THE CLASSIFICATION OF PUNCTURED-TORUS GROUPS

YAIR N. MINSKY

Abstract. Thurston’s ending lamination conjecture proposes that a finitelygenerated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus.

As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers’ conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate.

Contents

1. The ending lamination conjecture and its consequences 1 2. The Farey triangulation and the torus 6 3. Geometric tameness and end invariants 11 4. The pivot theorem 13 5. Simplicial hyperbolic surfaces 16 6. The structure of Margulis tubes 19 7. Geometry of Farey neighbors 22 8. Connectivity and a-priori bounds 25 9. Controlled surfaces and building blocks 26 10. Proof of the pivot theorem 33 11. Model manifolds and geometric limits 36 12. Proofs of the main theorems 47 References 52

1. The ending lamination conjecture and its consequences

The general classification problem for discrete groups of Möbius transformations remains tantalizingly open, although a conjectural picture of the solution has been in place since the late 70’s, and is roughly as follows. In the representation space for a given group G into PSL_{2}(C) the discrete, faithful elements are expected (barring

Date: May 1, 1998. To appear in Annals of Math.

This work was partially supported by an NSF postdoctoral fellowship and a fellowship from the Alfred P. Sloan Foundation.

trivial cases) to comprise the closure of an open set of structurally stable representations. In a component of the structurally stable set all representations are quasi-conformally conjugate and hence parametrized by a Teichmüller space, or a quotient of one. On the boundary of this set one obtains geometrically infinite groups and groups with new parabolics, and these are expected to be parametrized by what remains of the Teichmüller parameter, together with a combinatorial invariant known as an ending lamination (see Abikoff [1] for an overview).

In this paper we verify this conjectural picture for punctured-torus groups, which are the simplest of all classes of Kleinian groups with a non-trivial deformation theory. The primary component of the solution is the proof of Thurston’s “ending lamination conjecture” in this case (Theorem A).

A punctured-torus group is a free, discrete two-generator group Γ of (orientation preserving) Möbius transformations with the added condition that the commutator of the generators is parabolic. We should think of Γ as the image of a representation ρ : π_{1}(S) → PSL_{2}(C), where S is a once-punctured torus (to keep the representation in mind we often call this a marked group). The commutator condition means that the loop surrounding the puncture determines a cusp of the three-manifold H^{3}/Γ, and in general a representation of a surface group taking cusps to cusps in this way is called type-preserving.

To such a representation one may associate an ordered pair of end invariants (ν_{−}, ν_{+}), lying in (\overline{D} × \overline{D}) \ Δ, where \overline{D} is the closed unit disk whose interior D is identified with the Teichmüller space of S, and whose boundary S^{1} is identified with the space of measured laminations on S. Δ is the diagonal of S^{1} ×S^{1}. We also identify S^{1} with R̂ = R ∪ {∞} by stereographic projection, and let Q̂ = Q ∪ {∞}. When both ν_{±} lie in D ∪ Q̂ the group is geometrically finite, and this case has been well-understood through work of Ahlfors, Bers, Kra, Marden, Maskit and others. When either invariant lies in R \ Q the group is geometrically infinite (and the invariant is called a “lamination on S”), and the existence of the invariant in this case is due to Thurston and Bonahon. Thurston’s ending lamination conjecture states that these invariants suffice to determine the group up to isometry. See section 3 for more precise definitions, and [70, 71] for discussions of the conjecture for more general groups. In this paper we shall prove:

Theorem A. (Ending Lamination Theorem) A marked punctured-torus group ρ : π_{1}(S) → PSL_{2}(C) is determined by its end invariants (ν_{−}, ν_{+}), uniquely up to conjugacy in PSL_{2}(C).

In other words, the map ν : ρ ↦ (ν_{−}, ν_{+}) is injective. It can also be shown to be surjective, as a consequence of Bers’ Simultaneous Uniformization Theorem [10], and of Thurston’s Double Limit Theorem [85]. We will further obtain the following theorem about the deformation space D(π_{1}(S)) of all punctured-torus Kleinian groups, modulo conjugation in PSL_{2}(C):

Theorem B. (Deformation space topology) The map

ν^{−1} : (\overline{D} × \overline{D}) \ Δ → D(π_{1}(S))

is a continuous bijection.

In addition every Bers slice is a closed disk, and every Maskit slice is a closed disk with one boundary point removed.

(See section 12.3 for definitions of Bers and Maskit slices). Note that this does not imply ν^{−1} is a homeomorphism and in fact ν itself is discontinuous! (See Section 12.3, and Anderson-Canary [6].) However, as the interior D × D of the space of invariants maps precisely to the set of structurally stable (or, in particular, quasi-Fuchsian) representations, from a dynamical point of view we have proved that “structural stability is dense” for this family of representations. In particular this gives a positive answer (for punctured-torus groups) to Bers’ conjecture in [11] that all degenerate groups in a Bers slice are limits of quasi-Fuchsian groups.

A final application, also with a dynamical flavor, is the following rigidity theorem:

Theorem C. (QC Rigidity) If the actions of two punctured-torus groups on the sphere are conjugate by a homeomorphism, then they are conjugate by a quasiconformal or anti-quasiconformal homeomorphism, according as the original homeomorphism preserves or reverses orientation.

The core of the proof of Theorem A is the Pivot Theorem 4.1, which is the main step to getting quasi-isometric control of the group in terms of what amounts to the continued-fraction expansions of the end invariants. In particular, the presence of very short geodesics in the quotient manifold is predicted precisely by the presence of high coefficients in the expansion, as has long been conjectured. The statement is given (see §4) in terms of the combinatorics of the Farey triangulation in the disk (see §2).

1.1. Historical comments. Ahlfors and Bers (see [4, 10, 11]) first studied the deformation theory of quasi-Fuchsian groups (in any genus) and showed that they are parametrized by a product of Teichmüller spaces (D × D in our case). Maskit [60] further studied the groups that arise on the boundary of these deformation spaces when the domains of discontinuity are “pinched” and new parabolics arise – this corresponds in our discussion to the case when ν_{+} or ν_{−} are rational points in the boundary R̂. Keen-Maskit-Series [54] gave a proof of Theorem A in this case. Jørgensen made some very careful studies of quasi-Fuchsian punctured-torus groups in [46], in particular obtaining a combinatorial description in terms of the Farey triangulation which is very closely related to the results we obtain in Theorem 4.1. He also studied some degenerate groups, in particular with Marden in [50] applying the triangulation to show that two particular degnerate groups are not quasiconformally conjugate. These ideas have been helpful to the writing of this paper. Degenerate groups were first shown to exist by Bers, and then in greater generality by Thurston, who analyzed them geometrically and introduced the ending lamination invariant (section 3). Bonahon showed that Thurston’s theory applied in fact to all Kleinian surface groups. Our analysis takes these developments as its starting point.

The problem has also been studied by McMullen [67] who showed that cusps (representations corresponding to a rational ν_{+}) are dense in the boundary of a Bers slice. Wright [89] has carefully analyzed the combinatorics of limit sets for groups lying at the boundary of a punctured-torus Maskit slice, and produced some very good computer pictures of such limit sets and of the boundary itself, giving considerable evidence to support the above conjectures. Keen and Series [53] have given geometric coordinates for the interior of punctured-torus Maskit slices, in particular generalizing some of Wright’s findings.

Recently, Bowditch [16] has given an analysis of trace functions on the Farey graph arising from general (not necessarily discrete) representations of π_{1}(S) with parabolic commutator, using algebraic methods. In particular he has established a considerably stronger version of our Lemma 8.1, by a completely different proof. Alperin, Dicks and Porti [5] have used the Farey graph to analyze the geometry of the Gieseking manifold, which is a specific punctured-torus bundle over the circle. In particular they have given an alternate proof of the theorem of Cannon-Thurston [25] in this case.

1.2. Summary of the proof. To simplify this discussion let us consider here a punctured-torus group ρ for which both ν_{+} and ν_{−} are irrational points in R̂. In other words, the manifold N = H^{3}/ρ(π_{1}(S)) has two simply degenerate ends and no domain of discontinuity. This allows us to avoid a number of special cases in the argument having to do with boundaries of convex hulls or accidental parabolics. In fact, the reader is strongly advised to make this assumption on a first reading of the proof itself.

The problem of proving Theorem A reduces to showing that the end invariants ν_{±} describe the group, or manifold, up to quasi-isometry in H^{3}. Then any two representations with the same invariants are conjugate by a quasi-conformal homeomorphism, and one can use Sullivan’s rigidity theorem [83] to show that the conjugating map is, in fact, a Möbius transformation.

However, the end invariants only give asymptotic information: for example, simple closed curves in S are represented by rational numbers in R̂ (Section 2.1), and ν_{±} are characterized by the property that any infinite sequence of curves whose corresponding geodesics in N have uniformly bounded lengths gives rise to a sequence of rational numbers accumulating onto ν_{±}. To know the quasi-isometry type of the manifold we need at the very least to determine which such sequences can arise.

Farey graph and pivot sequence: In §2 we discuss the Farey triangulation C, a well-known triangulation of the disk with vertices in Q̂ ≡ Q ∪ {∞}, which can be interpreted in terms of slopes and intersection numbers of simple closed curves on S. The two irrational points ν_{±} determine, via the combinatorial structure of C, a biinfinite sequence P = {α_{n}} ⊂ Q̂, closely related to continued fraction approximations, such that α_{n} → ν_{±} as n → ±∞. We call these vertices pivots. A good starting point for reading the paper is Section 4, where we define P and state the Pivot Theorem 4.1. This theorem asserts that the pivots indeed have bounded length in the manifold, and furthermore gives an explicit recipe for estimating their complex translation lengths from the combinatorial data of P.

Connectivity: The main idea that leads to the Pivot Theorem is an application of the fact that paths in C correspond to continuous families of simplicial hyperbolic surfaces in the 3-manifold. With this we prove Lemma 8.1, which states that the set of vertices in C whose geodesics in the 3-manifold satisfy a certain length bound is connected. From here it is easy to obtain an a-priori bound on the lengths of all of the pivots, which we do in Lemma 8.2.

Bounded homotopies and control on Margulis tubes: The second idea is roughly this: two homotopic Lipschitz maps of the same non-elementary hyperbolic surface into a hyperbolic 3-manifold are connected by a homotopy of bounded length (the bound depending on the Lipschitz constant). We prove a version of this via the “figure-8 argument” in Section 9.5, and this allows us to constrain the geometry of Margulis tubes in the manifold. In particular, any Margulis tube in N can be encased, homologically, by a pair of surfaces with controlled geometry and a homotopy between them which has bounded tracks in the complement of the Margulis tube. This discussion is carried out in Section 9 via a mechanism we call a “building block”. In particular each block B contains a solid torus U whose boundary torus ∂U is mapped to the boundary of the corresponding Margulis tube. It follows, for example, that there is a uniform bound on the diameter of the boundary torus of the Margulis tube.

Halfway surfaces: The block construction also depends on an analysis we carry out in Lemma 7.1 of Section 7, to describe the possible geometric configurations of axes for generator pairs in punctured-torus groups. In particular when a pair of generators have bounded lengths we can find a simplicial hyperbolic map of S into N in which both generators are simultaneously bounded. We call these halfway surfaces, and they are used to begin the building block construction in Section 9. The proof of Lemma 7.1 is carried out by fairly standard use of trace identities, although one can give a more geometric argument (as was done in a previous draft of this paper).

These ingredients are put together in Section 10, where the proof of the Pivot Theorem is completed. The blocks, one per pivot, are glued end to end to produce a “model manifold” M = ⋃_{n} B_{n} and a Lipschitz map f : M → N, that in particular takes the solid torus U_{n} in each block to the Margulis tube it is meant to model. A subtle point to emphasize here is that, for each solid torus individually it is not a priori clear that the map gives a faithful model, and for example the map restricted to each boundary torus ∂U_{n} is not at first known to be homotopic to a homeomorphism. This issue is settled by a global argument showing that the map f is proper and has degree 1 (Lemma 10.1).

Once this map is in place we have enough control to estimate a Teichmüller parameter for the boundary torus of each Margulis tube, with respect to a natural marking of the torus. As described in Section 6.2, this is exactly what we need to determine the quasi-isometry type of each Margulis tube, and give the statement of Theorem 4.1.

The construction of the model manifold is actually completed in Section 11, where we must face the fact that our map f : M → N is only Lipschitz, and not bilipschitz. What we actually need is that f lifts to a quasi-isometry of the universal covers, and this is the goal of Theorem 11.1. To prove this theorem, we must switch to a different mode: instead of explicit constructions and plausibly computable bounds, we appeal to compactness arguments. In particular we must consider the possible geometric limits of sequences of punctured-torus manifolds with basepoints. These limits conform to the already well-known picture of “drilled holes” developed by Thurston [85] and Bonahon-Otal [15], in which an infinite sequence of new rank-2 cusps can appear in the limiting manifold. (Note that for higher-genus surface groups much more dramatic limits can occur – see Brock [18].) The geometric limits we obtain come equipped with their own model manifolds, and this control in the limit suffices to give us the uniform bounds we need.

The proofs of the main theorems A, B and C are carried out in Section 12, and are fairly straightforward given what has gone before. A subtle issue which arises in the proof of Theorem B involves the continuity of the end invariants under algebraic limit, which in fact does not hold in general. We are led back to consideration of the geometric limit apparatus of Section 11 in order to resolve this point.

Sections 2, 3, 5 and 6 are essentially expository. In Section 2 we discuss the Farey graph and some elementary cartography of the Teichmüller space of the torus, and give a lemma on quasiconformal maps. In Section 3 we state the definitions of the end invariants ν_{±}. In Section 5 we discuss simplicial hyperbolic surfaces and pleated surfaces, which will play a central role in almost all of our arguments. In Section 6 we discuss Margulis tubes. In particular we state some well-known bounds on the radii of such tubes, due to Brooks-Matelski and Meyerhoff, and develop in §6.2 the connection between the geometry of a Margulis tube and a parameter in Teichmüller space describing its quotient torus at infinity. In §6.3 we give some additional constraints, due to Thurston and Bonahon, on Margulis tubes which appear in surface groups.

1.3. Speculations on the general case. There are a few straightforward generalizations of the ideas in this paper. The quadruply-punctured sphere can be treated almost identically: in particular the combinatorics of the set of simple closed curves are again encoded by the Farey triangulation, and the figure-8 argument in Lemma 9.3 still applies.

Furthermore, let (M, P) be a “pared” manifold, that is let M be a compact 3-manifold and P a collection of tori and annuli on ∂M, and suppose that all the components of ∂M \ P are punctured tori or 4-punctured spheres which are incompressible in M. Suppose that M admits an embedding into a hyperbolic 3-manifold N which is a homotopy-equivalence and takes each component of P into a distinct parabolic cusp. Then the techniques of this paper apply directly to the ends of the resulting manifold, and a suitably restricted version of the ending lamination theorem holds (see [70] to see how this type of argument works). One may also extend the rigidity theorem (C) to this context; see e.g. Ohshika [75].

Beyond this, one must begin to consider the general problem for higher-genus surface groups. A number of very serious difficulties arise here. There is a natural simplicial complex generalizing the Farey graph, but its properties are considerably harder to understand. In Masur-Minsky [64] we study this complex from a point of view partially motivated by these ideas. The bound on diameter of Margulis tube boundaries also fails in general, and this is related to the fact that geometric limits of sequences of general surface groups can be much more complicated than what we obtain in Section 11.2. See Brock [18] for some of the type of phenomena that can occur. Acknowledgements. I am very grateful to Dick Canary, Curt McMullen and Jeff Brock, for many enjoyable and illuminating conversations on the subject of this paper, and to Caroline Series who pointed out to me the special nature of the punctured-torus case. Special thanks are due to Ada Fenick, who told me I had to finish writing it.

2. The Farey triangulation and the torus

Let H^{2} denote the upper half plane with boundary R. There is a classical ideal triangulation of H^{2}, defined as follows. For any two rational numbers written in lowest terms as p/q and r/s, say they are neighbors if |ps − qr| = 1. Allow also the case ∞ = 1/0. Joining any two neighbors by a hyperbolic geodesic, we obtain the Farey triangulation. (The proof that this is a triangulation is easy after considering the edges incident to ∞, and observing that the diagram is invariant under the natural action of SL_{2}(Z). See also Series [82, 81] or Bowditch [16]). Figure 1 shows this triangulation, after stereographic projection to the unit disk D. This picture is intimately related to the torus, as we shall now see.

Figure 1. The Farey graph in the unit disk, with some of the vertices labeled.

2.1. The complex of curves. Let C denote the set of free homotopy classes of unoriented simple non-peripheral curves on the punctured torus S. These are in one-to-one correpondence with Q̂ ≡ Q ∪ {∞}, after choosing an ordered basis for H_{1}(S). Let us do this explicitly in order to be careful about sign conventions:

Fix an orientation for S and choose a positively oriented ordered basis (α, β) for H_{1}(S). This is equivalent to choosing two oriented simple closed curves which cut S into a positively oriented rectangle. Any element of H_{1}(S) can be written as (p, q) = pα + qβ in this basis, and we associate to this the slope −p/q ∈ Q̂. Note that this ratio forgets the orientation of the curve, as well as integer multiples. Thus it exactly describes an element of C(S). (The proof that every element of C is obtained in this way is left to the reader). The determinant ps−rq which appeared above is easily seen to be just the oriented intersection number i(·, ·) in S. Let α · β = |i(α, β)| denote unoriented intersection number, which is defined on C.

Thus the Farey graph reflects the combinatorial structure of C, and from now on we shall identify the two. Remarks: 1. C is a special case of the Hatcher-Thurston complex [41], and is related closely to the complexes of curves introduced by Harvey [40] and studied by Harer [38, 39] and Ivanov [43, 42, 44]. See also Bowditch-Epstein [17] for another perspective. 2. A pair of vertices joined by an edge can also be considered as representing a pair of generators for π_{1}(S), up to conjugation and inverses, as in Jørgensen [46] and Jørgensen-Marden [50]. (See also Section 7) 3. The same construction works for the regular torus – the difference there is that we don’t need to worry about peripheral curves or non-simple curves that, say, wind around the puncture.

2.2. Neighbors and Dehn twists. Given α ∈ C, its neighbors may be indexed by the integers {β_{n}}_{n∈Z} according to their counterclockwise order around S^{1} \ {α}. Denote by D_{α} the positive Dehn twist around α, defined for example by the convention that positive twists around a vertical curve increase slope. Note that “positive” makes sense after a choice of orientation on S, but without orienting α – see figure 2. Then the indices are defined, after arbitrary choice of β_{0}, by β_{n} = D_{α}^{n}(β_{0}).

Figure 2. Positive Dehn twists (a) in the punctured torus, and (b) in the Farey triangulation.

2.3. Teichmüller space. The interior D of the disk, or half plane H^{2}, also has a well-known interpretation in terms of the torus: it parametrizes the Teichmüller space T (S) of conformal, or hyperbolic, structures on S. (Recall that the conformal structures on the regular and once-punctured torus are the same – although the regular torus admits no hyperbolic metric). In this interpretation, the circle R̂ ≡ R ∪ {∞} is Thurston’s compactification of T (S) using projective measured laminations – where the rational points correspond to simple closed curves, as above, and the irrational points to laminations with infinite leaves (see [31, 79]).

Explicitly, fix a marking (α, β) of the torus, as above. To a point z in the upper half plane we associate the lattice generated by 1 and z, whose quotient is a torus with induced conformal structure ν. The position of the puncture is irrelevant since the torus has a transitive family of conformal automorphisms. An orientationpreserving identification of our fixed torus S with this torus is determined by taking the curve α to the image of [0, 1] and β to the image of [0, z].

For later convenience let us denote by

z = τ(S, ν, α, β)

the relationship between a marked conformal torus and its Teichmüller parameter. (We also write τ(S, α, β) if the conformal structure on S is understood).

Let us also recall the Teichmüller metric which is defined as \frac{1}{2} log K where K is the best dilatation constant for a marking-preserving quasiconformal homeomorphism between two marked tori. We will use the fact that this is exactly equal to the hyperbolic metric on H^{2}. Shortest curves. Extremal lengths (see Ahlfors [3]) can be computed directly in the Euclidean metric inherited by the torus, as length squared over area. Note in particular that α, which was identified with ∞ ∈ R̂, has extremal length 1/ Im z in the structure parametrized by z. When this quantity is very small it gives a good estimate for the hyperbolic length of α (see Maskit [62]).

