν]. Let q_{0} ∈ P_{μ,ν}^{+} ∪F and let Q_{μ}^{q_{0}} be the μ-quakebend plane based at q_{0}. There exists a neighborhood U of q_{0} in Q_{μ}^{q_{0}} such that if q ∈ U and λ_{ν}(q) ∈ R^{+}, then q ∈ P_{μ,ν} ∪ P_{ν,μ} ∪F.

Proof: This is just theorem 26 applied in the quakebend plane Q_{μ}^{q_{0}}. We can prove it either by applying proposition 25 to see that for q ∈ Q_{μ}^{q_{0}} near q_{0}, we have q ∈ P_{μ} ∪F, and then applying theorem 26 to ν; or by noting that since λ_{μ} is constant on Q_{μ}^{q_{0}} and real valued at q_{0}, we can apply theorem 26 first to μ and then to ν. □

Remark 33 The condition λ_{γ}(q) ∈ R^{+} is key in proposition 28 and in theorem 31. We can always find a pleated surface Π whose pleating locus σ contains the geodesic γ. In general, however, σ properly contains γ and has leaves spiralling into γ, and thus carries no transverse measure. Then, even though [δ_{γ}] is near [pl^{±}] in P ML, the pleated surface P_{γ} realizing γ (see [7],[45]) is not necessarily embedded; moreover, even if it is, neither of the half spaces it bounds in H^{3} will be convex. The point is that the condition σ = γ is equivalent to λ_{γ}(q) ∈ R^{+}.

8.1 Consequences of theorem 26

From theorem 26 we obtain the following local extension of the picture of Fuchsian space described in section 3.

Theorem 34 Let μ, ν ∈ ML, i(μ, ν) > 0, p ∈ F. Then there is a neighborhood U of p in QF such that

(1) if p ∉ F_{μ,ν} then P_{μ,ν} ∩U = ∅, while

(2) if p ∈ F_{μ,ν} then the intersection of the R-loci λ_{μ}^{−1}(R) ∩ λ_{ν}^{−1}(R) with U is

exactly

(P_{μ,ν} ∪ P_{ν,μ} ∪F) ∩ U.

In the second case, let p = p(μ, ν, c) ∈ F_{μ,ν}, let Q_{μ}^{p} be the quakebend plane along μ based at p and let V = U ∩ Q_{μ}^{p} . Then λ_{ν}|_{V} has a simple critical point at p and λ_{ν}^{−1}(R^{+}) ∩ (V − F) has exactly two components, one lying in P_{μ,ν} and the other in P_{ν,μ}.

Proof: Part 1 follows since for p ∉ F_{μ,ν}, there exists a neighborhood U of p in the quakebend plane Q_{μ}^{p} based at p such that λ_{ν}^{−1}(R^{+}) ∩ U ⊂ F. The first statement in part 2 is immediate from Theorem 26.

By Corollary 21, λ_{ν}|_{E_{μ,c}} has exactly one critical point at p and it is simple. Thus the second statement in part 2 is a restatement of corollary 32 with q_{0} = p. □

We note that this theorem provides an alternative proof of theorem 13.

We can also now prove the density theorem 4 of the introduction. First, we need a bound on the bending angle in a quakebend plane.

Proposition 35 Suppose μ ∈ ML, q ∈ P_{μ} ∪F and let Q_{μ}^{q} be the quakebend plane along μ based at q with parameter τ = τ_{μ}. Given K > 0, there exist B_{2} > B_{1} > 0 such that if |ℜτ| < K and B_{2} > ℑτ > B_{1}, then Q_{μ}^{q}(τ) ∉ P_{μ}.

We need the restriction B_{1} < ℑτ < B_{2} rather than simply ℑτ > B_{1} because of the periodicity of the twist parameter τ_{μ} for rational laminations. The period is 2πi when μ = δ_{γ}, γ ∈ S. The statement Q_{μ}^{q}(τ) ∉ P_{μ} means that either Q_{μ}^{q}(τ) ∉ QF or that Q_{μ}^{q}(τ) ∈ QF but [pl^{+}(Q_{μ}^{q}(τ))] ≠ [μ]. We show that, under the hypotheses of the proposition, Q_{μ}^{q}(τ) fails to be in P_{μ} because the surface obtained by bending along μ is not embedded. In this situation, it may or may not be true that Q_{μ}^{q}(τ) ∈ QF. The proof is given in appendix 12.3, see also [31] theorem 6.2.

As an immediate corollary we have

Proposition 36 Suppose q ∈ QF, q ∈ P_{μ,ν} ∪F. Then the holomorphic function λ_{ν}(q) is non-constant on Q_{μ}^{q} ∩ QF.

Proof: Since q ∈ P_{ν} we know λ_{ν}(q) ∈ R^{+}. By construction λ_{μ}(q) = c > 0 for all q ∈ Q_{μ}^{q}. Suppose that λ_{ν}(q) = d > 0 for all q ∈ Q_{μ}^{q} ∩ QF. By theorem 26, P_{μ,ν} is open in Q_{μ}^{q}.

Now suppose that q_{n} = Q_{μ}^{q}(τ_{n}) ∈ P_{μ,ν} and that τ_{n} → τ_{∞}. Since l_{μ}(q_{n}) = c and l_{ν}(q_{n}) = d for all n, it follows from theorem 15 that q_{n} → q_{∞} ∈ QF. By theorem 11, q_{∞} ∈ P_{μ,ν} ∪F. Clearly, q_{∞} = Q_{μ}^{q}(τ_{∞}) and so P_{μ,ν} is closed in Q_{μ}^{q} − F. Therefore P_{μ,ν} is a connected component of Q_{μ}^{q} − F and must be one of the half planes ℑτ_{μ} > 0 or ℑτ_{μ} < 0, contradicting proposition 35. □

Finally we can prove theorem 4.

Theorem 4 The rational pleating varieties P_{μ,ν}, μ, ν ∈ ML_{Q} are dense in QF.

Proof: Let q ∈ QF and let μ ∈ [pl^{+}(q)], ν ∈ [pl^{−}(q)]. By theorem 23, λ_{μ}(q), λ_{ν}(q) ∈ R^{+}. Clearly, we may as well assume μ ∉ ML_{Q}. Find a sequence {μ_{n}} ∈ ML_{Q}, μ_{n} → μ. By Hurwitz’s theorem in QF, we can find points q_{n} → q with λ_{μ_{n}}(q_{n}) ∈ R^{+} and so by theorem 31, q_{n} ∈ P_{μ_{n}} for large enough n. If ν ∈ ML_{Q} we are done, otherwise find {ν_{n}} ∈ ML_{Q}, ν_{n} → ν. By proposition 36, λ_{ν_{n}} is non-constant on Q_{μ_{n}}^{q_{n}} ∩ QF and we can apply Hurwitz’s theorem again in Q_{μ_{n}}^{q_{n}} ∩QF to find q′_{n} near q_{n}, such that q′_{n} → q and such that λ_{ν_{n}}(q′_{n}) ∈ R^{+}. By theorem 31 again, q′_{n} ∈ P_{μ_{n},ν_{n}} for large enough n. □

9 Pleating rays and planes

In this section, we apply the local and limit pleating theorems to prove our main results theorems 6 and 2 of the introduction.

Recall from section 4.2 the definition of the pleating ray

P_{μ,ν,c} = {q ∈ P_{μ,ν} : l_{μ}(q) = c},

where (μ, ν) ∈ ML × ML, and c > 0. Pleating rays are the basic building blocks out of which we construct pleating planes and the BM-slices mentioned in the introduction. Notice that, because of theorem 23, we can equally well define

P_{μ,ν,c} = {q ∈ P_{μ,ν} : λ_{μ}(q) = c}.

Our results will justify the names “rays” and “planes”.

The main work is in the study of the pleating rays. Our strategy is as follows. We begin by applying the limit pleating theorem and the local pleating theorem to obtain some general results about P_{μ,ν} for arbitrary μ, ν ∈ ML. We then prove theorem 6 in the case where [μ] = [δ_{γ}], [ν] = [δ_{γ′}] and (γ, γ′) is a marking for T_{1}. We show that in this case P_{δ_{γ},δ_{γ′},c}, which we call an integral pleating ray, is a straight line segment in the quakebend plane Q_{γ,c}. Using the integral rays we derive constraints on the rays P_{δ_{γ},ν,c} ⊂ Q_{|μ|,c} for arbitrary ν; using our general results we are then able to deduce theorem 6 in the general case. Finally, we apply theorem 6 to deduce theorem 2.