To get a clean picture for shortest hyperbolic lengths as a function of Teichmüller parameter, we need a little geometry. We remark first that the shortest closed geodesic is always simple, by an easy surgery argument. Now given a simple closed geodesic α of length ℓ on a hyperbolic punctured torus S, cut S along α to get a punctured cylinder with distance h between its boundaries. There is a unique way to cut this cylinder along the shortest geodesic γ between its boundaries and along geodesics orthogonal to the boundaries and heading into the cusp, to obtain two congruent pentagons, each with one ideal vertex and four right angles. A bit of hyperbolic trigonometry on this configuration yields (see Beardon [8] or Buser [20]):

cosh h/2 = coth ℓ/2.  (2.1)

If β is a geodesic in S that intersects α once, a symmetry argument shows that β must intersect the segment γ at its midpoint. Thus there are lifts α̃, β̃ and γ̃ to H^{2} which form a right triangle with legs h/2 and t, and hypotenuse ℓ′/2, where ℓ′ is the length of β and t is the distance along α̃ between its intersection with γ̃ and with β̃. The hyperbolic law of cosines gives us:

cosh ℓ′/2 = cosh h/2 cosh t/2.  (2.2)

We deduce from this a number of things. First, we see that t is a function of ℓ and ℓ′. Thus, if ℓ = ℓ′ we would find that the configuration obtained by cutting along β is equivalent to that obtained by cutting along α. We conclude that if ℓ = ℓ′ there is an orientation-reversing isometry of S that exchanges α and β.

Back in \overline{H}^{2}, suppose that α = ∞ and β = n ∈ Z. The orientation reversing homeomorphism of S interchanging them acts on H^{2} as the Möbius reflection through the axis {τ : |τ − n| = 1}. Thus if α and β have equal hyperbolic lengths for some τ ∈ H^{2} then τ lies on this semicircle. We know (by the collar lemma) that α is the shortest curve in S when Im τ is sufficiently large, and that a different curve can only become shortest at a point where it and α have equal lengths. It follows that the locus of H^{2} where α has strictly shorter hyperbolic length than any of its neighbors is exactly H(α) = {τ : ∀n ∈ Z, |τ − n| > 1}. Define H(γ) for other γ ∈ Q̂ via the action of SL_{2}(Z). We will see momentarily that in fact H(α) is the locus where α is strictly shortest among all geodesics.

After applying this discussion to all vertices we find that, for any Farey triangle Δ, if we divide up Δ into six regions by the axes of its reflection symmetries, then each vertex u has minimal hyperbolic length in the pair of regions that meet u, and is strictly minimal in the interior of the union of the pair.

There is always, for general reasons, some constant L_{0} > 0 bounding the length of the shortest geodesic of in all finite-area hyperbolic surfaces with a fixed topology (see e.g. Buser [20]). In fact L_{0} for the punctured torus is one of the few constants in this paper whose value we can compute precisely: If α and β are as above then, after possibly Dehn-twisting β about α a number of times, we may assume that t in (2.2) is at most ℓ/2 (this is easiest to see by considering the lifts of γ that are crossed by β̃ – successive ones are separated by at most ℓ/2 along the translates of α̃). Now if α has minimal length among its neighbors then ℓ ≤ ℓ′ and so, combining (2.1) and (2.2) and using t ≤ ℓ/2, we find that sinh ℓ/2 ≤ cosh ℓ/4. It follows that ℓ ≤ 4 sinh^{−1}(1/2) ≈ 1.9248. Thus we let this be L_{0} and it serves as an upper bound for the shortest α.

We can also compute, for ℓ ≤ L_{0}, that h ≥ h_{0} ≈ 1.609. Thus any curve which crosses α more than once has length at least 2h_{0} ≈ 3.218. We summarize our findings as follows:

Lemma 2.1. Let S be a hyperbolic punctured torus. The number L_{0} ≈ 1.9248 bounds the length of the shortest closed geodesic on S. If α and β have lengths bounded by L_{0} on S then they are Farey neighbors. In particular if they are both shortest on S then then they are Farey neighbors.

Finally, it will be useful to know that the same value of L_{0} bounds the shortest geodesic for any complete metric on S with curvatures bounded above by −1 where the hole is conformally a puncture: By a lemma of Ahlfors (see [2]) the hyperbolic metric in the same conformal class as such a metric is pointwise bigger. The author is grateful to Curt McMullen for pointing out this lemma. In particular, for the metrics which will arise later on from simplicial hyperbolic surfaces, the curvatures are −1 except for isolated cone singularities with cone angle 2π or more, and these can be represented in an isothermal coordinate as zeroes of the conformal factor. Ahlfors’ lemma applies in this generality. Teichmüller parameters for annuli. A marked annulus is an oriented annulus A with an arc β whose endpoints lie on distinct boundary components. A conformal structure on such an annulus yields a Teichmüller parameter τ(A, β) ∈ H^{2}, namely the point corresponding to the marked torus obtained by placing a Euclidean metric on A, gluing the boundaries together with an isometry that identifies the endpoints of β, and marking with the core curve of A and the image of β under the gluing.

Let T be a torus which is the union of a sequence of annuli A_{1}, …, A_{k} glued along their boundaries, and suppose that in a Euclidean metric on T the boundaries of the A_{i} are geodesic. Let α be a curve in the homotopy class of the boundary curves, and let μ be a curve crossing each annulus A_{i} in a single arc μ_{i} (we do not require that μ be a geodesic). Then α, μ give a marking for T, and we immediately have

τ(T, α, μ) = ∑_{i=1}^{k} τ(A_{i}, μ_{i}).  (2.3)

by considering the decomposition in the universal cover of T.

2.4. Quasiconformal Lemmas. Let us record some easy facts useful for estimating quasiconformal distortion in simple situations where we have mixed geometric and quasiconformal data.

Let h : T_{1} → T_{2} be an L-Lipschitz map of degree 1 between Euclidean tori, where T_{1} is a square torus of area 1, and T_{2} has area at least A_{0}. Then h is homotopic to a K-quasiconformal homeomorpism, where K depends only on L and A_{0}. This is easily seen by lifting to an isomorphism of lattices in R^{2}.

A slightly more subtle fact is the following, which will be applied in Sections 9 and 10 to control Margulis tubes at the boundary of the convex core. A Euclidean annulus is an annulus isometric to the product of a circle with an interval. Call the length of the circle the girth.

Lemma 2.2. Let h : A_{1} → A_{2} be a proper map between two Euclidean annuli of girth 1, which is K-quasiconformal on a Euclidean subannulus C of A_{1}, whose modulus is at least M_{0}, and such that the components B_{0} and B_{1} of A_{1} \ C have modulus bounded by M_{0}. Suppose also that h is L-Lipschitz on each B_{i}, and that h is an embedding on the boundary of A_{1} and L-bilipschitz on each boundary component.

Then h is homotopic to a K′-quasiconformal homeomorphism h′ : A_{1} → A_{2}, where K′ depends only on the constants L, K, M_{0}, and the homotopy is constant on the boundary.

Proof. (Sketch) For i = 0, 1 let B′_{i} be the Euclidean subannulus of A_{1} containing B_{i}, such that B′_{i} \ B_{i} has modulus m = M_{0}/2. We will replace h|_{B′_{i}} with a quasiconformal map h′ which is homotopic to h rel ∂B′_{i}, and whose quasiconformality constant depends only on the previous constants.

To see that this is possible, note that h|_{B′_{i}} is an element of a family of maps {g : B′_{i} → S^{1} × [0, ∞)} which satisfy the same quasiconformality and Lipschitz conditions that h|_{B′_{i}} does (here we are identifying A_{2} with an appropriate initial subannulus S^{1} × [0, T] of S^{1} × [0, ∞)).

Each such map g can be deformed rel boundary to a K(g)-quasiconformal map g′, by standard methods: uniformize both B′_{i} and g(B′_{i}) by the upper halfplane and check that the induced boundary map is quasisymmetric (this follows from the quasiconformality for one part of the boundary, and from the bilipschitz condition for the rest). Moreover, the family {g} is compact in the compact-open topology, and after proper normalization, so is the family of lifts to the universal covers. It follows that the constants K(g) have a uniform upper bound.

On A_{1}\(B′_{0}∪B′_{1}), we simply keep the same map h. This concludes the proof.

We remark also that in the argument, the process of lifting to the upper half plane is what allows us to do the qc extension in a way that keeps proper track of the twisting of the original map. The intuition here is that the amount of twist in the original map h is bounded by quasiconformality in C and by the Lipschitz condition in B_{i}. This is subtle to detect directly – in fact, the twist itself can go to infinity with fixed K if mod (A_{1}) → ∞. However, K′ is bounded independently of mod (A_{1}), and this is reflected in the quasisymmetry of the lift.

3. Geometric tameness and end invariants

In this section we describe how to associate to a punctured torus group an ordered pair of “end invariants” (ν_{−}, ν_{+}), each lying in the closed disk \overline{D}, or equivalently \overline{H}^{2}. This is a special case of end invariants for general (geometrically tame) Kleinian groups, coming from the work of Ahlfors, Bers and Maskit for geometrically finite ends (where the invariant is a collection of simple closed curves on the boundary and an element of the Teichmüller space of their complement), and from Thurston, Bonahon and Canary for geometrically infinite ends (where the invariant is a geodesic lamination). We omit a general discussion of this, referring the reader to [14, 22, 71, 86] for more details.

Let us now concentrate on the case of a punctured torus group ρ : π_{1}(S) → PSL_{2}(C) and associated manifold N = H^{3}/ρ(π_{1}(S)).

Let Ň denote N minus the ϵ_{0}-Margulis tube Q_{N} associated to the parabolic commutator (we call this the “main cusp”). This manifold has two ends; in fact, circumventing historical order we may note that Bonahon’s theorem [14] implies that N is homeomorphic to S × R, and Ň is homeomorphic to S_{0} × R, where S_{0} is S minus an open neighborhood of the puncture. (Remark: the ends can be defined without knowing Bonahon’s theorem, for example by considering the way that Ň is cut up by its relative Scott core [80, 65]. However, we prefer this simplified exposition). Let us name the ends e_{−} and e_{+}, where the following orientation convention is applied:

If M is an oriented manifold we orient ∂M by requiring that the frame (f, n) has positive orientation whenever f is a positively oriented frame on ∂M and n is an inward-pointing vector. Now fix the orientation on N induced from H^{3}, and choose a fixed orientation for S. This determines (up to homotopy through proper maps) an identification of N with S × (−1, 1) which induces the map ρ on fundamental groups, and such that the orientation of S agrees with that induced on S × {1}. Let e_{+} denote the end of Ň whose neighborhoods are neighborhoods of S_{0} × {1}, and e_{−} the other end.

Let Ω denote the (possibly empty) domain of discontinuity of Γ. Let \overline{N} denote the quotient (H^{3} ∪ Ω)/Γ. Any component of the boundary Ω/Γ is reached by going to one of the ends e_{+} or e_{−}, and this divides it into two disjoint pieces Ω_{+}/Γ and Ω_{−}/Γ (where Ω_{+}, Ω_{−} are the corresponding invariant subsets of Ω). There are three possibilities for each of these boundaries, corresponding to three types of end invariants (here let s denote either + or −):

1. Ω_{s} is a topological disk, and Ω_{s}/Γ is a punctured torus. This determines a point in the Teichmüller space of S, denoted by ν_{s}.

2. Ω_{s} is an infinite union of disks and Ω_{s}/Γ is a thrice-punctured sphere, obtained from the corresponding boundary of S × (−1, 1) by deleting a simple closed curve γ_{s}. In this case ν_{s} ∈ Q̂ denotes the slope of γ_{s}, as in section 2.1. The conjugacy class of γ_{s} in Γ is parabolic.

3. Ω_{s} is empty. In this case ν_{s} ∈ R\Q, and we describe its geometric significance below.

(This trichotomy is due to Maskit, see [59, 63]). Let C(N) denote the convex core of N, namely the quotient by Γ of the convex hull CH(Λ) of the limit set Λ of Γ. Each component of ∂CH(Λ) corresponds to a component of Ω via orthogonal projection from Ĉ to ∂CH(Λ) (see [30]), so that ∂C(N) divides naturally into ∂_{+}C(N) ∪ ∂_{−}C(N) where each ∂_{s}C(N) is a punctured torus, thrice punctured sphere or empty according to the above three cases. Each boundary component is a convex pleated surface in N with an induced hyperbolic metric.

In case 2, since all thrice-punctured sphere groups (with parabolic boundaries) are conjugate to a fixed Fuchsian group, the components of Ω_{s} are actually round circles, and the boundary component ∂_{s}C(N) is totally geodesic.

To define ν_{s} in case 3 we need to recall the theory of ends due to Bonahon and Thurston. For a simple closed curve γ in S let γ^{∗} denote its geodesic representative in N (more precisely γ determines a conjugacy class taken by ρ to a conjugacy class in Γ. If this class is non-parabolic it has a geodesic representative). Thurston showed [86] that if {γ_{n}} is a sequence of simple closed curves such that γ_{n}^{∗} are eventually contained in any neighborhood of e_{s}, then the slopes of γ_{n} converge in R to a unique irrational number. We say that such a sequence “exits the end”. Bonahon [14] showed that in fact for each end e_{s} that is not geometrically finite, that is, not in case 1 or 2, there is such a sequence of geodesics. (Thurston showed this for groups that are known to be limits of quasi-Fuchsian groups). We define ν_{s} to be this limiting irrational slope. Thurston calls ν_{s} an ending lamination in this case because it describes a geodesic lamination for any hyperbolic metric on S, obtained as a limit of the geodesics in S corresponding to γ_{j}.

The consequences of the existence of sequences of geodesics exiting an end will be more apparent once we introduce simplicial hyperbolic surfaces in section 5.

4. The pivot theorem

In this section we associate to any end-invariant pair (ν_{−}, ν_{+}) a pivot sequence, which is closely related to a continued-fraction expansion, and state our main structural theorem, Theorem 4.1, which translates the combinatorial structure of the sequence into geometric data about the associated representation ρ. The proof of Theorem 4.1 will be completed in Section 10, after a number of necessary tools are developed.

4.1. The pivot sequence. Letting s denote + or −, define a point α_{s} ∈ R̂ to be closest to ν_{s} in the following sense: If ν_{s} ∈ R̂ let α_{s} = ν_{s}. If ν_{s} ∈ D, let α_{s} ∈ C represent a geodesic of shortest length (hence at most L_{0} – see Lemma 2.1) in the hyperbolic structure corresponding to ν_{s}. In particular ν_{s} is contained in a Farey triangle Δ_{s} of which α_{s} is a vertex. Note that (non-generically) there may be two or three choices for α_{s}, in which case we choose one arbitrarily. Our constructions will work for any of the choices.

Now define E = E(α_{−}, α_{+}) to be the set of edges of the Farey graph which separate α_{−} from α_{+} in the disk. Let P_{0} denote the set of vertices of C which belong to at least 2 edges in E (except for one exceptional case, see (3) below). We call these vertices internal pivots (see figure 3). The edges of E admit a natural order where e < f if e separates the interior of f from α_{−}, and it is easy to see this induces an ordering on P_{0}. We therefore arrange P_{0} as a sequence {α_{n}}_{n=ι}^{p} where ι = −∞ if ν_{−} ∈ R \ Q, and ι = 1 otherwise, and p = ∞ if ν_{+} ∈ R \ Q, and is some finite nonnegative integer otherwise.

Figure 3. Sketch of a pivot sequence. For visibility the circle boundary has been stretched to two parallel lines and only the edges separating α_{−} from α_{+} are solid.

Note in particular the following special cases:

1. E is biinfinite and P_{0} = P is biinfinite: both ν_{±} lie in the irrational part of the boundary.

2. E = ∅ and P_{0} = ∅: α_{±} lie in the closure of a single Farey triangle. Note α_{+} and α_{−} may or may not be equal.

3. E is a singleton {e}: α_{±} lie in the closures of adjacent Farey triangles. In this case we redefine P_{0} to be a single point {α_{1}}, which is chosen arbitrarily from the two endpoints of e.

Case (1) is the case one should keep in mind throughout most of the paper, as it is easiest to deal with. Cases (2) and (3) are particularly simple types of geometrically finite groups, and hence in some sense we have nothing new to say about them, but they tend to complicate our exposition and notation. The geometrically finite cases are also responsible for introducing some real geometric subtleties, particularly in the behavior of geometric limits (see Section 11.2), and so we have no choice but to be careful.

We obtain the full pivot sequence P by appending to the beginning of P_{0} the vertex α_{−} if α_{−} ∈ C (hence ι = 1), and appending to the end of P_{0} the vertex α_{+} if α_{+} ∈ C (hence p < ∞). In these cases we define α_{0} ≡ α_{−} and α_{p+1} ≡ α_{+}, respectively.

With this numbering convention, we note that for any n ∈ {ι, …, p}, α_{n−1} and α_{n+1} are related by

α_{n+1} = D_{α_{n}}^{w(n)}α_{n−1},

and the integers w(n) so defined are called the widths of the pivots.

It will be useful to consider this from a different perspective: For each β ∈ C fix an identification of D as H^{2} (by an orientation-preserving Möbius transformation) such that β is identified with ∞, and its neighbors with the integers Z. Such a normalization is unique only up to integer translation; the ambiguity will turn out not to matter. Let ν_{+}(β) and ν_{−}(β) denote the points of \overline{H}^{2} = H^{2} ∪ R̂ to which ν_{±} are taken by this identification. Similarly let α_{i}(β) be the images of α_{i} by this identification, for i ∈ Z or i = ±.

One easily checks that (except in case (3)) β is in P_{0} if and only if α_{−}(β) and α_{+}(β) are separated by at least two integers. The width w(n) can be written α_{n+1}(α_{n}) − α_{n−1}(α_{n}), and can also be estimated as follows:

For any x ≠ y ∈ R which are separated by an integer, let ⌊x, y⌋ be defined as k − j where j is the integer closest to x in the closed interval spanned by x and y, and k is the integer closest to y in this interval. In particular we note the sign of ⌊x, y⌋ equals the sign of y − x, and |y − x| − 2 < |⌊x, y⌋| ≤ |y − x|. With this notation we have

w(n) = ⌊α_{−}(α_{n}), α_{+}(α_{n})⌋.  (4.1)

This easily gives the estimate

|w(n) − (Re ν_{+}(α_{n}) − Re ν_{−}(α_{n}))| ≤ 2.  (4.2)

We can also define widths for the first and last pivots of P, if these exist. Suppose α_{+} ≠ α_{−}. When α_{+} = α_{p+1} and ν_{+} ≠ α_{+}, we compute w(p + 1) using (4.1), but replacing the second term by Re ν_{+}(α_{p+1}). When α_{−} = α_{0} and ν_{−} ≠ α_{−}, we compute w(0) using (4.1), but replacing the first term by Re ν_{−}(α_{0}). If α_{+} = α_{−} we make both replacements in (4.1) to obtain w(0). In all cases, the estimate 4.2 applies. Remark. The connection of this to continued-fraction expansions is easiest to state if ν_{−} = ∞ and ν_{+} ∈ R. Then |w(n)| are exactly the continued-fraction coefficients of the fractional part of ν_{+}. The other cases are similar, and we omit the details.

4.2. Statement of the pivot theorem. For any element γ of PSL_{2}(C), let λ(γ) = ℓ+iθ denote its complex translation length; it is determined by the identity tr^{2} γ = 4 cosh^{2} λ/2, and the normalizations ℓ ≥ 0 and θ ∈ (−π, π]. (More about this choice in Section 6.2.) Note that λ is invariant under conjugation and inverse. Geometrically, ℓ (if positive) gives the translation length of γ along its axis, and θ gives the rotation.

Thus, fixing a discrete faithful representation ρ : π_{1}(S) → PSL_{2}(C), we obtain a function on C which we write λ(α) ≡ λ(ρ(α)).

The Pivot Theorem will give us quasi-isometric control of the complex translation lengths of the pivots of a punctured-torus group:

Theorem 4.1. (Pivot theorem) There exist positive constants ϵ, c_{1} such that, if ρ is a marked punctured-torus group with associated pivot sequence,

1. If ℓ(β) ≤ ϵ then β is a pivot.

2. If α is a pivot then

\frac{2πi}{λ(α)} ≈ ν_{+}(α) − \overline{ν}_{−}(α) + i

Where “≈” denotes a bound c_{1} on hyperbolic distance in H^{2} between the left and right sides.

Remarks: 1. The quantity ω(α) = 2πi/λ(α) is a convenient way to encode the geometry of α and its Margulis tube. In particular note that λ lies in the righthalf plane {Re z > 0}, and ω lies in the upper half-plane {Im z > 0}. Both are Teichmüller parameters for the torus (Ĉ \ F ix(α))/α, with respect to different markings; see Section 6.2 for more on this. The hyperbolic distance estimate on ω is natural because, being also a Teichmüller distance estimate, it implies a bilipschitz estimate on the action of α in H^{3} – see Lemma 6.2.

2. Note in particular that |ω(α_{n})| is bounded away from zero by (2), which implies a universal upper bound on the length of all pivots.

3. The real part of the estimate is just Re ν_{+}(α_{n}) − Re ν_{−}(α_{n}), which is an estimate for w(n), via (4.2). Furthermore, if α_{n} is an internal pivot we note that Im ν_{±}(α_{n}) ≤ 1, by the discussion in Section 2.3. It follows that (2) is equivalent, for internal pivots, to

ω(α_{n}) ≈ w(n) + i.  (4.3)

In terms of λ = ℓ + iθ, (4.3) translates to:

\frac{c_{2}}{w(n)^{2}} ≤ ℓ(α_{n}) ≤ \frac{c_{3}}{w(n)^{2}}  (4.4)

and

|w(n) − \frac{2π}{θ(α_{n})}| ≤ c_{4}  (4.5)

with suitable constants c_{i} independent of ρ or α_{n}. It is easiest to see this by noting that the map ω ↦ λ is an isometry between the Poincaré distance on the upper half-plane and that on the right half-plane.

4. For a non-internal pivot α = α_{±}, the imaginary part of ν_{+}(α) or ν_{−}(α) is large. One direction of the estimate (2) is then a variation of an inequality of Bers [11], reflecting the fact that a curve which is very short on the domain of discontinuity is very short in the 3-manifold. See Lemma 6.4 for more on this.

5. Simplicial hyperbolic surfaces

An important role in the theory of hyperbolic 3-manifolds is played by images of surfaces which are, in some sense, “hyperbolic”. Thurston introduced this technique with his pleated surfaces (see Thurston [86, 87] and Canary-Epstein-Green [24]), and Bonahon and Canary [14, 22] have used a related construction known as simplicial hyperbolic surfaces. Sometimes the difference between these is merely technical, but in our case we find that the simplicial hyperbolic surfaces have a particular advantage: one can have better explicit control over continuous families of such surfaces. We will, however, briefly use pleated surfaces, in Sections 6 and 8, so we will discuss them here as well.

5.1. Definitions. Let S be a (possibly punctured) surface and N a hyperbolic 3-manifold. A proper map f : S → N is a simplicial hyperbolic surface if the following hold: A neighborhood of each puncture is mapped to a cusp of N. There is a triangulation T of S (with some edges terminating at punctures) such that f takes each edge to a geodesic, and each triangle of T to a totally geodesic immersed triangle (with punctures going to ideal vertices). The sum of corner angles around any non-ideal vertex is at least 2π.

In particular f induces on S a singular hyperbolic metric: that is, a complete metric σ of finite area, smooth with curvature −1 away from finitely many singularities (at the vertices) where atoms of negative curvature may be concentrated.

We consider two simplicial hyperbolic surfaces to be equivalent if they differ only by precomposition with a homeomorphism of S isotopic to the identity.

In the case of the punctured torus we will consider simplicial hyperbolic surfaces adapted to a curve α ∈ C (Canary calls these surfaces with a distinguished edge). Given α in C, realize it as a specific curve on S (still called α), choose a vertex x ∈ α and let β be a simple curve meeting α transversely only at x. These curves cut S into a punctured quadrilateral, and adding four edges from the vertices to the puncture yields a triangulation T (see figure 4).

Figure 4. A triangulation of the punctured torus with one ideal vertex (in the center) and one real vertex

A simplicial hyperbolic surface f : S → N is adapted to α if it is simplicial with respect to this T and takes α to its geodesic representative α^{∗} in N. It is easy to see that such maps exist in the homotopy class determined by ρ [14, 22] and any two with isotopic triangulations differ only by “sliding” the vertex around α^{∗} (perhaps more than once around) – that is, move (2) in Section 5.2 below.

A pleated surface is a map f : S → N which induces a non-singular hyperbolic metric on S, with respect to which it is totally geodesic on the complement of a geodesic lamination (a closed set foliated by geodesics). The leaves of the lamination are also mapped geodesically. One can think of this heuristically as a simplicial hyperbolic surface where the triangulation has infinite-length edges and no vertices. For example one can show that, starting with a simplicial hyperbolic surface adapted to a curve α and performing the sliding operation infinitely many times, one obtains a pleated surface in the limit for which α is part of the lamination. In particular the singular hyperbolic metrics converge to the hyperbolic metric on the pleated surface.

Using simplicial hyperbolic surfaces we can obtain the following slightly stronger characterization of the ending lamination:

Lemma 5.1. There exists L_{0} > 0 for which, if ρ is a punctured torus group with (irrational) ending lamination ν_{s} (here s denotes + or −), there is a sequence γ_{n} ∈ C converging to ν_{s}, such that the corresponding geodesics are eventually contained in any neighborhood of the end e_{s}, and ℓ(γ_{n}) ≤ L_{0}.

Proof. Let L_{0} be the bound on the shortest geodesic in a simplicial hyperbolic punctured torus, from Section 2.3. The Bonahon-Thurston theory gives a sequence of curves δ_{n} whose geodesic representatives are eventually in any neighborhood of e_{s}. Let f_{n} : S → N be simplicial hyperbolic surfaces adapted to δ_{n} (one could equally well use pleated surfaces) and let γ_{n} be the shortest curve in the metric induced by f_{n} on S. The statement follows for these curves. One needs only to check that γ_{n}^{∗} indeed exit the end e_{s}. Let Ŝ denote a fixed embedded cross-section of N. For large enough n, f_{n}(γ_{n}) lies in a neighborhood of e_{s} disjoint from Ŝ. Let A_{n} be a homotopy from f_{n}(γ_{n}) to γ_{n}^{∗} which has geodesic tracks. The bound on length of f_{n}(γ_{n}) implies that, either A_{n} has bounded tracks, or it has very short circumference for most of its length. If γ_{n}^{∗} is not contained in a neighborhood of e_{s} disjoint from Ŝ then A_{n} must meet Ŝ, but then the translation length of γ_{n} in a fixed compact set is small; this can only hold for finitely many n.

5.2. Interpolation of simplicial hyperbolic surfaces. We recall at this point the elementary moves between simplicial hyperbolic surfaces discussed in Canary [23].

Canary uses three moves. In each case we begin with a simplicial hyperbolic surface f_{0} : S → N with triangulation T_{0} and vertex v, adapted to a curve α_{0}. A move replaces this data with f_{1}, T_{1}, v, α_{0}, and gives a homotopy f_{t} : S → N, t ∈ [0, 1], connecting the two maps so that each f_{t} is still a simplicial hyperbolic surface.

1. Diagonal switch: Let Q be a quadrilateral in T_{0} with diagonal d. In T_{1}, d is replaced by the opposite diagonal d′. The maps f_{t} agree with f_{0} everywhere but on the interior of Q, where for t ∈ (0, 1) the triangulation contains both diagonals and a new vertex where their interiors intersect. (See figure 5.)

2. Vertex slide: The vertex v is “pushed” once around α_{0}. The new triangulation is actually isotopic to the old one, but not rel v.

3. Geodesic switch: here T_{0} and T_{1} are equal but the map f_{1} is adapted to the other closed curve in the triangulation. (See figure 6.)

Figure 5. A sequence of two diagonal switches which effects a Dehn twist D_{α_{0}} on the triangulation. (Two fundamental domains in the abelian cover are shown, and α_{0} is the vertical curve.)

Let us describe move (3) in more detail. Lift the image f_{0}(v) of the vertex to ξ ∈ H^{3}. If α_{0} and α_{1} are the two closed curves of the triangulation, the lifts of α_{0} and α_{1} based at ξ determine group elements A and B, respectively. (compare section 7). Let P be the common perpendicular of T_{A} and T_{B}. The homotopy f_{t} is in three parts. Lifted to H^{3}, it moves ξ first along T_{A} to P, then along P to T_{B}, and then (if desired) along T_{B} to a new position. The rest of the triangulation varies accordingly, so that at each point the edges α_{0} and α_{1} map to geodesics and the map is simplicial hyperbolic.

Figure 6. An intermediate stage in move (3). Pictured in the universal cover are T_{A} and T_{B}, and a portion of the simplicial hyperbolic surface near ξ.

Any two triangulations of the punctured torus (in fact any surface) can be connected by such elementary moves. In particular figure 5 shows how a Dehn twist is effected. This is sufficient for traversing the edges of our graph C. A consequence of this is the following lemma, proven in [23]:

Lemma 5.2. If f_{0}, f_{1} are two homotopic simplicial hyperbolic surfaces then they may be connected by a continuous family f_{t}, t ∈ [0, 1] such that f_{t} is a simplicial hyperbolic surface for each t.

6. The structure of Margulis tubes

6.1. Thick-thin decomposition. The ϵ-thin part M_{thin(ϵ)} of a manifold M (see e.g. [88, 9, 7]) is the subset where the injectivity radius is at most ϵ/2. (Our epsilons always denote length of the shortest non-trivial closed path through a point, rather than injectivity radius). We denote by M_{thick(ϵ)} the closure of the complement of M_{thin(ϵ)}. The Margulis lemma [52], or Jørgensen’s inequality [48], implies that for each dimension n there is a value of ϵ, known as the Margulis constant, below which the ϵ-thin part of a hyperbolic n-manifold is of a standard type. In particular in dimensions 2 and 3 in the orientable case, every component, known as a Margulis tube, is either a tubular neighborhood of a closed geodesic, or the quotient of a horoball by an abelian parabolic group.

Let ϵ_{0} be a Margulis constant for H^{3}. Let T_{ϵ}(g) denote the ϵ-Margulis tube in N = H^{3}/Γ for a conjugacy class g in Γ. Similarly with a fixed representation ρ in mind and a conjugacy class α in π_{1}(S), let T_{ϵ}(α) denote T_{ϵ}(ρ(α)).

Let r(α) be the radius of T_{ϵ_{0}}(α), that is the distance from the geodesic core to the boundary. Recall that λ(α) denotes the complex translation length ℓ + iθ of α. Let us record some preliminary bounds relating these quantities, due to Meyerhoff [68] and Brooks-Matelski [19].

Lemma 6.1. There is a function R(ϵ) with lim_{ϵ→0} R(ϵ) = ∞, such that for an ϵ_{0}

Margulis tube T_{ϵ_{0}}(α) in any hyperbolic 3-manifold, the distance d(ϵ) between ∂T_{ϵ}(α) and ∂T_{ϵ_{0}}(α) for any ϵ ∈ [ℓ(α), ϵ_{0}] is bounded below by

d(ϵ) ≥ R(ϵ).  (6.1)

In particular for ϵ = ℓ(α) this gives r(α) ≥ R(ℓ(α)).

Proof. (Sketch.) In fact R(ϵ) can be written as \frac{1}{2} log(1/ϵ) − c for some positive c. Consider first the lower bound on r(α). Brooks-Matelski apply Jørgensen’s inequality to show that r(α) is at least log c′/|λ(α)|, for a fixed c′ > 0.

In general, |λ| may be much larger than ℓ because of the rotational part. Brooks-Matelski show that some function R(ℓ(α)) exists by a limiting argument. Meyerhoff uses a pigeonhole principle argument to find an iterate of α with small rotational and translational part, and applies Jørgensen’s inequality to this iterate. He also gives a finer estimate using an argument of Zagier.

To get (6.1) for general ϵ, it suffices to observe that the radial projection from an r_{1}-equidistant surface of a geodesic in H^{3} to an r_{2}-equidistant surface, where r_{2} > r_{1}, expands by at most cosh r_{2}/ sinh r_{1} (and a similar bound for horospheres in the parabolic case).

6.2. Teichmüller parameters for Margulis tubes. The shape of the Margulis tube T_{ϵ_{0}}(α) is determined completely by the complex length λ(α), but this dependence is subtle in general (for example if ℓ is extremely small compared to θ then this shape will be sensitive to the continued-fraction expansion of θ/2π).

To improve our picture a little, we will find it useful to define a reciprocal invariant

ω(α) = \frac{2πi}{λ(α)}  (6.2)

whose geometric interpretation is as follows: Assuming α is non-parabolic, a representative g of the conjugacy class ρ(α) acts on H^{3} ∪ (Ĉ \ F ix(g)) with quotient a solid torus H, with torus boundary F_{∞}. Let m be the unique homotopy class in F_{∞} of a meridian of H, and let α′ be homotopic to the core α (More about the choice of α′ in a moment). The pair (α′, m) determines a marking of F_{∞}, so together with the conformal structure inherited from Ĉ, we obtain a point in the Teichmüller space of the torus, viewed as the upper half plane H^{2} (see Section 2.3). This point is exactly ω, if α′ is chosen properly:

The choice of α′ is only determined up to Dehn twist around m, which gives a parabolic action ω ↦ ω/(1 + ω) on the upper half-plane H^{2}, corresponding to the action λ ↦ λ + 2πi in the right half-plane. The normalizing convention θ ∈ (−π, π] which we imposed in Section 4 corresponds to the condition that α′ is of minimal length among the possible choices (strictly minimal when |θ| < π and the boundary choice θ = π taking care of the cases where there is not a unique shortest choice). This constrains ω to the region {ω : |ω − 1| ≥ 1, |ω + 1| > 1}. This is certainly the natural choice when ℓ is small, which will always be the case for us.

Let F_{0} denote the torus boundary ∂T_{ϵ_{0}}(α), with its induced Euclidean metric. The natural identification between F_{0} and F_{∞} via radial projection from the axis induces a marking on F_{0}. If r(α) is sufficiently large, moreover, we find that F_{0} and F_{∞} are uniformly quasiconformally diffeomorphic via this projection (in fact the constant is coth r). Thus, letting ω(F_{0}) denote the Teichmüller parameter of F_{0} with the same normalization, we find that ω and ω(F_{0}) are within bounded hyperbolic (i.e. Teichmüller) distance in H^{2}.

This means that we can use the (marked) geometry of F_{0} or F_{∞} interchangeably to describe the geometry of T_{ϵ_{0}}(α) up to quasi-isometry. In fact we note:

Lemma 6.2. An approximation of ω(α) up to hyperbolic distance D in H^{2} yields an approximation of T_{ϵ_{0}}(α) up to K(D)-bilipschitz diffeomorphism.

Proof. As observed above, if two values of ω differ by a bounded distance then the corresponding loxodromic elements are quasiconformally conjugate at infinity with bounded constant. An extension theorem like Douady-Earle [29] suffices to relate the corresponding Margulis tubes. Alternatively, one can just demonstrate this by hand, working in cylindrical coordinates around the axis.

When α is parabolic we have λ = 0, so we write ω = ∞i. We think of this as the limit obtained when Im ω → ∞. (The limit of a sequence of Margulis tubes with Im ω bounded and Re ω → ∞ is more subtle – see Section 11.2). The torus F_{∞} becomes an infinite annulus with core homotopic to α.

6.3. Margulis tubes in surface groups. As Thurston first observed, there are special constraints on the geometry of Margulis tubes in hyperbolic 3-manifolds homotopy-equivalent to a surface, which arise from the presence of pleated surfaces near every point in the convex hulls of these manifolds. In this section we fix a surface S of finite type and let ρ : π_{1}(S) → PSL_{2}(C) be a discrete faithful typepreserving representation, or “marked surface group”. As usual N = H^{3}/ρ(π_{1}(S)).

We record these facts, due to Thurston and Bonahon, in the following lemma, whose proof we sketch.

Lemma 6.3. There are constants ϵ_{1}, a_{0} and A_{0} depending only on S such that, for any marked surface group ρ : π_{1}(S) → PSL_{2}(C), and any primitive element γ ∈ π_{1}(S) such that ℓ(ρ(γ)) ≤ ϵ_{1},

1. γ represents a simple curve in S.

2. The translation length of γ on ∂T_{ϵ_{0}(γ)} is at most a_{0}.

3. The area of ∂T_{ϵ_{0}}(γ) is at least A_{0}

Proof. (Sketch) It follows from the work of Bonahon [14] and Thurston [86] that any point in the convex core C(N) is within bounded distance of a pleated surface f : S → N in the homotopy class of ρ. (In fact Thurston’s geometric tameness property, which Bonahon establishes for any surface group, implies that there is a continuously parameterized family of surfaces sweeping out C(N), each of which is either a pleated surface or a bounded deformation of one). Let f be such a surface. If X is a component of the ϵ_{0}-thick part of S (in the induced metric) then π_{1}(X) is non-abelian and thus, since f is π_{1}-injective, the image f(X) cannot be contained in T_{ϵ_{0}}(γ). Since there is a uniform bound on the diameter of X depending only on the topological type of S, Lemma 6.1 then implies that for ϵ_{1} sufficiently small (depending on the function R(ϵ)) only the ϵ_{0}-thin part of a pleated surface can touch the ϵ_{1}-thin part of N. Conclusion (1) follows. See also Otal [76], who shows furthermore that a sufficiently short geodesic in N is actually unknotted in a natural sense.

For part (2), let f : S → N be a pleated surface which meets T_{ϵ_{1}}(γ). Since the boundary of the ϵ_{0}-thin part of S (in the induced metric) must again map outside T_{ϵ_{1}}(γ), the translation length of γ on ∂T_{ϵ_{1}}(γ) is at most ϵ_{0}. Since the distance from ∂T_{ϵ_{1}}(γ) to ∂T_{ϵ_{0}}(γ) is uniformly bounded above, we obtain a bound on the translation distance on ∂T_{ϵ_{0}}(γ) as well.

For part (3), let γ′ be the geodesic core of the Margulis tube in S corresponding to γ. there exists a geodesic arc β ⊂ S with both endpoints on γ′, whose intersection with the thick part of S is a nonempty arc of length bounded by D, where D depends again only on the topological type of S. Since β is not deformable rel endpoints into γ′, its image cannot be deformed into T_{ϵ_{0}}(γ) in N.

Lifting to the universal cover, it follows that within distance D of a lift T̃ of the tube T_{ϵ_{0}}(γ) there is a translate T̃′. The translates of T̃′ by the isometry g stabilizing T̃ are all disjoint. Since the radii of these tubes are bounded below in terms of ϵ_{1} via (6.1), one can show that their radial projections to ∂T̃ each have area at least A(D, ϵ_{1}) > 0. Since the projections are all disjoint this gives a lower bound on the area of the quotient torus. This proves (3).

We can reinterpret these results in terms of the reciprocal invariant ω.

Lemma 6.4. If ρ is a surface group and α is a curve of length ℓ(α) ≤ ϵ_{1}, then:

Im ω(α) ≥ c_{1}  (6.3)

where c_{1} depends only on the topological type of the surface. Now suppose that ρ is a punctured-torus group, and let ν_{+}(α) and ν_{−}(α) be the end invariants of ρ, in a normalization of H^{2} where α = ∞. Then for any α ∈ C,

Im ω(α) ≥ Im ν_{+}(α) + Im ν_{−}(α).  (6.4)

The last inequality (6.4) is actually just a variation of Bers’ inequality 1/ℓ(α) ≥ \frac{1}{2}(1/ℓ_{+}(α) + 1/ℓ_{−}(α)) (see [11]). Bers’ inequality holds in much greater generality, but we state it in the case of the punctured torus to indicate how it is expressed using our notation for ν_{±}(α).

Im ω is just the modulus of the torus F_{∞} cut along the Euclidean geodesic α′ representing α (see discussion in Section 6.2). Equivalently it is the reciprocal of the extremal length of α′ in F_{∞}.

As we will also see, Im(ω) estimates the area of the torus F_{0}. In Sections 8 and 10 we will establish one of the central geometric facts about punctured torus groups – that Im ω is also bounded above if the Margulis tube of α lies in the convex core of N, or equivalently that such a Margulis tube boundary has bounded area.

Proof of Lemma 6.4. Assume that ℓ(α) ≤ ϵ_{1}. The Euclidean translation length of α on F_{0} is bounded below by ϵ_{0}, by definition, and above by a_{0}, as a consequence of Part (2) of Lemma 6.3. Since α′ is chosen to be the shortest representative of α in F_{0}, the same bounds hold for α′.