9.1 Pleating rays

In the four lemmas which follow, μ, ν are arbitrary laminations in ML and, as usual, Q_{μ}^{q} denotes the μ-quakebend plane through q ∈ P_{μ} ∪F.

Lemma 37 Let q ∈ P_{μ,ν}. The set P_{μ,ν} ∩Q_{μ}^{q} is a union of connected components of the R-locus of λ_{ν} in (QF − F) ∩ Q_{μ}^{q}.

Proof: We have to show that P_{μ,ν} ∩Q_{μ}^{q} is open and closed in the R-locus of λ_{ν} in (QF − F) ∩ Q_{μ}^{q}. The openness is the local pleating theorem 26 and closure follows by the continuity theorem 11. □

If ν = kδ_{γ}, γ ∈ S, we obtain a stronger result. Let V ∈ G represent γ. In this case, by proposition 24, trace Tr V is defined and holomorphic on all of Q_{μ}^{q} (including the part outside QF), and we obtain a version of lemma 37 for the R-locus of λ_{γ} in Q_{μ}^{q}. Define the hyperbolic locus of γ in Q_{μ}^{q} as {q ∈ Q_{μ}^{q} : Tr V ∈ R, | Tr V | > 2}.

Lemma 38 Let ν = kδ_{γ} ∈ ML_{Q} and let q ∈ P_{μ,ν}. Let V ∈ G represent γ. Then the set P_{μ,ν} ∩Q_{μ}^{q} is a union of connected components of the hyperbolic locus of Tr V in Q_{μ}^{q} − F.

Proof: The openness follows as above, using the local pleating theorem 26. The closure follows from theorem 15. The point is first, that length and trace are related by the trace formula Tr V = 2 cosh(l_{γ}/2), and second, that if we reach a limit point at which | Tr V | > 2, then l_{γ} > 0 so that by the second part of theorem 15 we must still be in QF. (See [18] proposition 5.4 for a more elementary proof without using theorem 15.) □

This is a strong result. The point is, that starting from a point we know is in QF, the lemma asserts that if we move along branches of the hyperbolic locus, then we stay in QF until we reach a boundary point of ∂QF at which | Tr V | = 2. This observation is what makes it possible to use the pleating invariants for computations of ∂QF, see theorem 3 of the introduction.

With the notation of lemma 37, set c = λ_{μ}(q). Clearly, P_{μ,ν} ∩Q_{μ}^{q} = P_{μ,ν,c}. As usual, we let p_{μ,ν,c} ∈ F_{μ,ν} be the minimal point for the length function l_{ν} on the earthquake path E_{μ,c}. The following lemma makes essential use of theorem 15.

Lemma 39 Let q ∈ P_{μ,ν} and let c = λ_{μ}(q). The image of each component of P_{μ,ν} ∩Q_{μ}^{q} under the map λ_{ν} is an interval of the form (0, ∞), (0, d) and (d, ∞) where d = f_{μ,ν}(c) = l_{ν}(p_{μ,ν,c}). Moreover, there is at most one component of P_{μ,ν} ∩Q_{μ}^{q} whose image is (0, d); the closure of such a component meets F exactly in p(μ, ν, c).

Remark 40 As we shall see in theorem 6, in fact P_{μ,ν} ∩Q_{μ}^{q} has a unique component, and the image of this component is (0, d).

Proof: Let K be a connected component of P_{μ,ν,c}. By theorem 23, λ_{ν}|_{K} is real valued and, by proposition 36, it is non-constant on Q_{μ}^{q}. Since it is holomorphic, it is not locally constant and thus not constant on K. Therefore by lemma 37 the image I_{K} of λ_{ν}|_{K} is an open interval in R^{+}.

Suppose that r ∈ R^{+} and that there is a sequence {q_{n}} ∈ K such that λ_{ν}(q_{n}) → r. Since λ_{μ}(q_{n}) = c, by theorem 15 a subsequence of {G(q_{n})} has an algebraic limit G_{∞}. Furthermore, since λ_{ν}(q_{n}) → r > 0, the group G_{∞} is represented by a point q ∈ QF such that λ_{ν}(q) = r. If q ∈ QF − F then by theorem 11, q ∈ K so that r ∈ I_{K}. On the other hand, if q ∈ F then by theorem 34, q = p(μ, ν, c) and r = λ_{ν}(q) = f_{μ,ν}(c) = d. Thus λ_{ν}(K) is open and closed in (0, d) ∪ (d, ∞). The result follows from theorem 34. □

Lemma 41 Let q ∈ P_{μ,ν} and let c = λ_{μ}(q). Let τ denote the quakebend parameter in the quakebend plane Q_{μ}^{q}. Suppose that the points q_{n} ∈ P_{μ,ν,c} are represented by the quakebend parameter τ_{n} and that λ_{ν}(q_{n}) → ∞. Then |ℜ(τ_{n})| → ∞.

Proof: Since q_{n} ∈ P_{μ,ν,c} we know λ_{ν}(q) is real. Moreover, λ_{ν}(q) ≤ l_{ν}(F^{+}(q_{n})); that is, λ_{ν}(q) is bounded above by the length of ν on the flat structure of ∂C^{+}/G(q_{n}). This flat structure is determined by the length of μ, which is fixed, and the earthquake parameter ℜ(τ_{n}). Thus if |ℜ(τ_{n})| is bounded, so is λ_{ν}(q_{n}). □

We can now start investigating the integral pleating rays. Suppose that [μ] = [δ_{γ}], [ν] = [δ_{γ′}] and (γ, γ′) is a marking for T_{1}. For simplicity, we write P_{γ} for P_{δ_{γ}} and so on. Let (λ_{V} , τ_{V,W}) ∈ C^{2} be complex Fenchel Nielsen coordinates relative to a marked pair of generators (V, W) corresponding to (γ, γ′). As in section 7.3, we denote by Q_{γ,c} the slice {(c, τ)} ⊂ C^{2}; Q_{γ,c} is the quakebend plane along γ that meets F along the earthquake path E_{γ,c}. We denote points in this slice simply by the parameter τ = τ_{δ_{γ}}. As usual, τ = 0 corresponds to the point p(γ, γ′, c) ∈ F, while ℑτ = 0 is the earthquake path E_{γ,c}.

For m ∈ Z, the pair (γ, γ^{m}γ′) is a pair of marked generators for π_{1}(T_{1}) corresponding to the pair of generators V, V^{m}W for G. Clearly P_{γ,γ^{m}γ′,c} ⊂ Q_{γ,c}. The generators V, V W are obtained from the pair V, W by the map induced by a Dehn twist about γ. The basepoint relative to which we measure the twist parameter changes and we find τ_{V,V W} = τ_{V,W} + λ_{V}; similarly, τ_{V,V^{m}W} = τ_{V,W} + mλ_{V}.

The following formula is derived in [39] for any pair (V,W) of marked generators for G:

cosh \frac{τ_{V,W}}{2} = ± cosh \frac{λ_{W}}{2} tanh \frac{λ_{V}}{2} (2)

By our conventions, ℜλ_{V} , ℜλ_{W} > 0, so that we should choose the + sign on F and hence everywhere in QF.

Applying this formula to the generators (V, V^{−m}W) we find

cosh \frac{τ − mλ_{V}}{2} = cosh \frac{λ_{V^{−m}W}}{2} tanh \frac{λ_{V}}{2} . (3)

In particular, at τ = mc we have

1 = cosh \frac{λ_{γ^{−m}γ′}}{2} tanh \frac{λ_{γ}}{2} . (4)

or equivalently

sinh \frac{c}{2} sinh \frac{λ_{γ^{−m}γ′}}{2} = 1. (5)

Proposition 42 Let (γ, γ′) be a marked pair of generators for π_{1}(T_{1}) and let c > 0. Then for m ∈ Z, P_{γ,γ^{−m}γ′,c} and P_{γ^{−m}γ′,γ,c} are the two line segments ℜτ = mc, |ℑτ| < 2 arccos tanh \frac{c}{2} in Q_{γ,c}. The two line segments ℜτ = mc, |ℑτ| ≥ 2 arccos tanh \frac{c}{2} in Q_{γ,c} have empty intersection with QF.

Remark 43 Which of the two segments corresponds to P_{γ,γ^{−m}γ′,c} and which to P_{γ^{−m}γ′,γ,c} depends on our convention for measuring τ and is not important here.

Proof: Because τ_{V,V^{−m}W} = τ_{V,W} − mc, we may restrict ourselves to the case m = 0. From lemma 38, P_{γ,γ′} is a union of connected components of the hyperbolic locus of γ′ in Q_{γ,c} − F, and by theorem 34 there is a unique component K whose closure meets the critical line F_{γ,γ′} in p(γ, γ′, c).