The modulus Im ω(F_{0}) is exactly Area(F_{0})/ℓ_{F_{0}}^{2}(α′). Thus, the length bounds on α′ imply that Im ω(F_{0}) and Area(F_{0}) are within bounded ratio.

Furthermore the bounded hyperbolic distance between ω(F_{0}) and ω(α) implies that Im ω(α) and Im ω(F_{0}) are within bounded ratio. The lower bound on Im ω(α) now follows from Part (3) of Lemma 6.3.

To conclude, let ρ be a punctured torus group, and let us summarize the proof of our variation (6.4) of Bers’ inequality. Recall that g is a representative of α. If Im ν_{+}(α) > 0 then Ω_{+} is nonempty, and its quotient by g is an annulus of modulus at least Im ν_{+} which embeds in the torus F_{∞}. The same is true for ν_{−}, and monotonicity of moduli (by the method of extremal length) implies that the modulus of the torus with distinguished curve α, namely Im ω, is at least the sum of the moduli of the two annuli. (Note that actually Bers’ inequality is slightly stronger since it involves the hyperbolic lengths ℓ_{±}(α), which satisfy π/ℓ_{±}(α) ≥ Im ν_{±}(α). However this inequality is nearly an equality for large Im ν_{±}, and so this form will suffice for us.)

We also remark that the same argument applies in the case that α is parabolic in Ω_{+} or Ω_{−}, so that Im ν_{+} + Im ν_{−} = ∞. In this case the conclusion is that the quotient Ĉ \ F ix(g) must have been an annulus rather than a torus, so that ρ(α) is parabolic.

A final corollary of this is the inequality

| tr(α)| ≥ c_{2}  (6.5)

for a uniform c_{2} > 0 over all curves α in a surface group, since tr → 0 implies λ → ±πi, which means ω → 2, contradicting the lower bound on Im ω. We will use this briefly in Section 10.

7. Geometry of Farey neighbors

The purpose of this section is to investigate some constraints on the geometric relationship between a pair of generators for a punctured torus group. In particular it will follow from Lemma 7.1 that if both of a pair of generators have bounded translation lengths, there is a point in H^{3} where their actions are simultaneously bounded.

A pair α, β ∈ C with α · β = 1 determines a pair of generators A, B for Γ, up to conjugation and inversion, as follows: Realize α and β as curves with a single intersection point x, choose an orientation for each and let A, B be the images under ρ of the resulting elements in π_{1}(S, x). Fixing a positive number D, let core_{D}(A) (similarly core_{D}(B)) denote the subset {p ∈ H^{3} : d(p, Ap) ≤ ℓ(A) + D}. Note that if ℓ(A) > ϵ_{0} then core_{D}(A) is a bounded tubular neighborhood of the axis of A, whereas if ℓ(A) ≤ ϵ_{0} then core_{D}(A) is a horoball or tubular neighborhood of the axis of A, contained in a bounded neighborhood of the Margulis tube of A.

The parabolic commutator condition serves to force these cores together:

Lemma 7.1. There exists D_{2} independent of the punctured-torus group ρ such that, for any generator pair A, B, core_{D_{2}}(A) ∩ core_{D_{2}}(B) ≠ ∅.

Proof. Assume first that neither A nor B is parabolic. The following familiar construction appears for example in Parker-Series [78]. Let P and Q be the axes of A and B respectively, and M their common perpendicular. Let M denote the halfturn around M. Then there are unique lines N orthogonal to P and L orthogonal to Q such that A = NM and B = ML, where N and L are the corresponding half-turns. Hence AB = NL, and its axis is the common perpendicular R of L and N (see figure 7. Note that L and N cannot coincide since the group is not elementary.)

Figure 7. The hexagonal configuration of axes determined by A and B.

The complex distance between N and M is given by μ = λ(A)/2 and between L and M by τ = λ(B)/2. Similarly σ = λ(AB)/2 gives the complex distance between L and N. Let δ denote the complex distance between P and Q. Note that there is an ambiguity of πi in these. However (see Fenchel [32] or Kourouniotis [56]), one can give orientations for all the lines, which resolve the ambiguities in such a way that the hexagon cosine law holds:

cosh σ = cosh μ cosh τ + sinh μ sinh τ cosh δ.  (7.1)

Furthermore, these choices determine matrix representatives for L, M and N, and hence for A, B and AB, for which the identities tr = 2 cosh λ/2 hold without sign ambiguity. The standard trace identity

tr^{2} A + tr^{2} B + tr^{2} AB − tr A tr B tr AB = 2 + tr[A, B]  (7.2)

can therefore be rewritten in terms of μ, τ and σ. Solving this simultaneously with (7.1), we arrive at the relation

sinh^{2} δ sinh^{2} μ sinh^{2} τ = \frac{tr[A, B] − 2}{4}.  (7.3)

Furthermore we have tr[A, B] = −2, as a consequence of the parabolic commutator condition. More precisely, the sign of tr[A, B] is well-defined since it is a commutator, and it must be −2 rather than +2 so that the group will be non-elementary (see e.g. [46, 50, 16]). It follows that

sinh^{2} δ sinh^{2} μ sinh^{2} τ = −1  (7.4)

(cf. (2.2) of Parker-Series). This will give us an estimate for the length Re δ of the segment G of M between P and Q.

We will next determine the subsegments of G belonging to core_{D}(A) and core_{D}(B) respectively, for appropriate D, and use the estimate on δ to show that these segments cover G, and in particular have the desired intersection point.

Figure 8. Projections of N and L to G yield segments G_{1}, G_{2} which are in the cores of A and B.

Let N^{+} denote the ray of N with endpoint on P whose projection to M is a subinterval G_{1} of G, and let Z_{1} be the geodesic from the endpoint of N^{+} at infinity to its projection on an endpoint of G_{1} (see Figure 8). Let δ_{1} denote the complex distance between P and Z_{1}. Another application of hyperbolic trigonometry to the quadrilateral X_{1} bounded by G_{1}, Z_{1}, N^{+} and P (see any of [32], [56] or [78] for how to derive this from the hexagon cosine law by considering degenerate hexagons) yields

sinh^{2} δ_{1} sinh^{2} μ = 1,  (7.5)
sinh^{2} δ_{2} sinh^{2} τ = 1.

Where G_{2}, δ_{2} are defined similarly with respect to Q and L.

We claim that

G_{1} ⊂ core_{D}(A)  (7.6)

where D = 4 sinh^{−1}(1), and similarly for G_{2} and core_{D}(B). It is easy to see (an ideal triangle is the extremal case) that for any triangle Δ(xyz) and point p on a side xy, the distance from p to at least one of the other two sides is no more than sinh^{−1}(1). (In fact this is the proof that hyperbolic space is “delta-hyperbolic” in the sense of Gromov). Similarly, if the triangle is a right triangle and p is on a leg, the same bound holds for the distance from p to the hypotenuse. Applying this to the quadrilateral X_{1} suitably subdivided into two triangles, we find that for any p ∈ G_{1}, dist(p, P ∪ N) ≤ 2 sinh^{−1}(1). If this holds for the distance to P then clearly dist(p, Ap) ≤ 4 sinh^{−1}(1) + ℓ(A). If it holds for the distance to N then, since A = NM and M fixes p, we have dist(p, Ap) ≤ 4 sinh^{−1}(1). Either way p ∈ core_{D}(A), so that (7.6) is established. The same argument applies to G_{2}.

Finally, we show that G\(G_{1}∪G_{2}) is (if nonempty) a segment of bounded length. Indeed, (7.4) and (7.5) together imply

sinh^{2} δ = − sinh^{2} δ_{1} sinh^{2} δ_{2}.  (7.7)

Now taking absolute values and applying the elementary inequality \frac{1}{2}(e^{| Re z|} − 1) ≤ | sinh z| ≤ e^{| Re z|} and a bit of algebra, we deduce

|G| ≤ |G_{1}| + |G_{2}| + log 3  (7.8)

where | · | denotes length in H^{3}.

Thus the \frac{1}{2} log 3-neighborhoods of G_{1} and G_{2} meet, and are contained in core_{D_{2}}(A) and core_{D_{2}}(B), respectively, where D_{2} = 4 sinh^{−1}(1) + log 3. This concludes the proof when A and B are loxodromic.

If one or both of them are parabolic, we can make a limiting argument to obtain a G which may be an infinite ray or all of M, and G_{1} and G_{2} which may be subrays. The G_{i} are still contained in core_{D}(A) and core_{D}(B), respectively, and the length bound on G \ (G_{1} ∪ G_{2}) holds as before. Thus the lemma follows in general.

8. Connectivity and a-priori bounds

In this section we exploit the connection between topological properties of hyperbolic surfaces, and combinatorial properties of the graph C, to obtain our initial bound on the length of pivots. We first prove Lemma 8.1, which shows that the set of curves in C satisfying a length bound is connected. As a consequence we obtain Lemma 8.2 which gives a preliminary bound on the lengths of all vertices in the pivot sequence.

8.1. Connectivity in the curve complex. Let L_{0} be the bound on the length of the shortest geodesic in a punctured torus with a singular hyperbolic metric (see Lemma 2.1). Fixing a punctured-torus group ρ, for any L ≥ L_{0} let C(L) denote the subgraph of C spanned by the vertices {α ∈ C : ℓ(α) ≤ L}.

Lemma 8.1. C(L_{0}) is connected.

Remark. Compare Theorem 1 of Bowditch [16], which has a similar conclusion but applies to the more general setting of punctured torus groups that are not necessarily discrete or faithful! Bowditch’s proof is completely different, employing a beautiful algebraic/combinatorial method.

Proof. Consider α_{0}, α_{1} ∈ C(L_{0}). Let g_{i} : S → N for i = 0, 1 be simplicial hyperbolic surfaces in the homotopy class of ρ, adapted to α_{i}. In order to streamline the proof (and constants), we make a brief appeal to the technique of pleated surfaces. As mentioned earlier, if an infinite sequence of vertex slide moves around α_{i} is performed starting with g_{i}, the resulting family of induced metrics converges to a smooth hyperbolic metric σ_{i} induced by a limiting map g′_{i} (a pleated surface), which still takes α_{i} to its geodesic representative.

Let β_{i} be a shortest curve in (S, σ_{i}). Then ℓ(β_{i}) ≤ ℓ_{σ_{i}}(β_{i}) ≤ L_{0}, so β_{i} ∈ C(L_{0}), and by lemma 2.1 we have α_{i} · β_{i} = 1, so that it suffices to show that β_{0} and β_{1} can be connected by a path in C(L_{0}).

Applying lemma 5.2, g_{0} and g_{1} can be connected by a continuous family g_{t} of maps, for t ∈ [0, 1], such that each g_{t} is a simplicial hyperbolic surface. Because of the way we constructed g′_{i}, we may extend this to a family (still called g_{t}) of simplicial hyperbolic surfaces interpolating between g′_{0} and g′_{1}. Let σ_{t} denote the (possibly) singular hyperbolic metric induced on S by g_{t}.

Let τ_{t} denote the hyperbolic metric in the conformal class of σ_{t}. Then τ_{t} = σ_{t} for t = 0, 1 and τ_{t} ≥ σ_{t} pointwise, by Ahlfors’ lemma (see section 2.3). Note also that the lengths ℓ_{τ_{t}} of homotopy classes vary continuously with t. For each t consider the curves of minimal τ_{t}-length (necessarily no longer than L_{0}). If there are two such curves β^{1} and β^{2}, then by Lemma 2.1 they are Farey neighbors.

Since {τ_{t}} is a compact family of metrics, the set B of curves β appearing as shortest curve for some τ_{t} is finite. If X(β) is the closed set of t values for which β ∈ B is shortest in τ_{t} we have shown that the adjacency graph of {X(β)}_{β∈B} is isomorphic to the subgraph of C spanned by B. Connectivity of the interval [0, 1] now implies that B is connected, and in particular there is a path between β_{0} and β_{1} of curves whose lengths in some τ_{t}, and hence in N, are at most L_{0}.

8.2. Bounds on pivots. As a consequence of the connectivity lemma 8.1, we may now obtain the simplest part of the Pivot Theorem:

Lemma 8.2. There is a fixed L_{1} such that ℓ(α_{n}) ≤ L_{1} for all pivots α_{n}.

Proof. Recalling the notation from Section 4, let e be an edge of E, the set of edges separating α_{−} from α_{+}. If ν_{−} is an irrational boundary point (so ν_{−} = α_{−}), Lemma 5.1 implies there exists some γ_{−} ∈ C(L_{0}) on the same side of e as ν_{−}. If ν_{−} is not irrational then α_{−} ∈ C is already in C(L_{0}), so let γ_{−} = α_{−}. The same applies for ν_{+} and γ_{+}. Now e separates γ_{−} from γ_{+}, and by Lemma 8.1 there is a path between them in C(L_{0}). This implies that e contains a vertex of C(L_{0}).

An internal pivot α_{n} by definition is the common point of at least two edges e_{0} and e_{1} of E (barring the exceptional case (3) in Section 4), which we may choose to be edges of a common Farey triangle. Thus, if ℓ(α_{n}) > L_{0}, the other vertices β_{0} and β_{1} of e_{0}, e_{1} have lengths bounded by L_{0}. Representing α_{n} and β_{0} by a generator pair A, B, we know that β_{1} can be represented by A^{±1}B. Now the trace identity (7.2) gives a bound for | tr A| in terms of | tr B| and | tr AB|, and in particular bounds ℓ(A) = ℓ(α_{n}) by some L_{1}.

In the exceptional case where E is a singleton {e}, we note that one vertex e is in C(L_{0}), and the other has as neighbors both of α_{−} and α_{+} and so it is also in C(L_{1}) by the same argument.

9. Controlled surfaces and building blocks

In this section we begin the construction of a geometric model for the quotient manifold of a punctured torus group. In Sections 9.2 and 9.3 we construct particular geometrically controlled maps of surfaces based only on some length bounds and end-invariant data. In Sections 9.4 and 9.5 we use these surfaces to construct “building blocks” which will later serve, in particular, to control the shape of Margulis tubes. Lemma 9.3 will summarize the basic properties of these blocks.

We remark that the cases of most interest here are those of “internal” blocks, and these are also the simpler cases. On a first reading one can suppress the discussion of boundary blocks (e.g. all of §9.3), which are intended to describe the boundaries of the convex core. There are at most two boundary blocks, and typically an infinite number of internal blocks.

To streamline the discussion in this section and those that follow, we frequently use terms like “bounded” or “lipschitz” to denote bounds which are uniform in the sense that they are independent of the particular punctured torus group.

9.1. Standard metrics. Let ν be a conformal structure on the punctured torus S, with marking α, β such that α is shortest in ν. Then Im(τ(S, ν, α, β)) ≥ \sqrt{3}/2 by the discussion in Section 2.3. Let σ^{e} be the Euclidean metric in the conformal class ν (incomplete at the puncture) such that α has length 1. We can cut S into an open annulus A and a closed punctured annulus B, with σ^{e}-geodesic boundaries homotopic to α, so that the puncture is on the center circle of B and B has width 1/2, and A has width at least (\sqrt{3} − 1)/2.

Let σ^{h} be the complete hyperbolic metric in the conformal class ν. By standard techniques of conformal mappings (see e.g. [26, Thm. 4.3]), we know that σ^{h} and σ^{e} are within bounded ratio of each other in B minus a disk of σ^{e}-radius 1/8 (say) around the puncture. It will be useful to fix a hybrid metric σ^{m} which equals σ^{e} outside a disk of σ^{e}-radius 1/4 around the puncture, equals σ^{h} inside a disk of σ^{e}-radius 1/8 around the puncture, and is within fixed bounded ratio of both in the remaining annulus.

Let us also assume ϵ_{0} is sufficiently small that the radius 1/8 disk contains the ϵ_{0}-Margulis tube for the cusp of σ^{h}.

9.2. Halfway surfaces. We can apply Lemma 7.1 on cores to conclude that, for any two neighbors in C with bounded lengths, there is a simplicial hyperbolic surface that maps them both to curves of bounded length.

Given α, β which are neighbors in C, consider the conformal structure ν_{α,β} whose Teichmüller parameter is τ(S, ν_{α,β}, α, β) = i. Let σ_{α,β}^{e}, σ_{α,β}^{h} and σ_{α,β}^{m} be the associated metrics as above. Note that σ_{α,β}^{e} makes S a square torus in which both α and β have length 1 and are orthogonal, and that σ_{α,β}^{h} and σ_{α,β}^{m} are within bounded ratio of each other. (The choice of a square torus is for convenience – any fixed modulus would do as well).

The following lemma gives conditions for mapping this surface in a controlled way into N:

Lemma 9.1. Fix L ≥ L_{0} and let α, β be a pair of Farey neighbors contained in C(L). Then there exists a simplicial hyperbolic surface f_{α,β} : S → N in the homotopy class of ρ, which is L′-Lipschitz with respect to the metrics σ_{α,β}^{m} and σ_{α,β}^{h}.

In particular, f_{α,β}(S) does not meet any ϵ_{2}-Margulis tube other than that of the main cusp.

The constants L′, ϵ_{2} depend only on L.

Proof. Apply the notation given in Section 7 for the lift to H^{3} and the group elements A and B corresponding to α and β. Let T denote the triangulation whose closed curves are α and β. Let f_{1} and f_{2} denote simplicial hyperbolic surfaces with triangulation T , adapted to α and β respectively. The type (3) elementary move from f_{1} to f_{2} involves a step in which the vertex of the triangulation (or rather its lift to H^{3}) travels from T_{A} to T_{B} along their common perpendicular. By Lemma 7.1, there is an intermediate map f_{t} for which the vertex is in the intersection of core_{D_{2}}(A) and core_{D_{2}}(B). At this point both A and B have translation distance bounded by L + D_{2}, and hence the images of the corresponding triangulation edges have this length bound.

If we realize T so that its edges are geodesic with respect to σ_{α,β}^{h}, and then parameterize f_{t} with constant speed on the edges, we find that it is Lipschitz on the 1-skeleton (this can be made to work on the edges going to the cusp, as well). The extension to the whole surface can easily be made (uniformly) Lipschitz with respect to σ_{α,β}^{h}. Thus we let this be the definition of f_{α,β}.

Since in particular there is a bound on the diameter of the non-cuspidal part of f_{α,β}(S), it follows from Lemma 6.1 that there is a corresponding ϵ_{2} so that the map avoids all ϵ_{2}-Margulis tubes other than the main cusp.

9.3. Boundary surfaces. In order to control the shape of Margulis tubes that are not contained in the convex core, we will need to construct a version of the controlled maps of the previous section, associated to boundary components of the convex core. In what follows we consider the + end, but of course the same considerations apply to both ends.

Suppose that e_{+} is geometrically finite and let S_{+} denote the surface at infinity Ω_{+}/ρ(π_{1}(S)). Unless α_{+} is parabolic, S_{+} is naturally identified with S and has conformal structure ν_{+}. Noting that α_{+} is shortest in ν_{+}, let σ_{+}^{h}, σ_{+}^{e} and σ_{+}^{m} denote the associated metrics as in section 9.1. Let A_{+} and B_{+} be the annulus and punctured annulus described in that section, with σ^{e}-geodesic boundaries isotopic to α_{+}.

If α_{+} is parabolic then S_{+} is a thrice punctured sphere, σ_{+}^{e} makes S_{+} a biinfinite Euclidean annulus with puncture and A_{+} is a union of two semi-infinite annuli. The other parts of the definition are unchanged.

Let C_{r}(N) denote the r-neighborhood of the convex core of N. Fixing a positive value for r, let Ĉ(N) denote the union of C_{r}(N) with all ϵ_{0} Margulis tubes other than that of the main cusp. (Note that there are at most two such tubes that are not already contained in C(N), corresponding to α_{+} and α_{−} if these are sufficiently short).

Let ∂_{+}Ĉ(N) and ∂_{+}C_{r}(N) denote the components of ∂Ĉ(N) and ∂C_{r}(N), respectively, that face the end e_{+}. Note that either ∂_{+}Ĉ(N) = ∂_{+}C_{r}(N), or ∂_{+}Ĉ(N) = U_{+} ∪ R_{+}, where U_{+} is an annulus in the Margulis tube boundary ∂T_{ϵ_{0}}(α_{+}), and R_{+} is a punctured annulus isotopic to the complement of α_{+} in S.

The following lemma gives the properties of the boundary maps we shall build:

Lemma 9.2. If e_{+} is geometrically finite, there exists a homeomorphism

Π_{+} : S_{+} → ∂_{+}Ĉ(N),

in the homotopy class determined by ρ, such that:

Π_{+} is K-bilipschitz with repsect to σ_{e}^{m} on S_{+} and the induced metric from H^{3} on ∂_{+}Ĉ(N). It takes A_{+} to U_{+} by a K-quasiconformal homeomorphism, where the constant K is independent of ρ. The corresponding statement holds for e_{−}.

Proof. We construct Π_{+} as follows. Recalling that we have fixed some small r, let Π_{r} : S_{+} → ∂_{+}C_{r}(N) be the orthogonal projection from infinity (see Epstein-Marden [30]). For any x ∈ S_{+} there is a geodesic ray m_{x} based at y = Π_{r}(x) and terminating in x at infinity (it may be helpful to consider this to be happening in the universal cover). This ray meets T_{ϵ_{0}}(α_{+}) if and only if y ∉ int(R_{+}), and in such a case let z be the (unique) point of intersection of m_{x} with U_{+} ⊂ ∂T_{ϵ_{0}}(α_{+}). We define Π̂_{r}(x) = z in this case, and Π̂_{r}(x) = Π_{r}(x) if Π_{r}(x) lies in R_{+}.

By Theorem 2.3.1 of [30], Π_{r} is a quasiconformal and bilipschitz homeomorphism between (S_{+}, σ_{+}^{h}) and ∂_{+}C_{r}(N), where the constants depend only on the choice of r. It is also easy to see that on any point z = Π̂_{r}(x) of U_{+} which is a distance r or more from ∂_{+}C_{r}(N), the tangent plane to U_{+} at z makes an angle with the ray m_{x} which is bounded away from 0 (this follows from the fact that ∂T_{ϵ_{0}}(α_{+}) is an equidistant surface from the core geodesic of α_{+}, which lies inside C(N).) Thus, by an elementary calculation, the map Π̂_{r} at x is quasiconformal with a bound again depending on r.

On the other hand the part of U_{+} which is within distance r consists of two annuli of bounded radius adjacent to ∂U_{+}, and they project to annuli Y_{1}, Y_{2} in ∂_{+}C_{r}(N) by a distance-decreasing map.

Since Π_{r} is bilipschitz with respect to σ_{+}^{h}, and the surfaces R_{+} and Y_{1}, Y_{2} are of standard shape (R_{+} is bilipschitz equivalent to the complement of a Margulis tube in a punctured torus, and Y_{i} are bilipschitz equivalent to Euclidean annuli of fixed height and girth), there is a bilipschitz homeomorphism k : S_{+} → S_{+} isotopic to the identity so that Π_{r}^{−1}(R_{+}) = k(B_{+}) and Π_{r}^{−1}(Y_{j}) = k(X_{j}) where X_{j} are Euclidean annuli in A_{+} of bounded modulus (j = 1, 2). Then letting Π′ = Π̂_{r} ∘ k, we have a map which is bilipschitz as desired from B_{+} to R_{+}, and from A_{+} to U_{+} is Lipschitz on bounded Euclidean neighborhoods of the boundary, and quasiconformal on the rest. Lemma 2.2 implies that Π′|_{A_{+}} is homotopic to a quasiconformal map with constants bounded in terms of the previous choices, and which is bilipschitz with respect to the Euclidean metrics on the annuli. We let Π_{+} be the resulting map.

9.4. Building Blocks. A building block B is a copy of S × [0, 1], with a certain metric defined on a subset of B. Blocks are associated to local configurations of pivots in C(L), and come in several varieties (there are also blocks with parabolics, which are topologically different).

Internal blocks. Let α, β, γ be distinct vertices of C(L) such that β and γ are neighbors of α. We define the associated internal block B = B_{α,β,γ} as follows:

Define two metrics σ_{0} = σ_{α,β}^{m} and σ_{1} = σ_{α,γ}^{m}, as in Sections 9.1 and 9.2. After isotopy we may assume that the open annulus A from Section 9.1 is the same for both, and that the two are equal on the complementary annulus B. Let A′ be an open 1/8-neighborhood of A in both metrics.

Within B let U = A × (1/4, 3/4) and U′ = A′ × (0, 1) be nested open solid tori, and let B^{0} = B \ U. We let ∂_{0}B and ∂_{1}B denote S × {0} and S × {1}, respectively.

Place the metrics σ_{i} on the boundaries S × {i}, respectively, and extend to a metric on B^{0} as follows: let S_{0} = S \ Q where Q is the ϵ_{0}-cusp of either metric, and B_{0} = B \ Q. Let Q_{B} = Q × [0, 1].

On S_{0} × [0, 1/4] we place the product metric σ_{0} × dt^{2} where t ∈ [0, 1/4], and on S_{0} × [3/4, 1] we similarly place σ_{1} × dt^{2}. On B_{0} × [1/4, 3/4], where both metrics are equal, we can place a product metric in the same way, using either metric. On Q_{B} we may place the geometry of a three-dimensional parabolic cusp: namely, identify Q_{B} = Q×[0, 1] with a segment of the ϵ_{0}-Margulis tube of a rank-1 parabolic group, so that ∂Q × [0, 1] is an annular section of length 1 of the boundary, and each slice Q × {t} is a totally geodesic 2-dimensional Margulis tube embedded orthogonally to the boundary.

Boundary blocks. Suppose e_{+} is geometrically finite, let α_{+} ∈ C be shortest in ν_{+}, as in Section 4, and let β be any neighbor of α_{+} which is also in C(L). To this pair we associate a boundary block B = B_{α_{+},β} as follows.

Suppose first that α_{+} is not parabolic. Let the metric σ_{0} be σ_{α_{+},β}^{m}, just as before. The metric σ_{1} is σ_{+}^{m} from Section 9.3. Again after isotopy we may assume that the annuli A and B of section 9.1 are the same for both metrics, as well as the cusps Q, and the metrics are equal on B except possibly on a small collar around the cusp, where they are in bounded ratio. Define A′ so that it excludes the cusp, and contains a 1/8 neighborhood of A in each metric (it can’t be exactly the 1/8 neighborhood on both because they are not quite equal), and define the solid torus U′ as before. However, we let U be A × (1/4, 1). The product metric on B \ U is defined as before, except on the neighborhood of the puncture where we interpolate between the two slightly different metrics.

If N has a parabolic cusp in its e_{+} end (so ν_{+} = α_{+} ∈ C) then we redefine the topological structure of B_{α_{+},β} to be (S × [0, 1]) \ (a × {1}), where a is a curve representing α, and note that in this case σ_{+}^{m} on the punctured cylinder S_{+} ≅ S \ a has infinite area. The rest of the construction is the same. In this case we say that B_{α_{+},β} is a boundary block with parabolic.

For the opposite end e_{−} we construct B_{α_{−},β} in just the same way, except that the roles of 0 and 1 are reversed, and U for example is A × (0, 3/4).

Finally, suppose α_{+} = α_{−}, that is the same curve is short on both boundaries. We define a double boundary block B ≡ B_{α_{±}} as follows: Let σ_{0} and σ_{1} be σ_{−}^{m} and σ_{+}^{m} respectively, again with a common decomposition into annuli A and B. Now we define U = A × (0, 1) and continue as before.

We can again have a block with parabolics in this case, but note that α_{+} = α_{−} can be parabolic at most on one of the two ends.

9.5. Block maps and the figure-8 argument. The following lemma gives us controlled maps of our blocks into the manifold N.

Lemma 9.3. Let B denote a block constructed as in the previous section. That is, B = B_{α,β,γ} for α, β, γ in C(L); or B = B_{α_{+},β}, B_{α_{−},β} or B_{α_{±}}. In the latter cases denote α = α_{+} or α = α_{−}.

There is a map H_{B} : B → Ĉ(N), in the homotopy class determined by ρ, with the following properties:

1. H_{B} is Lipschitz on B^{0}, with uniform constant K.

2. H_{B} respects the Margulis tube structure:

If ℓ(α) ≤ ϵ_{3} then H_{B} maps (U, ∂U) to (T_{ϵ_{0}}(α), ∂(T_{ϵ_{0}}(α)).

Furthermore, H_{B}(B^{0}) is outside every ϵ_{0}-Margulis tube whose core length is no more than ϵ_{3}, except for the main cusp.

3. For boundary blocks, H_{B} is boundary-preserving:

If α = α_{+} (respectively α = α_{−}) then H_{B} takes ∂_{1}B (resp. ∂_{0}B) to ∂_{+}Ĉ(N) (resp. ∂_{−}Ĉ(N)) by a K-bilipschitz homeomorphism of degree 1.

4. H_{B} respects the main cusp structure:

It takes (Q_{B}, ∂Q_{B}) to (Q_{N}, ∂Q_{N}) by a proper map, and takes B \ Q_{B} to N \ Q_{N}, where Q_{N} is the ϵ_{0} Margulis tube of the main cusp of N.

The constants K, δ and ϵ_{3} depend only on L.

Proof. We begin by defining H = H_{B} on ∂B. Take the case of an internal block. Then by construction the boundary metrics σ_{0} and σ_{1} are σ_{α,β}^{m} and σ_{α,γ}^{m} respectively, and Lemma 9.1 gives us homotopic maps h_{0} = f_{α,β} and h_{1} = f_{α,γ} that are Lipschitz with respect to σ_{0} and σ_{1} respectively, and avoid all ϵ_{2}-Margulis tubes except for the main cusp. It is easy to show (see e.g. [70, Lemma 4.1]) that, since π_{1}(B) is non-abelian and the maps are π_{1}-injective, there is a unique normalized geodesic homotopy H_{1} between h_{0}|_{B} and h_{1}|_{B}, that is, a homotopy whose tracks H_{1}|_{x×[0,1]}, x ∈ B, are geodesics parametrized at constant speed.

We can define a preliminary extension H_{2} : B \ U′ → N to be the unique map which restricts to h_{i} on S × {i} and which restricts on B′ × [0, 1] to the geodesic homotopy H_{1}. (Here B′ = S \ A′).

We can do the same thing for a noninternal block. Suppose for example we have B = B_{α_{+},β}. The map h_{0} is defined to be f_{α,β} as before (letting α = α_{+}), but h_{1} is taken to be the map Π_{+} : S → ∂_{+}Ĉ(N), constructed in Section 9.3. The case with parabolic is handled similarly.

If α = α_{+} = α_{−} and B = B_{α_{±}}, We define h_{0} = Π_{−} and h_{1} = Π_{+} and continue as above. In each case we extend h_{0} and h_{1} to the rest of B′ × [0, 1] with the geodesic homotopy.

We now prove that H_{2} is uniformly Lipschitz on B \ U′:

Let η be a “figure-8” curve in B′, as in figure 9. The unoriented free homotopy class of η only depends on the curve α, and there is a uniform bound on the length of the geodesic representative of η in either σ_{0} or σ_{1}. Since σ_{0} and σ_{1} are within uniformly bounded ratio in B′, we may choose η to have length bounded by some ℓ_{1} in both.

Figure 9. The figure-8 curve in S \ α

Think of η as the union of two simple loops η_{1} and η_{2} with a common point y. These determine distinct elements of π_{1}(S, y) in the conjugacy class of α. Choosing a lift of H_{2}({y} × {0}) to H^{3}, we represent η_{1} and η_{2} as distinct isometries A_{1}, A_{2}. The endpoints of the lift of the geodesic H_{2}({y}×[0, 1]) give us two points p, q ∈ H^{3} where the translation distances of both A_{1} and A_{2} are bounded above by ℓ_{1}. Bounding the length of the homotopy now reduces to bounding dist(p, q).

Suppose dist(p, q) ≥ D for some D. Let ε(D) denote a function of the form ce^{−D} where c is some uniform constant. The geodesic \overline{pq} has image A_{i}(\overline{pq}) (i = 1, 2) whose endpoints are at most ℓ_{1} from those of \overline{pq}. It follows by a bit of hyperbolic trigonometry that if, say, D > 3ℓ_{1}, then A_{i}(\overline{pq}) comes within ε(D) of the midpoint m of \overline{pq}, and the two geodesics have directions differing by at most ε(D) at their closest point, as measured in the tangent bundle to H^{3}. In particular A_{i}(m) is ε(D)-close to \overline{pq}. It follows that [A_{1}, A_{2}], which is non-trivial, translates m by no more than ε(D). However since A_{1} and A_{2} translate m no more than ℓ_{1}, when D is sufficiently large this violates Jørgensen’s inequality, and this gives us our upper bound. (See [70, Lemma 4.2] for a similar argument in a somewhat different setting.)

The bounds the length of the track H_{2}({y} × [0, 1]) and because of the bound on the length of η, we obtain a bound for the other tracks on η as well. Now since η is a deformation-retract of B and the thick part B′ \ Q of B′ in either metric has uniformly bounded diameter, we obtain a bound on the track length of H_{2} in B′ \ Q. Since H_{2} is parametrized at constant speed, this gives the desired Lipschitz bound on H_{2} in (B′ \ Q) × [0, 1]. In the cusp region Q, a bound is immediate from the fact that h_{0} and h_{1} are standard embeddings.

We next extend H_{2} to the rest of B.

Consider first the internal case. Let V′ be the boundary torus of U′. Consider the solid-torus covering space Ñ of N whose geodesic core is a homeomorphic lift of α^{∗}. The map H_{2}|_{V′} lifts to H̃_{2} : V′ → Ñ, and its image lies in a bounded neighborhood of the boundary torus F_{0} of the ϵ_{0}-Margulis tube of Ñ (a homeomorphic lift of T_{ϵ_{0}}(α)).

If ℓ(α) is at most some ϵ given by Lemma 6.1, the radius of the tube is at least 1 + diam(H̃_{2}(V′)), and hence the image is disjoint from a 1-neighborhood of the core of Ñ. Thus the radial projection taking Ñ minus its core to the torus F_{0} is uniformly Lipschitz on H̃_{2}(V′).

We can therefore use this projection, parametrized as a deformation retraction, to extend H̃_{2} to a collar V′ × [0, 1] so that H̃_{2} takes V′ × {0} to F_{0}, and so that the map is Lipschitz with uniform constant. Projecting downstairs to N we obtain an extension of H_{2} to the collar U′ \ U, which is still Lipschitz with some uniform constant since the collar is bilipschitz equivalent to V′ × [0, 1). The map is now defined on ∂U and extends to map the rest of U into T_{ϵ_{0}}(α) because it is in the right homotopy class. (Note that there is no Lipschitz condition to be satisfied on U, which in fact does not yet have a metric). Let H_{3} denote the resulting map.

If ℓ(α) ≥ ϵ, then the lifted map lies in a bounded neighborhood of the axis. Radial projection to the axis is then uniformly Lipschitz and we use this to extend the map to the the collar so that ∂U maps to the axis. Again, the map extends to some H_{3} on the rest of U for topological reasons.

The extension for boundary blocks works similarly – we lift to a cover in which T_{ϵ_{0}}(α), which is still a solid torus, lifts homeomorphically, and apply radial projection as before to show that the map on the subannuli of V′ which do not map to ∂Ĉ(N) can be pushed rel boundary to the boundary of the Margulis tube. We use this homotopy to define the map on U′ − U so that ∂U maps as required. It is easy to see that this can be made Lipschitz with uniform constants.

By construction, for some uniform ϵ_{3}, H(B \ U) stays out of the 2ϵ_{3}-thin part (minus main cusp). This is because B \ (U ∪ Q_{B}) (the thick part of the block) has bounded diameter and hence can penetrate only a bounded amount into any thin part. Thus for any curve γ of length less than ϵ_{3}, the region T_{ϵ_{0}}(γ) \ T_{2ϵ_{3}}(γ) retracts onto ∂T_{ϵ_{0}}(γ) by radial projection, which is uniformly Lipschitz. We may compose H_{3}|_{B\U} with this retraction to obtain a new map H_{4} which stays out of this Margulis tube, and still satisfies uniform Lipschitz bounds. Thus we have Part

A similar argument using the radial projection in the main cusp gives us an additional Lipschitz deformation of H_{4} to get H_{5} which satisfies Part (4). This will be our final map H_{B}.

Part (3) of the lemma follows from Lemma 9.2, with the statement about degree following from our orientation conventions applied to B and to N.

It is worth comment at this point that the restriction H|_{∂U} : ∂U → ∂T_{ϵ_{0}}(α) has not been shown to be homotopic to a homeomorphism. In fact one can construct blocks for which this map has degree different from 1. This will be remedied in the next section by a global argument.

10. Proof of the pivot theorem

We begin by piecing together a selected sequence of the blocks from the previous section.

Recall from Section 4 that the pivot sequence P is written {α_{n}}_{n∈J} where J is a finite, half-infinite or bi-infinite interval in Z. The subset P_{0} of internal pivots leaves out α_{−} = α_{0} and/or α_{+} = α_{p+1}, if these occur in P.

Hence for each internal pivot α_{n} the pivots α_{n−1} and α_{n+1} are defined. Since, by Lemma 8.2, all of P is contained in C(L_{1}), we may construct a block B_{n} = B_{α_{n},α_{n−1},α_{n+1}} and associated map H_{n} : B_{n} → Ĉ(N), as in Sections 9.4 and 9.5.

Note that L_{1} determines values for the constants ϵ_{2}, ϵ_{3}, as described in the previous section. Furthermore, a small positive constant r was chosen in Section 9.3, which determines the definition of Ĉ(N).

If α_{−} is in P and different from α_{+} we let B_{0} = B_{α_{−},α_{1}}, and if α_{+} is in P and different from α_{−} we let B_{p+1} = B_{α_{+},α_{p}}. If α_{−} = α_{+} (hence P_{0} is empty) we have just one (double boundary) block which we label B_{0} = B_{α_{±}}. Again we define the maps H_{n} appropriately, as in Section 9.5.

Next we observe that for consecutive n, n+1 ∈ J, there is a natural identification between ∂_{1}B_{n} and ∂_{0}B_{n+1} – namely, the identity on S induces an isometry of the metrics on these surfaces. Thus we can glue together all the blocks in sequence to obtain

M = ⋃_{n∈J} B_{n}

with a map H : M → Ĉ(N), restricting to H_{n} on each B_{n}. On the subset M^{0} = M \ ∪U_{n}, the map H is Lipschitz. (In fact on the complement of M^{0} we have not yet defined a metric).

Note that in the case of two degenerate ends the sequence is biinfinite and M is homeomorphic to S × R. In other cases it will have one or two boundary components, and in all cases M is homeomorphic to Ĉ(N).

Let us also denote by Q_{M} the “main cusp” of M, that is the union of cusps Q_{B_{n}} over all blocks B_{n}.

Lemma 10.1. The map H : M → Ĉ(N) is proper and has degree 1.

Proof. H takes boundary to boundary, the image of the thick part of each block has bounded diameter, and the pivot geodesics α_{n}^{∗} leave every compact set as |n| → ∞. It follows that H restricted to M \ Q_{M} is proper to Ĉ(N) \ Q_{N}. On each cusp block Q_{B}, H is already proper and in fact by construction the function x ↦ dist(H(x), ∂Q_{N}) is proper on Q_{B}. It follows that H is proper as a map from M to Ĉ(N). Therefore the degree of H is well-defined.

If Ĉ(N) has nonempty boundary then, by the last part of Lemma 9.3, H has degree 1 on the boundary and we are done. If not, then M = S × R, and we know that H acts as a homotopy equivalence in the homotopy class of an orientation preserving homeomorphism on S. Thus it has degree 1 if the order of the ends is preserved, degree −1 if it is reversed, and degree 0 if both ends of M are mapped to the same end of N. Because the pivot geodesics α_{n}^{∗} exit the end e_{+} when n → ∞ and e_{−} when n → −∞, our orientation convention applied to both M and N implies that H has degree 1.

With this map in place we can complete the proof of the Pivot Theorem 4.1. Let us show that only the pivots can have length less than ϵ_{3}: Since the map

H has non-zero degree, in particular it covers all of Ĉ(N) and hence meets the Margulis tube of every short geodesic. However, by the construction it can only meet the ϵ_{3}-Margulis tubes of the pivots. This proves part (1) of Theorem 4.1

We now consider the proof of part (2) of Theorem 4.1, which we recall asserts

ω(α_{n}) ≈ ν_{+}(α_{n}) − \overline{ν}_{−}(α_{n}) + i  (10.1)

for any pivot α_{n}, where ω(α_{n}) = 2πi/λ(α_{n}).