From equation (2) ,

cosh \frac{τ}{2} = cosh \frac{λ_{γ′}}{2} tanh \frac{λ_{γ}}{2} .

Thus the R-locus of λ_{γ′} in Q_{γ,c} is the set defined by cosh \frac{τ}{2} ∈ R, or equivalently, {ℜτ = 0} ∪ {ℑτ = 0}. The real axis ℑτ = 0 corresponds to E_{γ,c} = Q_{γ,c} ∩ F and we see easily (see lemma 38 ) that the connected components of the hyperbolic locus of γ′ in Q_{γ,c}−F which meet the real axis are the two segments 0 < |ℑτ| < 2 arccos tanh \frac{c}{2}. One of these segments must be the component K and the other is the corresponding component for P_{γ′,γ}. Each of these segments is mapped bijectively by λ_{γ′} to [0, 2 arccos tanh \frac{c}{2}).

Now on the imaginary axis, we have cosh \frac{λ_{γ′}}{2} ≤ (tanh \frac{c}{2})^{−1}, and hence by lemma 39, P_{γ,γ′,c} and P_{γ′,γ,c} have no other components.

Finally we have to show that no other points on the imaginary axis lie in QF. Equation (3) holds for groups in Q_{γ,c} even when they are outside QF. On this axis, therefore, we always have

−1 ≤ cosh \frac{λ_{γ′}}{2} tanh \frac{c}{2} ≤ 1.

In [39] proposition 6.2, it is shown by a direct argument that if λ_{γ′} ∈ R and the above inequality is strict, then the group generated by V, W is quasifuchsian and contained in P_{γ,γ′}. Moreover, in this situation, this group is determined by λ_{γ} and λ_{γ′} up to conjugacy. If equality holds, the group represents the unique point p(γ, γ′, c) ∈ F. These are the therefore the groups we have already discussed.

Since cosh \frac{λ_{γ′}}{2} ∈ R, the only other possibility is that λ_{γ′} is purely imaginary. In this case the corresponding group element would have to be elliptic which is impossible in QF. □

We can now obtain a bound on the pleating rays P_{γ,ν,c} for arbitrary ν ∈ ML.

Corollary 44 Let ν ∈ ML, i(ν, γ) > 0. Then |ℜτ| is bounded on each component of P_{γ,ν,c}, where τ denotes the quakebend parameter τ_{δ_{γ}} in Q_{γ,c}.

Proof: If along some component of P_{γ,ν,c} in Q_{γ,c}, |ℜτ| → ∞, the component would have to intersect infinitely many of the lines τ = mc + iθ, θ ∈ R. According to proposition 42, however, each such line is the union of the integral pleating rays P_{γ,γ^{−m}γ′,c}, P_{γ^{−m}γ′,γ,c}, the point p(γ, γ^{−m}γ′, c) ∈ F, and points not in QF. This is impossible. □

We can now prove theorem 6 on the structure of the pleating rays. Recall from section 7.3 that Q_{μ,c} is the quakebend plane along μ which meets F along the earthquake path E_{μ,c}.

Theorem 6 Let μ, ν be measured laminations on T_{1} with i(μ, ν) > 0 and let c > 0. Then the set P_{μ,ν,c} ⊂ QF on which [pl^{+}] = [μ], [pl^{−}] = [ν] and l_{μ} = c, is a non-empty connected non-singular component of the R^{+}-locus of the restriction of λ_{ν} to Q_{μ,c}. The restriction of λ_{ν} to P_{μ,ν,c} is a diffeomorphism onto its image (0, f_{μ,ν}(c)) ⊂ R^{+}.

Proof: We assume first that μ ∈ ML_{Q}; without loss of generality we may take μ = δ_{γ}, γ ∈ S. Let c > 0 and let K be a component of P_{γ,ν,c}. By corollary 44, |ℜτ| is bounded on K. By lemma 41, λ_{ν}|_{K} is bounded and hence by lemma 39 the image is the interval (0, d) where d = f_{γ,ν}(c). Moreover, there exist points τ_{n} ∈ K , τ_{n} → p(γ, ν, c) ∈ F_{γ,ν}.

Now by theorem 34, there is only one branch of λ_{ν}^{−1}(R^{+}) near p(γ, ν, c); thus if the degree of λ_{ν}|_{K} were greater than one, there would be points τ′_{n} ∈ K with λ_{ν}(τ′_{n}) → d, but with τ′_{n} → q_{∞} ∈ QF − F. Then, by lemma 39, λ_{ν}(K) ⊃ (0, ∞), which is impossible.

Now we remove the restriction that μ ∈ ML_{Q}. Suppose that q ∈ P_{μ,ν,c}. We have to replace the plane Q_{γ,c} by the plane Q_{μ}^{q}, in which we denote the quakebend parameter τ_{μ} by τ. Because there are no integral pleating rays if μ is irrational, we need another argument to bound ℜτ.

Choose a sequence ν_{n} ∈ ML_{Q} such that ν_{n} → ν. By theorem 20 the holomorphic function λ_{ν}(q) is continuous in ν and by proposition 36 it is nonconstant. Thus we can apply Hurwitz’s theorem in Q_{μ}^{q} to find q_{n} ∈ Q_{μ}^{q} such that q_{n} → q and λ_{ν_{n}}(q_{n}) ∈ R^{+}. By theorem 31, for large enough n, q_{n} ∈ P_{μ,ν_{n},c}. Now because ν_{n} ∈ ML_{Q}, we can apply the argument above with the roles of μ and ν_{n} reversed to deduce that λ_{μ}(q_{n}) < f_{ν_{n},μ}(λ_{ν_{n}}(q_{n})). Thus, since f_{ν_{n},μ} is monotonic decreasing, we have that λ_{ν_{n}}(q_{n}) < f_{ν_{n},μ}^{−1}(λ_{μ}(q_{n})). Since f_{ν_{n},μ}^{−1} = f_{μ,ν_{n}} (from the definition of f_{ν_{n},μ}) we conclude that λ_{ν_{n}}(q_{n}) < f_{μ,ν_{n}}(c).

Because ν_{n} → ν, by corollary 8 and theorem 20 we have f_{μ,ν_{n}}(c) → f_{μ,ν}(c) so that {λ_{ν_{n}}(q_{n})} is bounded by a constant depending only on μ, ν and c. The remainder of the argument is as before. □

As an immediate corollary we have

Corollary 45 If q ∈ P_{μ}, then G(q) is obtained from a group G(p), p ∈ F by a quakebend Q_{μ}^{p}(τ^{∗}) along μ. Moreover, there is a quakebend path σ : [0, 1] → C in QF from p to q, or, in the coordinate of Q_{μ}^{p}(τ), σ(0) = 0, σ(1) = τ^{∗} and Q_{μ}^{p}(σ(t)) ∈ QF, 0 ≤ t ≤ 1.

This settles the question about uniqueness of quakebend planes raised at the end of section 7.2.

Remark 46 In [18], we studied the Maskit slice for punctured tori in terms of pleating rays with a similar definition to the above. In particular, theorem 7.2 of [18], asserts a non-singularity result similar to that in theorem 6. It has been pointed out to us by Y. Komori that our proof in [18] in the case of rays ν ∉ ML_{Q} is incorrect. In fact, we need an openness result like theorem 26 above. The methods above also prove the important result, omitted in [18], that the range of the length function on an irrational ray in the Maskit slice is (0, ∞). We refer to [25] for a corrected version of the argument in [18].

9.2 Pleating planes

We are finally able to prove theorem 2 on the structure of the pleating varieties P_{μ,ν}. As in the introduction, let L_{μ,ν} : QF → C^{2} be the map q ↦ (λ_{μ}(q), λ_{ν}(q)).

Theorem 2 Let (μ, ν) be measured laminations on T_{1} with i(μ, ν) > 0. Then the set P_{μ,ν} ⊂ QF on which [pl^{+}] = [μ], [pl^{−}] = [ν] is a non-empty connected non-singular component of the R^{2}-locus in QF − F of the function L_{μ,ν}. The restriction of L_{μ,ν} to P_{μ,ν} is a diffeomorphism to the open region under the graph of the function f_{μ,ν} in R^{+} × R^{+}.

Proof: By theorem 23, the map L_{μ,ν|_{Pμ,ν}} takes values in R^{+} × R^{+}. That L_{μ,ν} restricted to P_{μ,ν} is injective follows immediately from the injectivity of λ_{ν} on each pleating ray P_{μ,ν,c}. Hence, P_{μ,ν} is a non-singular R^{2}-locus in QF − F. The statement about the image of L_{μ,ν} follows from theorem 6. □

We remark that a similar proof shows that P_{μ,ν} and P_{ν,μ} are the unique connected components of the R-locus of L_{μ,ν} in QF − F whose closure in QF meets F in F_{μ,ν}.

We also remark that if in theorem 2 we replace μ, ν by μ′ = sμ, ν′ = tν, s, t, ∈ R^{+}, then P_{μ,ν} is unchanged and the length function L_{μ′,ν′} is simply a rescaling of L_{μ,ν}:

L_{μ′,ν′}(q) = (sλ_{μ}(q), tλ_{ν}(q)).

Our main result, theorem 1, that a group in QF is characterized by its pleating invariants, uniquely up to conjugation in P SL(2, C), is an immediate consequence of theorem 2.

9.3 Relation to Otal’s theorem

In [37] and later [5], Bonahon and Otal study spaces of various topological types of 3-manifolds with a hyperbolic structure H^{3} /G such that ∂C(G) is a pleated surface with (in our terminology) a fixed rational pleating lamination. Translated to our situation, this means the study of a rational pleating plane P_{γ,γ′} for fixed γ, γ′ ∈ S. Write pl^{+} = θδ_{γ}, pl^{−} = θ′δ_{γ′}, θ, θ′ ∈ R. A special case of their results shows that the map Θ(q) = (θ(q), θ′(q)) is a homeomorphism from P_{γ,γ′} to an open neighborhood of (0, 0) in (0, π) × (0, π).

Our methods prove that the map Θ is open and proper; we have thus far however, been unable to derive injectivity by our methods. (For the special case i(γ, γ′) = 1, see [39], theorem 3.6.)

Note however that if q_{n} ∈ P_{γ,γ′}, q_{n} → p ∈ F, then Θ(q_{n}) → (0, 0) so the whole critical line F_{γ,γ′} appears on the boundary of this Bonahon-Otal embedding as a single point.

10 BM-slices

In this section we study what happens when we fix the pleating invariants on one side of ∂C. The slices thus defined turn out to be the complex extensions of the earthquake paths into QF.

The space of marked conformal structures on T_{1} can be identified with the space F. For q ∈ QF, let w^{±}(q) denote the marked conformal structures of Ω^{±}/G(q). Bers used the embedding q ↦ (w^{+}, w^{−}) of QF into F × \overline{F} to find holomorphic coordinates for F by fixing the second factor w^{−} and proving that w^{+} varies over F; this is called the Bers embedding of F. (Recall that the orientation and hence the marking on Ω^{−}/G(q) is reversed; this is why in the second factor we write \overline{F}.) Maskit, on the other hand, fixed a curve γ on T_{1} and studied the family of groups on ∂QF for which λ_{γ} = 0 and the corresponding element V ∈ G is an accidental parabolic. These groups are known as cusps. The conformal structure w^{−} is then fixed and represents a family of thrice punctured spheres; Maskit proved that the first coordinate w^{+} varies so as to define an embedding of F into C. We studied the pleating invariants for this Maskit embedding of F in detail in [18]. McMullen [31], defines coordinates for Bers embeddings of QF that extend to Maskit and generalized Maskit embeddings on ∂QF. On the Maskit embeddings his coordinates agree with the pleating invariants of [18].

In terms of Thurston’s ending invariants [45], [35], both constructions correspond to holding the ending invariant of one side fixed and allowing the other to vary. It is thus natural to ask what happens when, instead of fixing an ending invariant, we fix the pleating invariants of one side.

Let μ ∈ ML, c ∈ R^{+} and set

BM_{μ,c}^{+} = {q ∈ P_{μ}^{+} : λ_{μ}(q) = c}.

On BM_{μ,c}^{+}, neither the conformal structure on Ω^{+}/G nor the flat structure on ∂C^{+}/G are fixed. They are, however, constrained by the condition λ_{μ}(q) = c. We define

J : BM_{μ,c}^{+} → (P ML − {[μ]}) × R^{+},

by

J(q) = ([pl^{−}(q)], \frac{l_{pl^{−}}(q)}{i(μ, pl^{−}(q))}).

Since [pl^{−}(q)] ≠ [pl^{+}(q)], i(μ, pl^{−}(q)) > 0. The map J is continuous by theorem 11. Since for fixed μ ∈ ML, the functions l_{ν} and i(μ, ν) scale in the same way as we vary ν in its projective class in P ML, the entry in the second coordinate of J depends only on [pl^{−}]; it can therefore be written in terms of our pleating invariants as λ_{ν}(q)/i(μ, ν) for any choice of ν ∈ [pl^{−}].

Set

X (μ, c) = \{([ν], s) ∈ (P ML − \{[μ]\}) × R^{+} : 0 < s < \frac{f_{μ,ν}(c)}{i(μ, ν)}\}.

Identifying P ML − {[μ]} with R as in section 2.2, we can think of X (μ, c) as the region in R×R^{+} under the graph of the function [ν] ↦ \frac{f_{μ,ν}(c)}{i(μ,ν)}. As discussed above, this function is well defined and by corollary 8, it is continuous.

As before, we let Q_{μ,c} denote the quakebend plane along μ that meets F along E_{μ,c}. Clearly Q_{μ,c} = Q_{μ}^{p} for all p ∈ E_{μ,c}.

Theorem 5 Let μ ∈ ML and let c > 0. Then the closures in QF of precisely two of the connected components of Q_{μ,c}∩(QF−F) meet F. These components are the slices BM_{μ,c}^{±}. The intersection of the closure of each slice with F is the earthquake path E_{μ,c}; furthermore each slice is simply connected and retracts onto E_{μ,c} and the map J : BM_{μ,c}^{±} → X (μ, c) is a homeomorphism.

Proof: Noting that for ν ∈ ML, the pleating ray P_{μ,ν,c} depends only on the projective class [ν] of ν, it is clear from the definitions that

BM_{μ,c}^{+} = ⋃_{[ν]∈P ML−{[μ]}} P_{μ,ν,c}.

Since for [ν] ∈ P ML − {[μ]}, the closure of the pleating ray P_{μ,ν,c} in QF contains the point p(μ, ν, c), the closure of BM_{μ,c}^{+} in QF contains E_{μ,c}. It follows easily from theorems 15 and 26 that BM_{μ,c}^{+} is open and closed in Q_{μ,c} ∩ (QF − F). By theorem 34 there are no other components of Q_{μ,c} whose closure meets F.

For [ν] ∈ P ML − {[μ]}, by lemma 39, λ_{ν}|_{P_{μ,ν,c}} is a homeomorphism to the interval (0, f_{μ,ν}(c)). This proves J is a homeomorphism onto B_{μ,c}^{+}. Clearly therefore, BM_{μ,c}^{+} is simply connected and retracts to E_{μ,c} along rays. □

In analogy with theorem 4 we have

Theorem 47 The rational pleating rays P_{μ,ν,c} are dense in BM_{μ,c}^{+}.

Remark 48 As discussed above, holding the ending invariant of one side fixed and letting the ending invariant of the other side vary over the full Teichmüller space F, we obtain the Bers and Maskit slices. By contrast, the set of flat structures F^{−}(q) for points q ∈ BM_{μ,c}^{+} cannot be the full image of F. In fact, on each ray P_{μ,ν,c}, the length λ_{ν} is bounded above by f_{μ,ν}(c). Since by a theorem of Sullivan, [9], lengths on ∂C^{−} and Ω^{−} are in bounded ratio, those points on the earthquake path E_{μ,c} in F at which λ_{ν} is very large will not occur as F^{−}(q) for points q ∈ BM_{μ,c}^{+}. See also [31] for related phenomena.

11 Rational pleating planes and computation

We can now easily prove theorem 3 of the introduction.

Theorem 3 Let δ_{γ}, δ_{γ′} be rational laminations represented by non-conjugate elements V, V′ ∈ G. Then P_{γ,γ′} and P_{γ′,γ} are the unique components of the R^{2}- locus of the function Tr V × Tr V′ in QF − F whose closures meet F in F_{γ,γ′}. On P_{γ,γ′} ∪ P_{γ′,γ} the function Tr V × Tr V′ is non-singular and the boundary of P_{γ,γ′} ∪ P_{γ′,γ} can be computed by solving Tr V = ±2 and Tr V′ = ±2 on this component.

Proof: If V, V′ ∈ G represent γ, γ′ in S, then the R^{+}-loci in QF of Tr V, Tr V′ and λ_{γ}, λ_{γ′} agree. As a consequence of theorem 2, P_{γ,γ′} can be uniquely identified as the component of the R^{+} × R^{+}-locus of Tr V × Tr V′ which meets F in the critical line F_{γ,γ′}. □

As a consequence of this theorem, given any embedding QF → C^{2}, we can compute the position of P_{γ,γ′} and its boundary exactly, provided we can express Tr V and Tr W as holomorphic functions of the parameters and identify the critical line.