Let α_{n} be a pivot of length greater than ϵ_{3} (and no greater than L_{1}). Then λ(α_{n}) is bounded hyperbolic distance from 1 in the right half-plane, so ω(α_{n}) is bounded hyperbolic distance from 2πi in the upper half plane H^{2}. Thus, to establish (10.1) it suffices to find an upper bound for Im(ν_{+}(α_{n})) + Im(ν_{−}(α_{n})), and for | Re(ν_{+}(α_{n})) − Re(ν_{−}(α_{n}))|. The former comes directly from the variant of Bers’ inequality, given in part (6.4) of Lemma 6.4.

The difference of real parts is just w(n) up to an error of at most 2 by (4.2), where if α_{n} is internal then w(n) is the number such that α_{n+1} = D_{α_{n}}^{w(n)}α_{n−1}. Representing α_{n}, α_{n−1} by a generating pair A, B, we may represent α_{n+1} by A^{±w(n)}B. An easy computation shows that tr A^{k}B is of the form tr_{k} = ae^{kλ/2} + de^{−kλ/2}, where λ = λ(A). Since we have an upper bound on length (hence trace) of all pivots, we have an upper bound on tr_{0} and tr_{w(n)}. On the other hand we have a positive lower bound | tr_{k} | ≥ c_{2} > 0 by (6.5), and a lower bound of ϵ_{3} on Re λ, which gives | tr_{k} | definite exponential growth as a function of |k|. This implies an upper bound on |w(n)|.

If α_{n} is first or last in P, suppose without loss of generality it is first. If it has length at least ϵ_{3}, then in fact there is a second curve of bounded length on ∂_{−}C(N) besides α_{n}, and this plays the role of α_{n−1}. The rest of the computation goes through in the same way.

Now we come to the heart of the matter: consider a pivot α_{n} with length at most ϵ_{3}. In this case our strategy will be to estimate the Teichmüller parameter ω(F_{0}), where F_{0} is the boundary torus of T ≡ T_{ϵ_{0}}(α_{n}). Recall from Section 6 that ω(F_{0}) is within bounded distance of ω(α_{n}).

We estimate ω(F_{0}) by comparing it to the Teichmüller parameter for the torus ∂U_{n} in the model manifold M, which we can compute explicitly from the end invariants. Thus the argument is completed by showing that a quasiconformal map exists from one torus to the other, in the homotopy class preserving the markings.

Let us then fix a block B = B_{n}, and compute a Teichmüller parameter for the torus ∂U = ∂U_{n}.

We subdivide ∂U into a union of four annuli A_{0} ∪ A_{R} ∪ A_{1} ∪ A_{L}, as follows: Let t_{0} = 1/4 if B is internal or of type B_{α_{+},β}; in other cases let t_{0} = 0. Let t_{1} = 3/4 if B is internal or of type B_{α_{−},β}; in other cases let t_{1} = 1. With this convention, A_{0} = A × {t_{0}}, A_{1} = A × {t_{1}}, and A_{L} and A_{R} are the components of ∂A × [t_{0}, t_{1}].

We call A_{0}, A_{1} the bottom and top respectively, and A_{L}, A_{R} the sides of ∂U. Note that each annulus is Euclidean with geodesic boundary, so the union ∂U is a Euclidean torus (which is PL-embedded in B).

Next, we construct an explicit curve μ which represents the meridian of ∂U. Fix a neighbor β ∈ C of α = α_{n}. Realize β as a curve β_{0} which intersects B = S \ A in a geodesic arc orthogonal to the boundary with respect to the Euclidean metric σ_{0}^{e}, and intersects A in a σ_{0}^{e}-geodesic (recall from §9.1 that σ_{0}^{e} and σ_{0} are conformal, and agree except on a small disk around the puncture). Define β_{1} similarly with respect to σ_{1}^{e}. Since σ_{0}^{e} and σ_{1}^{e} are equal on B, we may choose the arcs β_{0} ∩B and β_{1} ∩B to be equal. Join the endpoints of (β_{0} ∩B)×{t_{0}} to those of (β_{1} ∩B)×{t_{1}} by vertical arcs μ_{L}, μ_{R} in A_{L}, A_{R} respectively, of length t_{1} − t_{0}. Let μ_{0} = (β_{0} ∩ A) × {t_{0}}, and μ_{1} = (β_{1} ∩A)×{t_{1}}. The closed curve μ = μ_{0} ∪μ_{R} ∪μ_{1} ∪μ_{L} is then null-homotopic in U.

By (2.3), the Teichmüller parameter of ∂U can be written

τ(∂U, α, μ) = τ(A_{0}, μ_{0}) + τ(A_{L}, μ_{L}) + τ(A_{1}, μ_{1}) + τ(A_{R}, μ_{R}).  (10.2)

By construction we have

τ(A_{L}, μ_{L}) = τ(A_{R}, μ_{R}) = (t_{1} − t_{0})i

which we note is between i and i/2.

For A_{0} and A_{1}, we claim

τ(A_{1}, μ_{1}) = τ(S, σ_{1}, α, β) − i/2,  (10.3)
τ(A_{0}, μ_{0}) = −\overline{τ}(S, σ_{0}, α, β) − i/2.

For the first of these, (2.3) implies that τ(S, σ_{1}, α, β) = τ(A, μ_{1}) + τ(B, β_{1} ∩ B), and by construction τ(B, β_{1} ∩ B) = i/2.

For the second, the proof is the same, except that we note the orientation of A_{0} as a subannulus of S and as a subannulus of ∂U are opposite, and hence τ is replaced with −\overline{τ}.

We next claim that

τ(S, σ_{1}, α, β) − \overline{τ}(S, σ_{0}, α, β) ≈ ν_{+}(α) − \overline{ν}_{−}(α) + i,  (10.4)

where “≈” again denotes uniformly bounded hyperbolic distance in H^{2}.

Recall that ν_{±}(α) is the value associated to ν_{±} under some fixed normalization of H^{2} taking α to ∞. The difference ν_{+} − \overline{ν}_{−} is invariant under real translations, so we may as well assume the normalization is one that takes β to 0 (that is, ν_{±}(α) = τ(S, ν_{±}, α, β)).

If α = α_{+}, then by construction σ_{1} is in the conformal class of ν_{+}, and τ(S, σ_{1}, α, β) is exactly ν_{+}(α).

If α = α_{n} ≠ α_{+}, then σ_{1} = σ_{α_{n},α_{n+1}}, which by construction has Teichmüller parameter α_{n+1} + i, where α_{n+1} is an integer in this normalization. Now, ν_{+}(α_{n}) by definition of the pivot sequence must be within Euclidean distance 1 of α_{n+1} – either ν_{+} is separated from α_{n} = ∞ by a Farey edge with integer endpoints adjacent to α_{n+1}, or it is in the horoball of diameter 1 tangent to R at α_{n+1} if α_{n+1} = α_{+}. It follows that α_{n+1} + i and ν_{+}(α_{n}) + i are within bounded hyperbolic distance.

A similar argument applies to ν_{−}, and the approximation (10.4) follows. Note that we need not be precise here about multiples of i.

Putting these estimates together with (10.2), we have the desired estimate on τ(∂U, α, μ).

Remark on the internal case. This computation may seem intricate because it takes care of all special cases at once, but for an internal pivot α_{n} we simply note that μ_{1} and μ_{0} differ by exactly w(n) twists around A, and that each of the four annuli making up ∂U have modulus exactly 1/2. Thus we obtain τ(∂U, α, μ) = w(n) + 2i.

It remains to relate the conformal structure on ∂U to that on F_{0}.

We claim first that the map H|_{∂U_{n}} : ∂U_{n} → ∂T_{ϵ_{0}}(α_{n}) has degree 1. This follows from the fact that H itself has degree 1, takes the pair (U_{n}, ∂U_{n}) to (T_{ϵ_{0}}(α_{n}), ∂T_{ϵ_{0}}(α_{n})), and takes nothing else into T_{ϵ_{0}}(α_{n}).

The map H is also uniformly Lipschitz on the torus ∂U_{n}, and the target torus F_{0} is itself Euclidean with area no less than A_{0} by part (3) of Lemma 6.3. When α_{n} is an internal pivot, ∂U_{n} is uniformly bilipschitz equivalent to a square torus. By the discussion in Section 2.4, this suffices to imply that H|_{∂U_{n}} is homotopic to a uniformly quasiconformal homeomorphism.

When α_{n} is α_{+} or α_{−} (or both), the torus ∂U_{n} contains high modulus annuli A_{+} and/or A_{−}. On these, the map H is already uniformly quasi-conformal. Since it is already Lipschitz on the remaining annuli, which have bounded modulus, we can apply Lemma 2.2 to show H is homotopic to a quasiconformal map, with a uniform bound on distortion.

Thus, the Teichmüller parameter ω(F_{0}) is approximated by ν_{+}(α_{n}) − \overline{ν}_{−}(α_{n}) + i up to bounded distance in H^{2}, and therefore so is ω(α_{n}). This concludes the proof of the pivot theorem.

11. Model manifolds and geometric limits

The Pivot Theorem and the constructions that led to its proof give us what looks like a fairly complete picture of a punctured torus manifold, as a sequence of standard blocks encasing Margulis tubes whose geometry is controlled by the pivot sequence. However, the map from the model to the hyperbolic manifold is so far only Lipschitz, with no bounds in the opposite direction. Such bounds are needed in order to determine the quasi-isometry type of our punctured torus group, and this is the purpose of the following theorem:

Theorem 11.1. (Model manifold) Let ρ : π_{1}(S) → PSL_{2}(C) be a marked puncturedtorus group with quotient manifold N. The end invariants (ν_{−}, ν_{+}) determine a manifold M homeomorphic to Ĉ(N), and a homotopy-equivalence f : M → Ĉ(N) whose lift f̃ to the universal covers is a quasi-isometry. The constants for the quasi-isometry are independent of ρ.

For us a map F : X → Y between metric spaces is a quasi-isometry if

\frac{1}{K} d(x, y) − δ ≤ d(F (x), F (y)) ≤ Kd(x, y) + δ

for all x, y ∈ X, where K and δ are positive constants.

11.1. Building the model. Much of the work of building the model has already been done by the block construction in Sections 9 and 10. In particular we have a manifold M and a proper map H : M → Ĉ(N) which is a degree-1 homotopy equivalence, and a metric on the submanifold M^{0} with respect to which H|_{M^{0}} is Lipschitz (with uniform constant).

What remains is to define the metric on the rest of M – that is, on the solid tori U_{n}, and adjust H on those parts to extend the Lipschitz condition. Each torus ∂U_{n} is Euclidean, and we have fixed a marking by α_{n} and the meridian curve μ_{n}. It has a Teichmüller parameter τ_{n} ≡ τ(∂U_{n}, α_{n}, μ_{n}) which we computed in the previous section. In particular for internal blocks τ_{n} is just w(n) + 2i.

The length of α_{n} on ∂U_{n} is exactly 1, and it is an easy exercise to show that there is a unique Margulis tube whose boundary is isometric to ∂U_{n} with the same marking. That is, the Euclidean structure on ∂U_{n} determines a complex translation length λ′ and a radius r. Furthermore the Teichmüller parameter at infinity ω′ = 2πi/λ′ for this tube will be within bounded Teichmüller distance of τ_{n}, and the distance becomes smaller for large w(n) (the radius r goes to ∞ as w does). Note that this is not quite an ϵ_{0} tube because the translation distance at the boundary is not exactly ϵ_{0}, but it is an ϵ′-tube where ϵ′ is within bounded ratio of ϵ_{0}.

We therefore extend the metric on U_{n} to make it isometric to this Margulis tube. The Pivot Theorem assures us, for a pivot α_{n} with length at most ϵ_{3}, that this Margulis tube admits a uniformly bilipschitz homeomorphism to the ϵ_{0}-Margulis tube of the pivot α_{n} in N itself. Our map H restricted to ∂U_{n} is only a Lipschitz map of degree 1, but admits a bounded homotopy to an affine isomorphism of Euclidean tori, and after realizing this homotopy in a collar we can extend to the rest of U_{n}, obtaining a map homotopic to H rel boundary on the solid torus. (The discussion is similar in the case of boundary blocks with parabolics, noting that τ_{n} is infinite in that case).

Note, by the way, that because of the corners in ∂U_{n} the resulting metric is not smooth, but this does not affect any of our arguments.

For pivots whose length is bounded below by ϵ_{3}, the widths w(n) and hence the meridian lengths are bounded, and we can obtain a metric on U_{n} in which the radius is bounded away from 0 and ∞. Therefore H on U_{n} can be homotoped rel boundary to a uniformly Lipschitz map taking U_{n} into a neighborhood of the geodesic α_{n}^{∗}.

Let f be the resulting map on M with its finished metric. We have so far proven that f is a degree 1 Lipschitz homotopy equivalence. To show that f lifts to a quasi-isometry, we will need some uniform estimates, which we will obtain by considering geometric limits. Our first goal then is Lemma 11.2, which gives us a uniform description of geometric limits of punctured-torus groups.

11.2. Geometric limits. We assume familiarity with the geometric topology on metric spaces with basepoints (N, x) (see [9, 24, 51, 86] and also [69] for applications in the spirit of this paper). For our purposes we use these basic definitions: a sequence (N_{i}, x_{i}) converges geometrically to (N, x) if and only if there is a sequence of numbers R_{i} → ∞ and K_{i} → 1, and maps h_{i} : (B(x, R_{i}), x) → (N_{i}, x_{i}) which are K_{i}-bilipschitz homeomorphisms to their images where B(x, R) denotes an Rneighborhood of x. Similarly a sequence of mappings f_{i} : (M_{i}, x_{i}) → (N_{i}, y_{i}) converges geometrically to f : (M, x) → (N, y) if we have convergence of (M_{i}, x_{i}) to (M, x) and (N_{i}, y_{i}) to (N, y) as above (with bilipschitz maps h_{i} and k_{i}, respectively), and if for any fixed R the maps h_{i}^{−1} ∘f_{i} ∘k_{i} : B(x, R) → N (defined for large enough i) converge in C^{0} to f.

An important phenomenon in hyperbolic geometry is that the set of pointed hyperbolic n-manifolds with a fixed positive lower bound on injectivity radius at the basepoint is compact in the geometric topology.

In this subsection, we try to understand geometric limits of punctured-torus manifolds, by considering geometric limits of their models and the maps f.

Convergence of combinatorial data. The model manifolds are determined by combinatorial data which it is convenient to describe as follows. Recalling that the pivot sequence is P is indexed by an interval J ⊂ Z, we have a sequence of integer widths W = {w(n)}_{n∈J}, determined by the pair (ν_{−}, ν_{+}) as in Section 4. Furthermore, if P has a first (or last) element we define v_{−} (or v_{+}) as a complex number with | Re v_{±}| ≤ 1 which equals ν_{±}(α_{±}) mod 1. More specifically, with notation as in Section 4, if α_{−} ≠ α_{+} we let v_{+} be such that w(p + 1) + v_{+} = ν_{+}(α_{+}) − α_{p}(α_{+}), and v_{−} be such that w(0) + v_{−} = α_{1}(α_{−}) − \overline{ν}_{−}(α_{−}). If α_{−} = α_{+} then we want w(0)+v_{−}+v_{+} = ν_{+}(α_{+})−\overline{ν}_{−}(α_{+}). In case e_{+} or e_{−} has an accidental parabolic we write v_{+} = ∞i or v_{−} = ∞i.

Note in particular that Im v_{+} and Im v_{−} determine the moduli of the annuli A_{+} and A_{−} in the corresponding boundary blocks, via (10.3). The triple Σ = (W, v_{+}, v_{−}) is called the combinatorial data (where v_{±} are ignored if undefined).

Note that the pivot sequence itself is forgotten by the combinatorial data, but can be recovered from the width sequence up to the action of SL_{2}(Z). This correponds to keeping track of the image Kleinian group but forgetting the marking, i.e. the representation ρ.

To understand limiting configurations, let us generalize slightly the definition of the combinatorial data Σ to allow the index set J to be any interval in Z, and to allow the value w(n) = ∞. With this definition, we consider a topology on the set of Σ’s as follows: We say that a sequence Σ_{i} = (W_{i}, v_{+}^{i}, v_{−}^{i}) converges to Σ if the index sets J_{i} converge to J as subsets of Z (that is, for any q > 0, eventually J_{i}∩[−q, q] = J ∩[−q, q]), and for each n ∈ J either w_{i}(n) → w(n) in Z, or w(n) = ∞ and |w_{i}(n)| → ∞. Finally we require if v_{+} is defined in Σ that v_{+}^{i} are eventually defined and v_{+}^{i} → v_{+} in the natural compactification of [−1, 1] + iR by ∞i, and similarly for v_{−}.

Convergence of model manifolds. A generalized combinatorial data set Σ determines a generalized model manifold M_{Σ}. For each n ∈ J, if w(n) is finite we construct a block B_{n} just as before, except that the pivots don’t have fixed names: for example for an internal block B_{n} we build B_{α,β,γ} where β and γ are neighbors of α and γ = D_{α}^{w(n)}β. When gluing successive blocks we must identify ∂_{1}B_{n} to ∂_{0}B_{n+1} by the unique orientation-preserving isometry taking the α curve on ∂_{1}B_{n} to the β curve on ∂_{0}B_{n+1}, and the γ curve on ∂_{1}B_{n} to the α curve on ∂_{0}B_{n+1}.

When w(n) = ∞ and the block is to be internal, we construct first the truncated block B_{n}^{0} isometric to B \ U. Note that without U there is no natural identification between top and bottom boundaries, and in fact all truncated internal blocks are isometric. There is a unique Margulis tube for a rank-2 parabolic group whose boundary torus is isometric to U, and we put the metric of this tube on U minus its core circle, and glue that into the block.

For a boundary block the same thing works, with the following provisos. We use Im v_{+} and/or Im v_{−} to determine the moduli of A_{+} and/or A_{−}, via (10.3), and the real parts together with the corresponding widths w(0) or w(p + 1) serve to determine the exact geometry of the torus ∂U and the choice of meridian. The case v_{+} = ∞i yields a pair of annuli each isometric to S^{1}×[0, ∞), as before, and a rank-1 parabolic Margulis tube (even if the corresponding width is infinite), and similarly v_{−} = ∞i. The one exception is the possibility that α_{+} = α_{−}, so that there is just one double boundary block B_{0}, and both v_{+} = v_{−} = ∞i. In this case we replace U with two copies of a rank-1 parabolic Margulis tube, glued to the annulus A_{L} and A_{R} respectively. Thus the manifold is homeomorphic to a thrice-punctured sphere cross [0, 1]. This case is not central to our discussion (it corresponds to a Fuchsian geometric limit), but we include it for completeness.

We can now describe geometric convergence for the model manifolds more clearly, and relate it to convergence for the combinatorial data Σ.

Let us adopt the following notation. Let M = ∪_{n∈J}B_{n} be a model manifold. Let B_{n}^{0} denote B_{n} minus the open solid torus U_{n}. Let B̌_{n} denote B_{n} minus the cusp region Q_{B_{n}} and let B̌_{n}^{0} denote B_{n}^{0} ∩ B̌_{n}. Then let M^{0} = ⋃_{n∈J} B_{n}^{0}, and similarly for M̌, M̌^{0}. For integer q > 0 let M[q] be the finite union of blocks B_{n} with |n| ≤ q. Let M^{0}[q] = M^{0} ∩ M[q] and define M̌[q], M̌^{0}[q] similarly. Note that M̌^{0}[q] is compact.

Now consider a sequence (M_{i}, x_{i}), with combinatorial data Σ_{i}. Let us assume, reindexing by shifting J_{i} if necessary, that 0 ∈ J_{i} for each i, and that x_{i} ∈ B_{i,0} (here we denote the n-th block in M_{i} as B_{i,n}). Suppose that Σ_{i} converge to Σ, and let M_{∞} = M_{Σ}. We claim that, if the x_{i} are chosen properly, a subsequence of (M_{i}, x_{i}) converge geometrically to (M_{∞}, x) for some x ∈ B_{∞,0}.