For the complex Fenchel Nielsen embedding this works as follows. We first note:

Proposition 49 Let (λ_{V} , τ_{V,W}) be complex Fenchel Nielsen coordinates for QF relative to a marked pair of generators (V, W). Suppose γ′ ∈ S with corresponding element V′ ∈ G. Then for fixed λ_{V}, the trace Tr V′ = ±2 cosh λ_{γ′} is a polynomial in cosh τ_{V,W} /2 and sinh τ_{V,W} /2.

Proof: From equation (3) we have

cosh \frac{λ_{W}}{2} = cosh \frac{τ_{V,W}}{2} /tanh \frac{λ_{V}}{2}

and

cosh \frac{λ_{V W^{±1}}}{2} = cosh \frac{τ_{V,W} ± λ_{V}}{2} /tanh \frac{λ_{V}}{2} .

Expanding cosh \frac{τ_{V,W} ±λ_{V}}{2} , the result follows in the special cases V′ = W and V′ = V W^{±1}. The results for general V′ follow from the recursive scheme in [47], see also [18], which allows us to express Tr V′ as a polynomial (with integer coefficients) in Tr V, Tr W and either Tr V W or Tr V W^{−1}. □

To find the critical line F_{γ,γ′} we proceed as follows. Fix c > 0 and consider the function Tr V′ = Tr V′(λ_{V} , τ_{V,W}). Along the earthquake path E_{γ,c}, t = τ_{V,W} is real and varies over all of R; λ_{V} is fixed and equal to c. By Kerckhoff’s theorem, the function λ_{γ′} has a unique critical point p = p(γ, γ′, c) ∈ F_{γ,γ′} along E_{γ,c}; clearly the same is true of the trace function Tr V′. Using proposition 49, the position of this point can be computed as a function of t. Moreover there are exactly two branches σ^{±} of the R-locus of Tr V′ in QF − F whose closures meet F at p.

By theorem 2, the pleating plane P_{γ,γ′} is the union of the pleating rays P_{γ,γ′,c}, c ∈ R^{+}. By theorem 6, the pleating ray P_{γ,γ′,c} is one of the two branches σ^{±}, each of which maps homeomorphically to (0, 2 cosh f_{γ,γ′}(c)/2) under Tr V′. Analytically continue Tr V′ along σ^{±}. Again by theorem 6, these branches are non-singular R-loci and remain in QF until they reach points τ^{∗} such that Tr V′(τ^{∗}) = ±2. The groups corresponding to such τ^{∗} are cusp groups on ∂QF for which γ′ is pinched and V′ is an accidental parabolic.

Drawing these rays for various c’s, we get a picture of the pleating planes P_{γ,γ′} and P_{γ′,γ}. Allowing γ′ to vary with c fixed gives us the slices BM_{γ,c}^{±}. By theorems 4 and 47, we can build up an arbitrarily accurate picture of QF. Pictures of various slices drawn this way have been obtained in [47] and [38].

In [25], similar ideas are used to draw a picture of the Earle slice of QF. This slice is an embedding of the Teichmüller space of T_{1} into QF consisting of groups for which the structures on Ω^{+} and Ω^{−} are related by a conformal involution which induces the rhombus symmetry on π_{1}(T_{1}).

11.1 Examples

We give two examples in which it is especially easy to compute the pleating plane.

Example 1. Take γ, γ′ to be generators of π_{1}(T_{1}), represented by the marked pair V, W ∈ G. By equation (2), cosh(\frac{λ_{W}}{2} ) = cosh(\frac{τ_{V,W}}{2} )/ tanh(\frac{λ_{V}}{2} ), so that on the earthquake path E_{γ,c}, cosh(\frac{λ_{W}}{2} ) = cosh(\frac{t}{2})/ tanh(c), t ∈ R. This function clearly has a unique critical point at the rectangular torus t = 0. Therefore the critical line F_{γ,γ′} is defined by the equation sinh(\frac{λ_{V}}{2} ) sinh(\frac{λ_{W}}{2} ) = 1 and the range of λ_{V} × λ_{W} is the region

\{(c, t) ∈ R^{+} × R^{+} : 0 < c < 2 sinh^{−1}(\frac{1}{sinh(\frac{c}{2})})\}.

Notice that under the rectangular symmetry (V, W) → (V, W^{−1}) the group is fixed but the marking is changed; clearly Ω^{+}(G(V, W)) = Ω^{−}(G(V, W^{−1})). Thus P_{γ,γ′} maps bijectively to P_{γ′,γ} while Fγ, γ′ = Fγ′, γ is fixed. This implies cosh \frac{λ_{V W}}{2} = cosh \frac{λ_{V W^{−1}}}{2} on Fγ, γ′. Solving this equation in F gives another way of finding the equation of the critical line.

Example 2. Let (V, W) be a marked pair of generators for G and let γ, γ′ be the curves represented by V W and V W^{−1}. Since G is a punctured torus group, the condition that the commutator [V, W] be parabolic is expressed by the well known Markov equation

Tr^{2} V + Tr^{2} W + Tr^{2} V W = Tr V Tr W Tr V W . (6)

Writing x = Tr V, y = Tr W, we can solve for z = Tr V W and z′ = Tr V W^{−1}. On the pleating plane P_{γ,γ′}, both z and z′ are real so that xy and x^{2} + y^{2} are real. It follows that x = ȳ. Further, on P_{γ,γ′}, x, y ∈ R if and only if G ∈ F. Thus in the real (z, z′) plane, the critical line

F_{γ,γ′} has equation zz′ = 2(z+z′); in other words the hyperbola (z−2)(z′−2) = 4. Rewriting in terms of the lengths 2 cosh^{−1} \frac{z}{2}, 2 cosh^{−1} \frac{z′}{2} we find the region T_{γ,γ′} is of the shape claimed.

We note that in this case, the critical line F_{γ,γ′} is the fixed line of the rhombic symmetry (A, B) → (B, A) in F, giving an alternative proof that on this line, λ_{A} = λ_{B}. It is also interesting to note in this example that the Earle slice studied in [25] is the holomorphic extension of the critical line F_{γ,γ′} into QF.

12 Appendix

12.1 The convergence lemma

For the proof of the convergence lemma 1, we need to recall some general facts about laminations. Let Σ be a hyperbolic surface and let α be a geodesic lamination on Σ. We call a set R ⊂ Σ a flow box for α if:

(1) R is a closed hyperbolic rectangle embedded in Σ, with one pair of opposite sides called “horizontal” and the other pair “ vertical”.

(2) The horizontal sides T, T′ of R are either disjoint from α or transversal to α. If a leaf γ of α intersects R then it intersects both T and T′ .

(3) The vertical sides of R are disjoint from α.

Label the sides of R in counterclockwise order 1, 2, 3, 4 so that 1, 3 are the horizontal sides and 2, 4 are the vertical ones. Suppose that β ∈ ML is any measured lamination on Σ. The underlying lamination |β| intersects R in a family of pairwise disjoint arcs. If such an arc joins a vertical to a horizontal side, we call it a corner arc; if it joins the two horizontal sides we call it a vertical arc and otherwise it is a horizontal arc. For i, j ∈ {1, …, 4}, let β(i, j) = β(j, i) denote the total transverse measure of the arcs joining side i to side j. Clearly, α(1, 3) = α(3, 1) = α(T) = α(T′), the transverse measure of the transversal T, while α(i, j) = 0 otherwise.

The following simple lemma applies to any hyperbolic surface Σ.

Lemma 11.1 Let ν_{0} ∈ ML and let R be a flow box for |ν_{0}|. Suppose ν_{0}(T) ≠ 0. Then for ν ∈ ML sufficiently near ν_{0}, the lamination |ν| has a vertical arc.

Proof: Note that because |ν| consists of pairwise disjoint simple geodesics, it does not have both horizontal and vertical arcs. Let V, V′ denote the vertical sides. Since ν_{0}(V) = ν_{0}(V′) = 0, both ν(V) and ν(V′) can be assumed arbitrarily small by taking ν sufficiently close to ν_{0} in ML. We can write ν(V) = ν(4, 1) + ν(4, 2) + ν(4, 3) and ν(V′) = ν(2, 1) + ν(2, 4) + ν(2, 3). All the terms on the right in these relations are non-negative so each is arbitrarily small.