In fact, consider M_{∞}^{0}[q] – it is composed of finitely many truncated blocks B_{n}^{0}. By our construction, any two truncated internal blocks are isometric, and any two truncated boundary blocks of the same type are bilipschitz equivalent with some uniform bound, except possibly for the long annuli A_{±} in their boundaries. Thus, for i large enough there is a natural embedding k_{i} : M_{∞}^{0}[q] → M_{i}[q] which is an isometry on each truncated internal block (and bilipschitz for boundary blocks minus the long annuli), and preserves the indexing n. Thus if we choose x_{i} ∈ B̌_{i,0}^{0} but not in the long boundary annuli, then k_{i}^{−1}(x_{i}) have a convergent subsequence with limit x ∈ B̌_{0}^{0}. Furthermore convergence of the v_{±} insures that the long boundary annuli converge geometrically, and their boundary identifications can vary in a compact set. Thus, letting q → ∞, we find that for some subsequence which we continue to index as if it were the whole sequence, the submanifolds (M_{i}^{0}, x_{i}) converge geometrically to (M_{∞}^{0}, x).

Note, the subtlety here is that the maps k_{i} do not necessarily extend across the solid tori U_{n} – the meridian curve μ_{n} of the solid torus may be different from the meridian of the corresponding torus in M_{i}. However, this information is encoded in the widths w_{i}(n). In particular, for internal blocks if w(n) ≠ ∞ then for large enough i w_{i}(n) = w(n), and then the map k_{i} extends to an isometry between the solid tori U_{n} and U_{i,n}. If w(n) = ∞ then w_{i}(n) → ±∞, the radii of the corresponding Margulis tubes go to infinity, and it is easy to check that the geometric limit (with basepoint on the boundary torus) is exactly the rank-2 parabolic Margulis tube which we have placed in M. (This is essentially Jørgensen’s original example of a geometric limit of cyclic groups which is not itself cyclic, see [47, 51]). The case of boundary blocks is similar, except that we use convergence of the v_{±} to give convergence of the long annuli. Furthermore the case where Im v_{+} or Im v_{−} go to ∞ yields a rank-1 parabolic rather than rank-2, regardless of the behavior of w_{i}(n).

We conclude that (M_{i}, x_{i}) converge geometrically to (M_{∞}, x). Furthermore, let us extend the definition of k_{i} to maps k_{i} : M_{∞}^{0} → M_{i} as follows: for each i choose q = q_{i} to be the largest q so that J_{i} ∩ [−q, q] = J_{∞} ∩ [−q, q] (so q_{i} → ∞ as i → ∞). We define k_{i} on M_{∞}^{0}[q_{i}] as before, and extend to the rest of M_{∞}^{0} by first collapsing the blocks outside M_{∞}^{0}[q] to ∂M_{∞}^{0}[q]. We then also have geometric convergence of k_{i} → k_{∞}, where k_{∞} : M_{∞}^{0} → M_{∞} is the inclusion map.

Note that, except in one case, a geometric limit of model manifolds with basepoints in the thick part must always be homeomorphic to product S ×I minus some sequence of curves γ_{k} × {k}, where I is an interval in R, each γ_{k} is a nonperipheral simple curve on S, and γ_{k} and γ_{k+1} are homotopically distinct. The exception is the case where for each i there is just one block, so α_{−}^{i} = α_{+}^{i}, and both v_{−}^{i} and v_{+}^{i} go to infinity. Then the limit is the exceptional thrice-punctured sphere case described above, and we leave the details to the reader. Convergence of model maps. The following lemma allows us to describe a geometric limit of any sequence of punctured torus groups in terms of a limit of models.

Lemma 11.2. Let N_{i} be a sequence of punctured-torus manifolds with associated models M_{i} and maps f_{i} : (M_{i}, x_{i}) → (Ĉ(N_{i}), f_{i}(x_{i})). Assuming that x_{i} are in the interior of the thick part M̌_{i}^{0}, these maps have a geometrically convergent subsequence, and the limit f_{∞} : (M_{∞}, x_{∞}) → (Ĉ_{∞}, y_{∞}) is a proper homotopy-equivalence of degree 1, where Ĉ_{∞} is a subset of N_{∞} homotopy-equivalent to it and containing its convex core.

Proof. Reindex as above, if necessary, so that the index sets J_{i} of the combinatorial data Σ_{i} always contain 0, and that x_{i} ∈ B̌_{i,0}^{0}. Then after possibly restricting to a subsequence we may assume that {Σ_{i}} converge to some Σ. By the previous discussion, after restricting to another subsequence, we may assume that (M_{i}, x_{i}) converge geometrically to some (M_{∞}, x_{∞}) (keeping x_{i} in the interior means it can’t lie on the long annuli A_{±} of boundary blocks). By Lemma 9.3, the injectivity radius of N_{i} at f_{i}(x_{i}) is uniformly bounded below by ϵ_{3}, and therefore we may further assume that (N_{i}, f_{i}(x_{i})) converge geometrically to (N_{∞}, y_{∞}), and that the subsets Ĉ(N_{i}) converge to some subset Ĉ_{∞} of N_{∞}. The maps f_{i} are uniformly Lipschitz, so after extracting a further subsequence we may assume that they converge to f_{∞}. We will abuse notation by continuing to index each subsequence as if it were the whole sequence.

Since our manifolds are K(π, 1)’s, showing that f_{∞} is a homotopy equivalence reduces to showing that (f_{∞})_{∗} : π_{1}(M_{∞}) → π_{1}(N_{∞}) is an isomorphism. The scheme for this proof is due to Thurston, in [86, 85].

On finite blocks, the map k_{i} defined above eventually factors through k_{∞} in this sense: Fix q > 0, and I large enough that, for i > I and n ∈ J_{∞} ∩ [−q, q], w_{i}(n) = w_{∞}(n) if the latter is not ∞. When w_{∞}(n) = ∞, U_{n} is missing its core circle and hence retracts to its boundary. It follows that for i > I we can extend k_{i}|_{M_{∞}^{0}[q]} to an embedding k_{i}^{q} : M_{∞}[q] → M_{i}[q] satisfying k_{i} = k_{i}^{q} ∘ k_{∞}.

The manifold M_{∞}[q] has the form of a product S × [0, 1] minus a number of level curves corresponding to those n where w_{∞}(n) = ∞, and the map k_{i}^{q} has the effect of filling in those missing curves by a Dehn surgery whose meridian curve is determined by w_{i}(n). Since w_{i}(n) → ∞ as i → ∞, there is a precise sense in which the map k_{i}^{q} is “eventually injective” on π_{1}:

Lemma 11.3. Let S be a surface of finite type and negative Euler characteristic. Let M be S × [s, t] minus a number of regular neighborhoods of level simple curves γ_{i} × {n_{i}}, where each γ_{i} is non-peripheral and successive curves γ_{i}, γ_{i+1} have essential intersection in S. Let M_{i} be a sequence of Dehn fillings on M such that the lengths of meridian curves on each torus boundary go to infinity.

1. If the images of α in all the π_{1}(M_{i}) are trivial, then α is already trivial in π_{1}(M).

2. If α ∈ π_{1}(M) is not in a conjugate of one of the boundary torus subgroups, and whose images in all the π_{1}(M_{i}) are m-th powers, then α is itself an m-th power.

The proof of this lemma is postponed to Section 11.4.

To apply the lemma, note that the condition on successive level curves follows automatically from the way in which we glue successive blocks. The condition on meridian lengths follows from the widths going to infinity.

We can now show that (f_{∞})_{∗} is injective. Let α be a non-trivial loop in M_{∞}. Then we can assume it lies in M_{∞}^{0}[q] for some q > 0, and applying part (1) of Lemma 11.3 we conclude that, for large enough i, k_{i}^{q}(α) is nontrivial in π_{1}(M_{i}[q]). In fact it is non-trivial in π_{1}(M_{i}) since π_{1}(M_{i}[q]) injects in π_{1}(M_{i}) by Seifert-van Kampen.

Now if f_{∞}(α) is trivial, it bounds a disk in N_{∞}. For high enough i the disk pulls back to N_{i}, and we conclude that f_{i}(k_{i}^{q}(α)) is also homotopically trivial. This contradicts the fact that, by construction, f_{i} are all π_{1}-injective, so we conclude that (f_{∞})_{∗} is injective.

We next prove that (f_{∞})_{∗} is surjective. Let Γ = (f_{∞})_{∗}(π_{1}(M_{∞})) and Γ̂ = π_{1}(N_{∞}), considered as subgroups of PSL_{2}(C). Up to suitably normalizing via conjugation in PSL_{2}(C), geometric convergence of the manifolds is equivalent to geometric convergence of the groups in the Gromov-Hausdorff topology (see [24]). Thus Γ̂ is the geometric limit of (f_{i})_{∗}(π_{1}(M_{i})). We need to show that Γ = Γ̂.

We also have a sequence of (surjective but non-injective) representations φ_{i} = (f_{i} ∘ k_{i})_{∗} from π_{1}(M_{∞}^{0}) to PSL_{2}(C), and φ_{∞} similarly, such that φ_{i} → φ_{∞} and φ_{∞}(π_{1}(M_{∞}^{0})) = Γ.

We first prove that if Γ is a finite-index subgroup of Γ̂ then Γ = Γ̂. Finite index implies that for any g ∈ Γ̂, some finite power g^{m} is in Γ. Thus g^{m} = φ_{∞}(h) for some h ∈ π_{1}(M_{∞}^{0}).

Choose q large enough so that h can be represented by a loop in M_{∞}^{0}[q], and h′ = (k_{∞})_{∗}(h) can thus be assumed to lie in π_{1}(M_{∞}[q]). Suppose that h′ is contained in a (conjugate of) a rank-2 cusp subgroup of M_{∞}. Since f_{∞} maps rank-2 cusps to rank-2 cusps with degree 1, the corresponding cusp subgroup of Γ̂ is contained in the image of φ_{∞}. g is contained in this cusp group since it commutes with φ_{∞}(h), so it must already be contained in Γ.

Now assume that h′ is not in a cusp group. By definition of geometric limit of groups, g = lim φ_{i}(g_{i}), for some sequence g_{i} ∈ π_{1}(M_{∞}^{0}). Thus lim_{i→∞} φ_{i}(g_{i}^{m}) = φ_{∞}(h). Since also lim_{i→∞} φ_{i}(h) = φ_{∞}(h) and the limit group is discrete, we conclude (recapitulating an argument of Jørgensen-Marden in [51]) that for large enough i, φ_{i}(g_{i})^{m} = φ_{i}(h). Note that since φ_{i} may not be injective, we cannot conclude that g_{i}^{m} = h. However we do know that

(k_{i})_{∗}(g_{i})^{m} = (k_{i})_{∗}(h),  (11.1)

since (f_{i})_{∗} is injective. Hence (k_{i}^{q})_{∗}(h′) = (k_{i})_{∗}(h) is eventually an m-th power, and since h′ is not in a cusp subgroup we can apply part (2) of Lemma 11.3 to conclude that h′ is itself an m-th power, i.e. h′ = (g′)^{m} for g′ ∈ π_{1}(M_{∞}). Since m-th roots are unique in a torsion-free Kleinian group (see also [45, Chap. VI]), we conclude that (f_{∞})_{∗}(g′) = g, and thus g ∈ Γ after all. Thus Γ = Γ̂ in this case.

Now consider the possibility that Γ has infinite index in Γ̂. In this case we use a variation of Thurston’s covering theorem (similar to an argument made in [85]).

Let N_{a} = H^{3}/Γ, so that there is a covering π : N_{a} → N_{∞} and a lift f_{a} : M_{∞} → N_{a} of f_{∞} with π ∘ f_{a} = f_{∞}, and f_{a} is a homotopy equivalence. Since f_{∞}(M_{∞}) = Ĉ_{∞}, which is the geometric limit of sets containing the convex cores of C(N_{i}), it contains the convex core of N_{∞} (see Kerckhoff-Thurston [55]).

Since C(N_{∞}) is a deformation retract of N_{∞}, every loop in N_{∞} can be deformed into f_{∞}(M_{∞}). It follows that the preimage π^{−1}(f_{∞}(M_{∞})) is equal to f_{a}(M_{∞}), and that if π is infinite-to-one on N_{a} it must already be infinite-to-one on f_{a}(M_{∞}).

However, restricted to each rank-2 cusp of M_{∞}, the map must clearly be finiteto-one. It follows that there exists a sequence of blocks B_{n_{j}} with |n_{j}| → ∞, whose images in N_{∞} meet a compact set. The map f_{∞} restricted to, say, ∂_{0}B_{n_{j}}, is a limit of a sequence of our “halfway surfaces”, and in particular is a uniformly Lipschitz map h_{n_{j}}^{∞} : S → N_{∞}, where the metric on S is the standard metric induced from any of the block boundaries, up to homeomorphism. Since these surfaces meet a compact subset of N minus its main cusp (and are standard in the cusp) there must be two of them h_{1}, h_{2} which admit a homotopy in N_{∞} with tracks shorter than the injectivity radius in this compact set. We may lift this homotopy to the covering N_{a} and conclude that a finite-volume 3-chain in N_{a} maps to a closed 3chain in N_{∞} (closed relative to the cusps). It follows that N_{∞} has finite volume, but this is impossible since it is the geometric limit of a sequence of manifolds with infinite volume. (See Canary [23] for a discussion of the covering theorem and other generalizations).

We conclude that Γ cannot be infinite index in Γ̂ either, and therefore (f_{∞})_{∗} is surjective.

The same argument also shows that f_{∞} is proper, for we know it takes boundary to boundary, and we have shown that the images of blocks B_{n} cannot accumulate in a compact subset of N_{∞}.

We now show that the degree is 1, which is a more subtle issue. Let us first make the following observations about ordering of ends and the orientation convention of section 3. Recall that an ordering for the ends (e_{−}, e_{+}) of S_{0} × R is determined by a choice of orientation for S_{0}. If ψ : (S_{0}, ∂S_{0}) → (S_{0} × R, ∂S_{0} × R) is an immersion inducing the identity on π_{1}(S_{0}), then we can also detect the ordering of the ends in this way: there are exactly two components of S_{0} × R \ ψ(S_{0}) which are unbounded (i.e. whose closures are not compact), and we can order them (U_{−}, U_{+}) according to the convention that an oriented path from U_{+} to U_{−} has intersection number +1 with ψ(S_{0}). One can easily check that U_{−} is a neighborhood of e_{−}, and U_{+} of e_{+}.

Now if h : (S_{0} × R, ∂S_{0} × R) → (S_{0} × R, ∂S_{0} × R) is proper and induces the identity on π_{1} then we can also check that if it induces a bijection of the ends which is order preserving then deg h = 1, if it induces an order reversing bijection then deg h = −1, and if it maps both ends to one then the degree is 0.

For any model map f : M → N recall that M̌ denotes the complement in M of the main cusp neighborhood Q_{M}, and similarly Ň is the complement in N of Q_{N}, and let Ĉ(Ň) = Ĉ(N) ∩ Ň. Let ∂′M̌ denote the cylindrical frontier of Q_{M} in M, and let ∂′Ĉ(Ň) similarly denote the frontier of Q_{N} ∩ Ĉ(N) in Ĉ(N). By our construction, f restricts to a map of pairs f : (M̌, ∂′M̌) → (Ĉ(Ň), ∂′Ĉ(Ň)), and in the case where the width sequence is biinfinite and the entries w(n) are all finite, each pair may be identified with (S_{0} × R, ∂S_{0} × R). Let S_{0}(n) denote a level surface of the form ∂_{0}B̌_{n} in M.

Returning to our sequence f_{i} → f_{∞}, consider first the case that the limiting width sequence w_{∞} is bi-infinite, and that w_{∞}(n) does not take on the value ∞. If deg f_{∞} is not 1, then at least one of the ends of M̌_{∞} must map to the opposite end of Ň_{∞}. This situation is covered by a special case of a theorem proved by Thurston for arbitrary genus, in [86], in which he also shows that the ends cannot flip in the limit. We will give a slightly different argument here, adapted to our techniques but following the same general logic.

Let (U_{−}, U_{+}) denote the unbounded components of Ň_{∞} \ f_{∞}(S_{0}(0)), ordered as above by the orientation convention. Then order reversal implies, without loss of generality, that as j → +∞, f_{∞}(S_{0}(j)) is eventually in U_{−}. Fix such a j. Let D be the maximal diameter of any bounded component of Ň_{∞} \ f_{∞}(S_{0}(0)). Let X_{∞} be a neighborhood of f_{∞}(S_{0}(0)) in Ň_{∞} of radius at least D + 1, and large enough to contain f_{∞}(S_{0}(j)). By the definition of geometric convergence, given K as close as we like to 1, for large enough i there is a region X_{i} in Ň_{i} and a K-bilipschitz homeomorphism h_{i} : X_{∞} → X_{i}. For large i, w_{i}(n) = w_{∞}(n) for |n| ≤ j, so the embedding k_{i} : M_{∞}^{0}[j] → M_{i}^{0}[j] extends to an isometry k_{i}^{j} : M_{∞}[j] → M_{i}[j], as in our previous discussion. Furthermore h_{i}^{−1} ∘ f_{i} ∘ k_{i}^{j} is ϵ-close to f_{∞} in the C^{0} topology, for ϵ small. Thus X_{i} must contain a (D + 1 − ϵ)/K-neighborhood of f_{i}(S_{0}(0)) (see e.g. [21]) so that (for K close to 1, ϵ close to 0) the unbounded components U_{±}^{i} of Ň_{i} \ f_{i}(S_{0}(0)) must each intersect X_{i} in sets of diameter greater than D. It follows that h_{i}(U_{±} ∩ X_{∞}) ⊂ U_{±}^{i}. Because h_{i} is orientation preserving, the ordering of + and −, as detected by intersection numbers, is preserved.

Thus, f_{i}(S_{0}(j)) must be contained in U_{−}^{i}. On the other hand since f_{i} itself does not reverse end order, for some k > j we have f_{i}(S_{0}(k)) ⊂ U_{+}^{i}. It follows that some block B_{i,m_{j}} with j < m_{j} < k has image under f_{i} that meets f_{i}(S_{0}(0)). Repeating this argument with arbitrarily high j we obtain a sequence of pivots α_{m_{j}}^{i_{j}} of the pivot sequence for N_{i_{j}}, with m_{j} > j, such that the geodesics (α_{m_{j}}^{i_{j}})^{∗}, or their ϵ_{0}-Margulis tubes, remain a bounded distance away from f_{i}(S_{0}(0)). Each (α_{m_{j}}^{i_{j}})^{∗} has uniformly bounded length by the Pivot Theorem, so after restriction to a subsequence their images under h_{i}^{−1} must converge to a geodesic β in N_{∞}, to which they are eventually homotopic. Since f_{∞} is a homotopy equivalence, β determines a a conjugacy class in π_{1}(M_{∞}) = π_{1}(S), which is eventually equal to the conjugacy class of α_{m_{j}}^{i_{j}}; but this contradicts the fact that, on the other hand, α_{m_{j}}^{i_{j}} must converge to ν_{+}^{∞} (compare Lemma 12.1), which is irrational in this case. We conclude that f_{∞} cannot reverse the order of the ends. (In Thurston’s argument the last contradiction is obtained in terms of realizability of the ending lamination.)

Proving deg f_{∞} = 1 in the remaining cases is easier. Suppose first that the limiting width sequence is not bi-infinite. Then Ĉ(N_{∞}), and Ĉ(N_{i}) for high i, have nonempty boundary, and recall that f_{i} : ∂M_{i} → ∂Ĉ(N_{i}) is constructed to be a uniformly bilipschitz orientation-preserving homeomorphism, which converges to f_{∞} : ∂M_{∞} → ∂Ĉ(N_{∞}). It follows that the degree is 1.

Finally, if w_{∞}(n) = ∞ for some n, then N_{∞} has a rank-2 parabolic cusp which is the image of a corresponding cusp U_{n} in M_{∞}. By our construction f_{∞} (and any f_{i}) has degree 1 over this cusp, and hence degree 1 globally since it is proper.

This concludes the proof of Lemma 11.2.

11.3. Lifting to a quasi-isometry. With Lemma 11.2 in hand, we may prove the following corollary, which essentially says that the map f does not contract too much, in a homotopic sense:

Lemma 11.4. Let f : M → N be a model for a punctured torus group. For any B > 0 there exists A > 0 such that, if β is a loop in N through f(x) of length no more than B, then there exists a loop α through x in M of length no more than A, such that f(α) is homotopic to β fixing the basepoint f(x).