If we assume |ν| has no vertical arc we have ν(T) = ν(4, 1) + ν(2, 1), ν(T′) = ν(3, 2) + ν(3, 4) and by the above we deduce that both are arbitrarily small. But this is a contradiction because ν(T) and ν(T′) are both near ν_{0}(T) which is a definite positive value. □

Now we need some facts specific to laminations on a punctured torus (see [45], 9.5.2). Let α ∈ S and cut T_{1} along α to obtain a punctured annulus A with boundary curves α_{1} and α_{2}. The leaves of any measured lamination ν, |ν| ≠ α intersect A in a union of arcs that either join α_{1} to α_{2} or join one of the boundary components to itself. It is easy to show, (see [45]), that the set of arcs joining a component α_{i} to itself has zero transverse measure. In particular, by minimality any transversal to any leaf of |ν| carries non-zero measure, so that all arcs of |ν| in A join α_{1} to α_{2}.

We also recall that on T_{1}, if ν ∉ ML_{Q}, the complement of |ν| is a punctured bigon B, and also that there is a horocyclic neighborhood of definite size about the cusp disjoint from the support of any measured lamination.

Now we can prove the convergence lemma 1.

Lemma 1 Suppose that ν_{0} ∈ ML−ML_{Q}, and that ν and ν_{0} are close in ML. Then |ν| and |ν_{0}| are close in the Hausdorff topology on GL.

Proof: First we show that given a long arc in |ν_{0}| there exists a long nearby arc in |ν|. Let L, ϵ > 0 be given. Since ν_{0} ∈ ML − ML_{Q}, all leaves have infinite length. Thus, given x ∈ |ν_{0}|, by choosing sufficiently short transversals we can find a flow box for which the leaf of |ν_{0}| through x is a vertical arc, the segments of length L on either side of x are contained in R, and the horizontal sides of R have length less than ϵ. We call a flow box of this kind, a good ϵ, L-flow box for x. Now standard hyperbolic geometry estimates show, that if two geodesics are a bounded distance apart over a long distance t, then in fact they are close to order e^{−t} along a large fraction of their length. Thus any vertical arc in a good ϵ, L-flow box is certainly close to leaves of |ν_{0}| over distance at least 2L. Clearly, |ν_{0}| can be covered by a finite number of flow boxes of this kind.

Now suppose we are given a long arc λ of a leaf of |ν_{0}|. Let x be the midpoint of λ and let R be a good ϵ, L-flow box for x. By lemma 12.1, we deduce that if ν ∈ ML is near ν_{0}, then ν has a vertical arc in R so that by the above, |ν| has long arc of a leaf near λ as required.

Next we claim conversely, that given a long arc in |ν| there exists a long nearby arc in |ν_{0}|. For a lamination λ, let T_{1}(λ) denote the the set of unit tangent vectors to leaves pointing along leaves of λ. Since there is a horocyclic neighborhood of definite size about the cusp disjoint from the support of any measured lamination on T_{1}, the set ∪_{λ∈GL}T_{1}(λ) is a compact subset of the unit tangent bundle T_{1}(T_{1}). Clearly, laminations λ and λ′ are close in the Hausdorff topology on closed subsets of GL if and only if T_{1}(λ) and T_{1}(λ′) are close in the Hausdorff topology on closed subsets of T_{1}(T_{1}).

If our claim is false, then there is a sequence of points v̄_{n} ∈ T_{1}(|ν_{n}|), ν_{n} ∈ ML_{Q} with ν_{n} → ν_{0} in ML, for which there are no nearby points of T_{1}(|ν_{0}|). A geodesic β through a limit point of the vectors v̄_{n} will be a limit of leaves of |ν_{n}|, but will not be a leaf of |ν_{0}|.

If β ∩ |ν_{0}| ≠ ∅, we obtain a contradiction. For if x ∈ β ∩ |ν_{0}|, the tangent directions to β and |ν_{0}| at x are distinct. Therefore we can find a good |ν_{0}| flow box R for x, such that the arc of β through x is only close to the leaf of ν_{0} through x for a short distance and thus cannot be either a vertical or a corner arc in R. But then all laminations |ν| with leaves close to β also contain arcs which must intersect R in horizontal arcs, contradicting lemma 12.1.

To complete the proof we must show β ∩ |ν_{0}| ≠ ∅. If not, then β is contained in the complement of |ν_{0}| in T_{1}. Since ν_{0} ∉ ML_{Q}, the complement of |ν_{0}| is a punctured bigon B. If β enters B through one vertex and leaves through the other it is homotopic to, and therefore coincides with, a leaf of |ν_{0}|; thus β must come in from one vertex of the bigon, go around the puncture and return back to the same vertex. Let α be a simple closed curve that intersects β and as above, cut T_{1} along α to obtain a punctured annulus A with two boundary curves α_{1}, α_{2}. Since β goes around the puncture, it crosses one of the α_{i} and returns through the same side of α_{i} (see the figure in [45], 9.5.2). It follows that any closed simple geodesic sufficiently close in the Hausdorff topology to β would also have an arc entering and leaving A across the same α_{i}. But any arc of a simple closed geodesic carries a non-zero transverse measure, and by the fact stated above, must join α_{1} to α_{2}. Hence β ∩ |ν_{0}| ≠ ∅. □

12.2 Proof of proposition 28.

Before beginning the proof, we need to review the definitions of the bending measure and intrinsic metric for paths on ∂C as given in [19]. We suppose that q ∈ QF, and that as usual ∂C = ∂C(q) is the convex hull boundary of H^{3}/G(q). We shall only indicate the dependence on q when needed in the proof. In fact, we shall only need to apply what follows to the component ∂C^{+}.

A support plane for ∂C at a point x ∈ ∂C is a hyperbolic plane P containing x such that C is contained entirely in one of the two half spaces cut out by P. The bending angle between two intersecting support planes P_{1}, P_{2} at points x_{1}, x_{2} ∈ ∂C is the absolute value of the angle θ(P_{1}, P_{2}) between their outward normals from ∂C.

Let Π(x) denote the set of oriented support planes at x ∈ ∂C and let

Z = {(x, P (x)) : x ∈ ∂C, P (x) ∈ Π(x)},

with topology induced from G = H^{3} ×G_{2}(H^{3}), where G_{2}(H^{3}) is the Grassmanian of 2-planes in H^{3}. Let Z^{+} be the obvious restriction of Z to ∂C^{+} and call the it approximating set for ∂C^{+}.

To define the bending measure and intrinsic metric, it suffices to define the measure and length of any path ω̄ on ∂C. Any such path lifts to a path ω: [0, 1] → Z as follows. Suppose x ∈ ω̄. Either Π(x) consists of a unique point, in which case there is nothing to do, or we add to the path an arc in which the first coordinate x is fixed but the second moves continuously on the line in G from the left to the right extreme support planes at x.

A polygonal approximation to ω is a sequence

P = {ω(t_{i}) = (x_{i}, P_{i}) ∈ Z}; 0 = t_{0} < t_{1} < … < t_{n} = 1,

such that P_{i} ∩ P_{i+1} ≠ ∅, i = 0, …, n − 1.

Let θ_{i} = θ(P_{i−1}, P_{i}) be the bending angle between P_{i−1} and P_{i}, i = 1, …, n and let d_{i} be the hyperbolic length of the shortest path from x_{i−1} to x_{i} in the planes P_{i−1} ∪ P_{i}.

The intrinsic metric on ∂C is given by

l(ω) = inf_{P} ∑_{i=1}^{n} d_{i} (7)

and the bending measure β on ∂C by

β(ω) = inf_{P} ∑_{i=1}^{n} θ_{i} (8)

where P runs over all polygonal approximations to ω.

In order to prove proposition 28, we shall also make similar polygonal approximations to the pleated surface P_{γ}(q). We shall prove the proposition by showing that polygonal approximations in Z^{+} = Z^{+}(q) to the convex hull boundary ∂C^{+} can be replaced by polygonal approximations to the pleated surface P_{γ}(q), and that the above approximating sums are simultaneously good approximations to the intrinsic metric of the flat structure ϕ_{γ}^{+}(q) and the transverse measure b_{γ}(q). Thus we also need to discuss polygonal approximations for P_{γ}(q).

The surface P_{γ}(q) is made up of planar pieces, precisely two of which meet along each bending line α (which projects to γ on T_{1}. Call a plane P a pseudosupport plane to P_{γ}(q) if either it is one of these planar pieces, or if it meets P_{γ}(q) along α and lies in the half space cut out by the planar pieces of P_{γ}(q) through α. The pseudo-support planes of P_{γ}(q) inherit natural orientations from the pleated surface map under which P_{γ}(q) is an immersed image of the hyperbolic disk D in H^{3}.

Let Π̃(x) denote the set of oriented pseudo-support planes at x ∈ P_{γ}(q) and let

W = W (q) = {(x, P (x))|x ∈ P_{γ}(q), P (x) ∈ Π̃(x)},

with topology induced from G as before. We define polygonal approximations in W (q) in the obvious way, and call W (q) the approximating set for P_{γ}(q).

We claim that the flat metric ϕ_{γ}(q) and the measure b_{γ} on P_{γ}(q) are defined by sums similar to those in (7) and (8), where the infimum is taken now over polygonal approximations in W (q).

Let ω be a path in W and let {(x_{i}, Q_{i})} be such a W -polygonal approximation. As in the proof of Proposition 4.8 of [19], we consider the segment of path ω_{i} in W (q) between x_{i−1} and x_{i}, and we work in a hyperbolic plane H through x_{i−1} and x_{i}, such that the shortest path σ from x_{i−1} to x_{i} in the planes Q_{i−1} ∪ Q_{i} is contained in the intersections of these planes with H. Let the segments of σ in Q_{i−1} and Q_{i} have lengths a_{1} and a_{2} respectively, so that a_{1} + a_{2} is an upper bound for the contribution to the sum giving the length of ω_{i}. Notice that even though we do not know that P_{γ}(q) bounds a convex half space, it follows easily from Gauss-Bonnet that ω_{i} does not intersect σ. Thus it is easy to check that inserting an extra pair (x, Q) ∈ W between x_{i−1} and x_{i}, the approximating sum for the length of ω_{i} decreases. Since by assumption [γ] ∈ ML_{Q}, there are in fact sufficiently fine polygonal approximations for which the sum in (7) actually equals the intrinsic metric on P_{γ}(q). A similar argument, on the lines of that in Proposition 4.8 of [19], shows that the sums (8) decrease on inserting extra support planes and that there are sufficiently fine sums which actually equal the measure b_{γ}.

We are now ready to prove proposition 28.

Proposition 28 Given ν_{0} ∈ ML − ML_{Q}, and q_{0} ∈ P_{ν_{0}}^{+} ∪F, let F^{+}(q_{0}) ∈ F and pl^{+}(q_{0}) be the flat structure and bending measure of ∂C^{+}(q_{0}) respectively. (If q_{0} ∈ F, then pl^{+}(q_{0}) = 0 and F^{+}(q_{0}) is the Fuchsian group representing q_{0}.) Then, given neighborhoods V of F^{+}(q_{0}) in F and W of pl^{+}(q_{0}) in ML, there exist neighborhoods U of q_{0} in QF and X of [ν_{0}] in P ML such that if q ∈ U, [δ_{γ}] ∈ X ∩ P ML_{Q} and λ_{γ}(q) ∈ R^{+}, then the flat structure ϕ_{γ}(q) of P_{γ}(q) is in V and the transverse measure b_{γ}(q) is in W.

Proof: Let ν_{0}, q_{0} be as in the statement of the proposition. Suppose that for some q near q_{0} and [δ_{γ}] near [ν_{0}], we have λ_{γ}(q) ∈ R^{+}. Let P_{γ}(q) be the associated pleated surface with approximating set W (q) ⊂ G as above. Let Z^{+}(q_{0}) and Z^{+}(q) be the approximating sets for ∂C^{+}(q_{0}), ∂C^{+}(q) respectively.

We claim that for every (x, P (x)) ∈ Z^{+}(q_{0}) and q ∈ QF near q_{0}, there is a nearby pair (y, P (y)) ∈ W (q), and conversely. This will follow immediately if we can show that, for every geodesic in |pl^{+}(q)|, there is a geodesic in the bending locus of P_{γ}(q) with nearby endpoints in H^{3}, and vice versa. Now, the crucial condition λ_{γ}(q) ∈ R^{+} implies that the bending locus of P_{γ}(q) is exactly γ = γ(q). Thus, applying lemma 1 to the laminations ν_{0} and kδ_{γ} for a suitable choice of k > 0 on the surface ∂C^{+}(q_{0}), we see that |ν_{0}(q_{0})| and γ(q_{0}) are close in the Hausdorff topology on closed subsets of ∂C^{+}(q_{0}). Lifting to H^{3}, this means that the endpoints x_{0}, x′_{0} of any lift of a leaf of |ν_{0}(q_{0})| are close to the endpoints x, x′ of a lift of γ(q_{0}) and vice versa since the geodesic representative of γ on ∂C^{+} has the same endpoints as the geodesic γ in H^{3}. It follows that the H^{3} geodesics with x_{0}, x′_{0} and x, x′ also have long close arcs. Finally, moving to a nearby point q in QF, the endpoints of geodesics which project to the leaves of |ν_{0}(q)| are close to the endpoints of geodesics which project to |ν_{0}(q_{0})|, and similarly for endpoints of geodesics which project to γ(q) and γ(q_{0}). The claim follows.

We now consider the key estimates which were the basis of the continuity results proved in [19]. Call a polygonal approximation an (α, s)-approximation if

max_{1≤i≤n} θ(P_{i−1}, P_{i}) < α

and

max d_{ω}(x_{i−1}, x_{i}) < s

where d_{ω} is distance along ω measured in the intrinsic metric on ∂C. We have

Proposition [19], Prop. 4.8 There is a universal constant K, and a function s(α) with values in (0, 1), such that if P is an (α, s(α))-approximation to a path ω in Z, where α < π/2, then

| ∑_{P} d_{i} − l(ω)| < Kαl(ω)

and

| ∑_{P} θ_{i} − β(ω)| < Kαl(ω).

To complete the present proof, it suffices to check that similar estimates hold if polygonal approximations in Z^{+}(q_{0}) are replaced by approximations in W (q). The estimates work in exactly the same way; the only point to note is that we need the same local convexity property implied by Gauss Bonnet as above. □

12.3 Proof of proposition 35

Proposition 35 Suppose μ ∈ ML, q ∈ P_{μ} ∪F and consider the quakebend plane Q_{μ}^{q} along μ based at q with parameter τ_{μ}. Given K > 0, there exist B_{2} > B_{1} > 0 such that if |ℜτ| < K and B_{2} > |ℑτ| > B_{1}, then Q_{μ}^{q}(τ) ∉ P_{μ}.

Proof: Our proof will show that if τ_{μ} is inside the range described the proposition, then the pleated surface obtained by bending by τ along μ cannot be embedded and thus that Q_{μ}^{q}(τ_{μ}) ∉ P_{μ}. The group Q_{μ}^{q}(τ_{μ}) may or may not be in QF.

We use the definitions of support planes and bending angles from the proof of proposition 12.2. From the definition, the bending angle between two intersecting support planes P_{1}, P_{2} to ∂C at points x_{1}, x_{2} is an upper bound for the bending measure of a transversal to |μ| joining x_{1}, x_{2} which lies between the “roof” formed by P_{1} and P_{2} and the H^{3} geodesic from x_{1} to x_{2}.

We make the following claims.

(1) There exists ϵ > 0 such that if x_{1}, x_{2}, x_{3} ∈ ∂C lie in a ball of radius ϵ in B^{3}, and if P_{1}, P_{2}, P_{3} are support planes to ∂C at x_{1}, x_{2}, x_{3} respectively, then either P_{1} ∩ P_{3} ≠ ∅, or both P_{1} ∩ P_{2} ≠ ∅ and P_{2} ∩ P_{3} ≠ ∅.

(2) Given ϵ > 0, μ ∈ ML, μ ≠ 0, and a compact subset V ⊂ F, there is a constant a > 0 such that if ϕ ∈ V, then there is a transversal κ to |μ| with hyperbolic length l(κ) < ϵ in the structure ϕ and transverse measure μ(κ) > a.

Proof of claim 1. A support plane P to ∂C meets Ĉ in a circle which contains points of the limit set Λ and which bounds a disk D(P) containing no points of Λ. Therefore if P_{1} ∩ P_{3} = ∅, the discs D(P_{1}) and D(P_{3}) are disjoint. To prove the claim amounts to showing that in this case, both D(P_{1})∩D(P_{2}) and D(P_{3})∩D(P_{2}) are non-empty. Without loss of generality, we may suppose that x_{1}, x_{2}, x_{3} are within hyperbolic distance ϵ of the origin O in B^{3} so that the planes P_{i} are close to equatorial planes through O. The result is then obvious.

Proof of claim 2. Choose γ ∈ S with i(γ, μ) > 0. There are constants c_{1}, c_{2}, d_{1}, d_{2} such that c_{1} < μ(γ) < c_{2} and d_{1} < l(γ) < d_{2} for ϕ ∈ V. Subdividing γ into N segments with d_{2}/N < ϵ, the result is clear with a = c_{1}/N.

Now, working in the quakebend plane Q_{μ}^{q}, with parameter τ = τ_{μ}, consider the set of groups for which |ℜτ| < K. The corresponding flat structures F^{+}(τ) are independent of ℑτ and thus lie in a compact set V ⊂ F. Choose a transversal κ as in claim (2). Let x_{1}, x_{3} be its initial and final points and x_{2} its midpoint, and let P_{i} be a support plane at x_{i}. Using claim 1, either P_{1}, P_{3}, or both pairs P_{1}, P_{2} and P_{2}, P_{3}, intersect. Thus at least one of the segments (x_{1}, x_{3}), (x_{1}, x_{2}) or (x_{2}, x_{3}) of κ, for definiteness say the segment κ_{1} joining (x_{1}, x_{2}), has μ(κ_{1}) > a/2.

Consider the point in Q_{μ}^{q} with parameter τ. The bending measure pl^{+}(τ)(κ_{1}) of κ_{1} is k + ℑτμ(κ_{1}), where k = pl^{+}(q)(κ_{1}) is the bending measure of κ_{1} at the base point q. The bending angle between P_{1}, P_{2} is not in the range (π, 2π). From the above, a/2 < μ(κ_{1}) < c_{2}. Putting this together gives an upper bound for pl^{+}(τ)(κ_{1}), and we obtain the required bound on |ℑτ|. □

References

[1] L. Bers. Uniformization, moduli, and kleinian groups. Bull. London. Math. Soc., 4, (1972) 257–300.

[2] J. Brock. Continuity of Thurston’s length function. Geom. and Funct. Analysis,10(4), (2000) 741–797.

[3] F. Bonahon. Bouts des variétés de dimension 3. Ann. Math., 124(1), (1986) 71–158.

[4] F. Bonahon. Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Ann. Fac. Sci. de Toulouse., 5(2), (1996) 233– 297.

[5] F. Bonahon and J. P. Otal.. Laminations mesurées de plissage des variétés hyperboliques de dimension 3. Preprint, 2001.

[6] R. D. Canary Algebraic convergence of Schottky groups. Trans. A.M.S., 337(1), (1993) 233–297.

[7] R. D. Canary, D. B. A. Epstein and P. Green. Notes on notes of Thurston. In D. B. A. Epstein, editor, Analytical and Geometric Aspects of Hyperbolic Space, LMS Lecture Notes 111, 3–92. Cambridge University Press, 1987.

[8] A. J. Casson and S. A. Bleiler. Automorphisms of surfaces after Nielsen and Thurston. LMS Lecture Notes 9. Cambridge University Press, 1988.

[9] D. B. A. Epstein and A. Marden. Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. In D. B. A. Epstein, editor, Analytical and Geometric Aspects of Hyperbolic Space, LMS Lecture Notes 111, 112–253. Cambridge University Press, 1987.

[10] A. Fahti, P. Laudenbach, and V. Poénaru. Travaux de Thurston sur les surfaces, Astérisque 66-67. Société Mathématique de France, 1979.

[11] W. Fenchel and J. Nielsen. On discontinuous groups of motions. Unpublished manuscript.

[12] F. Gardiner and H. Masur. Extremal length geometry for Teichmüller space. Complex Variables Theory Appl., 16(2-3), (1991) 209–237.

[13] T. Jørgensen. Once punctured tori. Previously unpublished manuscript. In Y. Komori, V. Markovic and C. Series, editors, Kleinian Groups and Hyperbolic 3-Manifolds, L.M.S. Lecture notes, to appear.

[14] T. Jørgensen and A. Marden. Algebraic and geometric convergence of Kleinian groups. Math. Scand., 66, (1990) 47–72.

[15] M. Kapovich. Hyperbolic manifolds and discrete groups. Progress in Mathematics Vol. 183, Birkhäuser, 2000.

[16] L. Keen. Teichmueller spaces of punctured tori: I,II. Complex Variables, 2, (1982) 199–211, 213–225.

[17] L. Keen, B. Maskit, and C. Series. Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets. J. Reine und angewandte Math., 436, (1993) 209–219.

[18] L. Keen and C. Series. Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori. Topology, 32(4), (1993) 719–749.

[19] L. Keen and C. Series. Continuity of convex hull boundaries. Pacific J. Math., 168(1), (1995) 183–206.

[20] L. Keen and C. Series. How to bend pairs of punctured tori. In J. Dodziuk and L. Keen, editors, Lipa’s Legacy, Proceedings of the Bers Colloquium 1995, Contemp. Math. 211, 359–388. AMS, 1997.

[21] S. Kerckhoff. The Nielsen realization problem. Ann. of Math, 117(2), (1983) 235–265.

[22] S. Kerckhoff. Earthquakes are analytic. Comment. Mat. Helv., 60, (1985) 17–30.

[23] S. Kerckhoff. Lines of Minima in Teichmüller space. Duke Math J., 65, (1992) 187–213.

[24] S. Kerckhoff and W .Thurston. Non-continuity of the action of the modular group at Ber’s boundary of Teichmüller space. Invent. Math., 100, (1990) 25-47.

[25] Y. Komori and C. Series. Pleating coordinates for the Earle embedding. Ann. Fac. Sci. de Toulouse, X, (2001) 69-105.

[26] C. Kourouniotis. Bending in the space of quasi-fuchsian structures. Glasgow Math. J., 33(1), (1991) 41–49.

[27] C. Kourouniotis. Complex length coordinates for quasi-fuchsian groups. Mathematika, 41(1), (1994) 173–188.

[28] A. Marden. The geometry of finitely generated Kleinian groups. Ann. Math., 99, (1974) 607-639.

[29] B. Maskit. Moduli of marked Riemann surfaces. Bull. A.M.S., 80, (1974) 773–777.

[30] C. McMullen. Cusps are dense. Ann. of Math, 133(1), (1991) 773– 777.

[31] C. McMullen. Complex earthquakes and Teichmüller theory. J. Amer. Math. Soc., 11, (1998) 283–320.

[32] C. McMullen. Renormalization and 3-manifolds which fiber over the circle. Annals of Math. Studies 142. Princeton University Press, 1996.

[33] R. Mañe, P. Sad and D. Sullivan. On the dynamics of rational maps. Ann. Sci. Ec. Norm. Sup., 16, (1983) 193–217.

[34] Y. Minsky. Harmonic maps into hyperbolic 3-manifolds. Trans. A.M.S., 332, (1992) 539–558.

[35] Y. Minsky. The classification of punctured-torus groups. Ann. of Math., 149, (1999) 559–626.

[36] K. Ohshika. Divergent sequences of Kleinian groups. In I. Rivin, C. Rourke, C. Series, eds, Analytical and Geometric Aspects of Hyperbolic Space, Geometry and Topology Monographs vol.1, 419-450. Int. Press, 1999.

[37] J. P. Otal. Sur le coeur convexe d’une variété hyperbolique de dimension 3. Unpublished preprint.

[38] J.R. Parker and J. Parkkonen. Horocyclic and pleating coordinates for punctured tori. In I. Rivin, C. Rourke, C. Series, eds, Analytical and Geometric Aspects of Hyperbolic Space, Geometry and Topology Monographs vol.1, 451-478. Int. Press, 1999.

[39] J. R. Parker and C. Series. Bending formulae for convex hull boundaries. J. Anal. Math., 67, (1995) 165–198.

[40] R. C. Penner with J. Harer. Combinatorics of Train Tracks. Annals of Math. Studies 125. Princeton University Press, 1992.

[41] R. M. Porter. Computation of a boundary point of Teichmüller space. Bol. Soc. Math. Mexicana, 24(1), (1979) 15–26.

[42] S. P. Tan. Complex Fenchel-Nielsen coordinates for quasifuchsian structures. Int’l. J. Math, 5(2), (1994) 239–251.

[43] W. Thurston. Hyperbolic structures on 3-manifolds I: deformation of acylindrical manifolds. Ann. of Math., 124, (1986) 203–246.

[44] W. Thurston. Hyperbolic structures on 3-manifolds II. eprint at front.math.ucdavis.edu/search/author:Thurston+category:GT.

[45] W. Thurston. Geometry and Topology of Three-Manifolds. Princeton lecture notes, 1979.

[46] S. Wolpert. The Fenchel-Nielsen deformation. Ann. Math., 115(3), (1982) 501–528.

[47] D. J. Wright. The shape of the boundary of Maskit’s embedding of the Teichmüller space of once punctured tori. Unpublished preprint.