Proof. For any fixed x ∈ M there is such an A(x, B), just because f is a homotopy equivalence and there is a finite number of homotopy classes of loops with a given length bound passing through a given point in a hyperbolic manifold. If there were no universal A(B) we could find a sequence x_{i} in M so that the best A(x_{i}, B) go to infinity. Clearly x_{i} would not go out a cusp because there f is uniformly bilipschitz. Therefore we may assume that x_{i} ∈ B̌_{n_{i}}^{0} where |n_{i}| → ∞. Thus if we consider the sequence of manifolds with shifted basepoint (M, x_{i}) we may apply lemma 11.2 to obtain a geometric limit f_{∞} : (M_{∞}, x_{∞}) → (N_{∞}, y_{∞}) which is still a homotopy equivalence. Therefore there is some A which works in the limit. Now pulling back large neighborhoods of x_{∞} and y_{∞} to the approximants, we see that A+δ for some δ > 0 works there too. (Compare a similar argument in lemma 4.5 of [69]).

Finally we claim that a model map f : M → N is a quasi-isometry. Restricted to each Margulis tube U_{n}, f is by construction a bilipschitz homeomorphism to its image. Since each internal truncated block minus its Margulis tube, B̌_{n}^{0}, has bounded diameter, we must prove that the separation between the images of any two blocks, f(B̌_{n}) and f(B̌_{n′}), as measured in the truncated manifold Ň, is roughly proportional to |n − n′|. This will prove that the restriction of f to M̌ → Ň is a quasi-isometry, and by construction the extension to the main cusp neighborhoods will be a quasi-isometry too.

Again let S_{0}(n) denote ∂_{0}B̌_{n}. Let us prove the following claim: there is some m_{0} > 0 such that, whenever n ∈ J and n + m_{0} ∈ J, there is some 0 < m ≤ m_{0} such that f(S_{0}(n+m)) separates f(S_{0}(n)) from e_{+} in Ň. (Equivalently we can make the claim for m_{0} < 0 and e_{−}). Suppose this is not the case. Then there are sequences m_{i} → +∞ and n_{i} → ±∞ such that for every 0 < m ≤ m_{i} f_{i}(S_{0}(n_{i} + m)) fails to separate f_{i}(S_{0}(n_{i})) from the e_{+} end of Ň. In other words, if (U_{−}^{i}, U_{+}^{i}) are the unbounded components of Ň \ f(S_{0}(n_{i})), ordered by our orientation convention, then we have for all 0 < m ≤ m_{i} that f(S_{0}(n_{i} + m)) is not contained in U_{+}^{i}.

Choosing basepoints x_{i} in B_{n_{i}} and possibly restricting to a subsequence, we obtain by Lemma 11.2 a geometric limit f_{∞} : M_{∞} → N_{∞}, which is a proper degree 1 homotopy equivalence. Note that in the limit the combinatorial data is a biinfinite sequence since n_{i} → ±∞ – thus we need not consider boundary blocks.

We claim that Ň_{∞}^{0} \ f_{∞}(S_{0}(0)) has two unbounded components U_{±}^{∞}. (This is in contrast to Ň_{∞} which may have infinitely many ends corresponding to rank-2 cusps.) Recalling that f_{∞} restricts to a map of pairs f_{∞} : (M̌_{∞}^{0}, ∂M̌_{∞}^{0}) → (Ň_{∞}^{0}, ∂Ň_{∞}^{0}), we can perform a series of integer Dehn fillings on the torus boundaries of M̌_{∞}^{0} to obtain a manifold homeomorphic to S_{0} × R. After performing fillings with the same integers on the torus boundaries of Ň_{∞}^{0}, we can extend f_{∞} to a map between the filled manifolds that is still a proper, degree 1 homotopy equivalence. It follows that the filled Ň_{∞}^{0} is also S_{0} × R (see [14, 86]) and hence has two ends, and that the extended f_{∞} is, again, order-preserving on the ends. Thus Ň_{∞}^{0} also has two ends and we define U_{±}^{∞} to be the components of Ň_{∞}^{0} \ f_{∞}(S_{0}(0)) which are neighborhoods of these ends, ordered again by our intersection-number convention.

We now see that there exists m_{∞} > 0 such that f_{∞}(S_{0}(m_{∞})) ⊂ U_{+}^{∞}. We may now apply to Ň_{∞}^{0} the same argument as in the proof of Lemma 11.2 to see that there is a large neighborhood X_{∞} of f_{∞}(S_{0}(0)) in Ň_{∞}^{0} and a sequence of nearisometries h_{i} : X_{∞} → X_{i}, where X_{i} are neighborhoods of f(S_{0}(n_{i})) in N, and h_{i} takes U_{+}^{∞}∩X_{∞} into U_{+}^{i}. (We have to make X_{∞} include all the bounded components of Ň_{∞}^{0} \ f_{∞}(S_{0}(0)), as before). Moreover if we choose X_{∞} large enough we may push f_{∞}(S_{0}(m_{∞})) via h_{i} into X_{i} and conclude that f(S_{0}(n_{i} + m_{∞})) ⊂ U_{+}^{i}. But for i large enough m_{∞} < m_{i}, a contradiction.

This proves the claim that m_{0} exists as above. It follows that for our original model f : M → N we can find a sequence s = (⋯ < n_{1} < n_{2} < ⋯) with n_{i+1} ≤ n_{i} + m_{0} and inf s = inf J, sup s = sup J, such that f(S_{0}(n_{j})) separates f(S_{0}(n_{i})) from f(S_{0}(n_{k})) in Ň_{∞} for all i < j < k. All these immersed surfaces are disjoint and any two with adjacent indices are a definite distance apart (again by a geometric limit argument). This forces the separation between any f(S_{0}(n)) and f(S_{0}(n′)) to be at least a_{1}|n − n′| − a_{2} for some constants a_{i} > 0.

On the other hand we know f is Lipschitz, so the separation is bounded above by another linear function of |n − n′|. This concludes the proof that f is a quasiisometry.

We now have a map f which satisfies the following conditions: It is a surjective homotopy equivalence, Lipschitz, a quasi-isometry, and lemma 11.4 holds. It follows by lemma 3.1 of [70] that the lift f̃ to the universal covers is a quasi-isometry. This completes the proof of theorem 11.1 (Model manifold), modulo a leftover lemma:

11.4. Proof of lemma 11.3. A version of this lemma is asserted by Thurston in [85] without proof. Although the statement is purely topological we present here a proof which uses hyperbolic geometry in what seems like an essential way. (We have considered applying the boundary-slope techniques of Gordon-Litherland and Gordon-Luecke et. al. [35, 36, 28], but without success).

In fact consider the following more general situation. Let M be the interior of a compact 3-manifold \overline{M}, such that M admits a complete hyperbolic metric σ. Suppose that F is a collection of torus components of ∂\overline{M}, and let M_{i} be a sequence of Dehn fillings of \overline{M}, obtained by attaching solid tori to the components of F so that the lengths of meridians on each component of F go to infinity as i → ∞.

In a complete non-positively curved manifold, a null-homotopic curve of length K bounds a disk of diameter at most K/2 (use a ruled disk). Let α be a curve in M which is null-homotopic in M_{i} for infinitely many i. Let K denote the length of α, and let H denote a union of horoball beighborhoods of the cusps, one for each component of F, that exclude a K/2-neighborhood of α. Then M_{i} is obtained from M \ H by gluing in a union of solid tori H_{i} to the torus boundaries of H. Since the meridians of H_{i} have lengths going to ∞ with i, it follows from the “2π-theorem” of Gromov-Thurston (see [37], and Bleiler-Hodgson [13] or Moriah-Rubinstein [72] for a detailed proof) that for large enough i one can put a metric σ_{i} on M_{i}, which has sectional curvatures pinched between some fixed negative constants, and equals the hyperbolic metric σ in M \ H. If α is null-homotopic in M_{i} it bounds a disk of σ_{i}-diameter at most K/2, as above, but it follows that this disk is contained in M \ H, and hence α was already null-homotopic in M. This proves the first assertion of the lemma.

For the second assertion, we consider a manifold with sectional curvatures bounded above by some −κ < 0. Let A : S^{1} × [0, 1] → N be a ruled annulus, that is, a map such that A|_{{θ}×[0,1]} is a geodesic for each θ ∈ S^{1}. Then the induced metric on the annulus also has sectional curvatures bounded by −κ. Standard comparison theorems imply that Area(A) ≤ C(κ)ℓ(∂A). Furthermore, given K, ϵ > 0 there exists D(κ, ϵ, K) such that, if ∂A has length at most K and A has diameter at least D, then the shortest curve in A homotopic to a boundry component has length at most ϵ.

Let −κ be the upper bound on the sectional curvatures of the metrics σ_{i} (in fact κ can be taken arbitrarily close to 1 as i → ∞). Let ϵ be less than the σ-length of the shortest non-cuspidal curve in M. Let α be a curve whose image in π_{1}(M_{i}) is an m-th power for infinitely many i, and which is not homotopic into a cusp. Let K be the σ-length of α, and choose the horoball neighborhoods H so that they avoid a D(κ, ϵ, 2K)-neighborhood of α.

If α_{i}^{∗} is the σ_{i}-geodesic representative of α in M_{i}, let A be a ruled annulus with one boundary mapping to α and the other to α_{i}^{∗}. After a small perturbation we may assume A is transverse to ∂H, and consider the intersection locus A^{−1}(∂H) in the annulus. If a component is homotopically trivial in the annulus and its image is a power of a meridian of H_{i}, then the area enclosed by this curve is at least the area of a meridian disk of H_{i}, and this goes to ∞ with i (see Moriah-Rubinstein [72]). It follows from the upper bound on Area(A) that eventually every such component bounds a disk that can be pushed out of H_{i}.

Furthermore if A meets H at all then its diameter is at least D, and hence it contains a non-trivial curve of length at most ϵ. This curve must then be cuspidal, so A intersects ∂H in a nontrivial curve. Combined with the previous observation we conclude that α can be deformed to ∂H within M \ H, in contradiction to the choice of α.

It follows that A misses H, so α is homotopic to α_{i}^{∗} within M \ H. This means that α_{i}^{∗} is already the geodesic representative of α in the metric σ on M. Since α is an m-th power in M_{i}, α_{i}^{∗} runs m times around the geodesic representing the root, and therefore is an m-th power in M as well. This proves the second assertion.

To apply this to our situation, note that the condition that γ_{i} and γ_{i+1} intersect implies that S × [s, t] minus the curves γ_{i} × {n_{i}} is acylindrical and hence hyperbolic, by Thurston’s geometrization theorem (actually more simply in the case of the punctured torus, one can do it using Klein-Maskit combination theorems, combining a sequence of punctured torus groups with accidental parabolics across totally geodesic thrice-punctured sphere boundaries).

11.5. The exterior of the convex core. There is one last detail to tidy up: We have a proper degree 1 map f : M → Ĉ(N), which in the case of two degenerate ends can be written as f : M → N and lifts to f̃ : M̃ → H^{3}. When there is a geometrically finite end, Ĉ(N) ≠ N and we would like to get a map which covers all of N in this case too.

Recall that Ĉ(N) is the union of an r-neighborhood of the convex core C(N), and the Margulis tubes of short non-internal curves, if any. There is a submanifold C(M) in M which corresponds to C(N), defined as follows: suppose B is a boundary block with solid torus U. Then ∂U is composed of four annuli A_{0}, A_{1}, A_{L} and A_{R} as in Section 10 where A_{1} or A_{0} (or both) lie in ∂M. (in the case of a block with parabolic, one of A_{0} or A_{1} is actually two annuli). Thus Y = ∂U \ ∂M is a union of one or two annuli. Recalling that U has been given the metric of a Margulis tube, let C(Y) denote the convex hull of Y in U with respect to this metric. Let C(M) be the submanifold of M obtained by replacing U with C(Y) for each of the (0, 1 or 2) boundary blocks.

We can now append to C(M) a collar of the form ∂C(M) × [0, ∞) with metric

ds^{2} cosh^{2} t + dt^{2},  (11.2)

where ds^{2} is the metric on ∂C(M) and t ∈ [0, ∞). Let \overline{M} denote C(M) union this collar. We extend f to this collar to obtain \overline{f} : \overline{M} → N via the following observations:

For each boundary block as above, we note that in N, the boundary of the Margulis tube T corresponding to U meets C(N) in one or two annuli, whose rneighborhood is the image f(Y). Since, in the universal cover, the lifts of T and C(N) are both convex, it follows that C(N)∩T is the convex hull in T of C(N)∩∂T.

One can now apply the methods of Epstein-Marden [30] to see that f restricted to ∂C(Y) is homotopic rel boundary to a bilipschitz map from ∂C(Y) to T ∩ ∂C_{r}(N). (The constants can in fact be made independent of the group, but this does not matter to us).

Epstein-Marden also show in [30] that the exterior N \C(N) is bilipschitz equivalent to a metric of the form (11.2), where ds^{2} is the metric on ∂C(N). It therefore follows, after an appropriate interpolation in a collar ∂C(M) × [0, 1], that \overline{f} can be made a proper degree 1 map from \overline{M} to N, which is bilipschitz in ∂C(M) × [1, ∞), and lifts to a quasi-isometry ˜ \overline{f} : ˜ \overline{M} → H^{3}.

12. Proofs of the main theorems

12.1. The ending lamination theorem. We are now ready to prove Theorem A. We may proceed as in [70]. If N_{1} and N_{2} have the same ending invariants we obtain liftable quasi-isometries f_{1}, f_{2} from the same model manifold \overline{M} to N_{1} and N_{2}. Thus the map f̃_{1} ∘ f̃_{2}^{−1}, where f̃_{2}^{−1} is any quasi-inverse of f̃_{2}, gives a quasiisometric conjugacy of the two group actions in H^{3}. This extends at infinity to a quasi-conformal conjugacy of the two group actions on the sphere (see Mostow [73], and in a more general context [34, 27]). Furthermore, if the domain of discontinuity is non-empty the equality of ending invariants implies that the map can be made conformal from Ω_{1} to Ω_{2} (on the quotient surfaces there is a bounded homotopy to a conformal map, and this can be lifted to a homotopy which is constant on the limit set). Sullivan’s theorem [83] states that a quasi-conformal conjugacy of finitely generated Kleinian groups which is conformal on the domains of discontinuity is in fact an element of PSL_{2}(C), so we are done.

12.2. The rigidity theorem. In this section we prove Theorem C. Consider two punctured torus groups Γ_{1}, Γ_{2} whose actions on the sphere are topologically conjugate: there exists a homeomorphism ψ : Ĉ → Ĉ such that Γ_{2} = ψΓ_{1}ψ^{−1} in homeo(Ĉ). We wish to prove that ψ can be replaced by a quasiconformal or antiquasiconformal map.

Note first that it suffices to consider orientation-preserving ψ (if not, let ψ′ = R ∘ ψ where R is a Möbius inversion and apply the argument to Γ′_{2} = RΓ_{2}R^{−1}). Thus we would like to show that Γ_{2} = ΦΓ_{1}Φ^{−1} where Φ is quasiconformal.

The conjugacy gives an identification of Γ_{1} and Γ_{2} as groups, so we may view them both as representations ρ_{1}, ρ_{2} of a fixed copy of π_{1}(S). Let (ν_{−}^{i}, ν_{+}^{i}) denote the ordered pair of ending invariants of ρ_{i} where i = 1, 2.

Let α_{j} be a sequence of elements of C, with geodesic representatives α_{j}^{i} in N_{i} = H^{3}/Γ_{i} (i=1,2). We can characterize whether α_{j}^{i} leave every compact set in N_{i} as follows: Let F_{i}(α_{j}) be the set of pairs of fixed points (x, y) ∈ Ĉ of the elements in the conjugacy class in Γ_{i} determined by α_{j}. Then α_{j}^{i} leaves every compact set if and only if d_{j}^{i} = sup{d(x, y) : (x, y) ∈ F_{i}(α_{j})} go to zero as j → ∞ (where d(·, ·) denotes spherical distance). This property is preserved by a homeomorphism of the sphere (compare Ohshika [75]). Thus a sequence converges to an ending lamination of Γ_{1} if and only if it converges to an ending lamination of Γ_{2}.

We conclude that any irrational ending invariant of Γ_{1} appears as an invariant of Γ_{2}. The same holds for a rational invariant by an easier argument, since ψ must conjugate parabolics to parabolics. Suppose, say, ν_{+}^{1} ∈ R̂. If ν_{+}^{1} = ν_{+}^{2} (in which case, if ν_{−}^{1} ∈ R̂ then ν_{−}^{1} = ν_{−}^{2}), we can apply the ending lamination theorem directly to conclude that Γ_{1} and Γ_{2} are quasiconformally conjugate, and we are done.

Suppose then that ν_{+}^{1} = ν_{−}^{2} (necessarily then ν_{−}^{1} = ν_{+}^{2} if ν_{−}^{1} ∈ R̂), and let us derive a contradiction. If R is a Möbius inversion then the conjugate representation ρ′_{2} defined by ρ′_{2}(g) = Rρ_{2}(g)R^{−1} has reversed ending invariants (ν_{+}^{2}, ν_{−}^{2}). (See section 3 to see how orientation induces an ordering on ν_{±}.) Applying the ending lamination theorem to ρ_{1} and ρ′_{2}, we get a qc-homeomorphism F conjugating Γ_{1} to Γ′_{2} = ρ′_{2}(π_{1}(S)) and inducing the same identification as R∘ψ. Thus the composition Φ = F^{−1}∘R∘ψ is an orientation-reversing homeomorphism conjugating Γ_{1} to itself, and acting as the identity on the group.

This can of course happen if Γ_{1} is quasi-Fuchsian and Φ exchanges the components of Ω_{Γ_{1}}, but we claim now that this is the only case. Since the conjugation is the identity on the group, Φ must restrict to the identity on Λ_{Γ_{1}}, which is the closure of the set of fixed points of Γ_{1}. Thus if Λ_{Γ_{1}} = Ĉ we have Φ = id and it can’t be orientation-reversing.

Suppose Ω_{Γ_{1}} is non-empty. For any component O of Ω_{Γ_{1}}, Φ is the identity on ∂O so either Φ(O) = O or Φ(O) = Ĉ − \overline{O}. The latter is impossible when Γ_{1} is not quasi-Fuchsian since Ĉ − \overline{O} is empty or contains some of the limit set. The former implies that Φ descends to an orientation-reversing homeomorphism of the quotient surface O/Γ_{1} which induces the identity on its fundamental group, and this is also impossible. This contradiction implies that, in fact, the ending laminations must have matched up correctly to begin with.

Note that, in case both ν_{±}^{1} are in D or in Q̂, the same holds for ν_{±}^{2}, and in this case (the geometrically finite case) the theorem follows from Marden’s isomorphism theorem [58].

12.3. Topology of the deformation space. In this section we give the proof of Theorem B. Let R = R(π_{1}(S)) denote the space of representations of π_{1}(S) into PSL_{2}(C) with parabolic commutator, modulo conjugation of the image in PSL_{2}(C). In the natural topology on R (known as the “algebraic topology”), the subset D of discrete, faithful representations is closed (by a theorem of Chuckrow, see [49]) and has non-empty interior (precisely the quasi-Fuchsian representations, by theorems of Marden [58] and Sullivan [84]).

Let Δ denote the diagonal of S^{1} × S^{1}. The ending invariant construction gives a map

ν : D → (\overline{D} × \overline{D}) \ Δ

associating to a representation ρ its ordered pair of ending invariants. Points in Δ are never obtained: the same element cannot be parabolic on both ends of the manifold, and more generally the two ending laminations must be distinct (see Thurston [86] or Bonahon [14]).

It is perhaps surprising that the map ν is not continuous. As pointed out to the author by Dick Canary, one can construct sequences of representations ρ_{i} which converge to some ρ, but for which the invariants (ν_{−}^{i}, ν_{+}^{i}) converge to the diagonal Δ (for example take some α ∈ C and consider the sequence (D_{α}^{i}ν_{0}, D_{α}^{2i}ν_{0})). The construction, which is related to the Kerckhoff-Thurston example in [55], appears in Anderson-Canary [6]. See also McMullen [66] for a further discussion of this phenomenon.

Nevertheless the following fact does hold:

Lemma 12.1. Let {ρ_{i}} be a sequence of marked punctured-torus groups, whose ending invariants (ν_{−}^{i}, ν_{+}^{i}) = ν(ρ_{i}) converge to a pair (ν_{−}^{∞}, ν_{+}^{∞}) which is not in Δ. Then, ρ_{i} converge in D to a representation ρ, and its ending invariants are ν(ρ) = (ν_{−}^{∞}, ν_{+}^{∞}).

Remark. The conclusion that a convergent subsequence exists (up to conjugation in PSL_{2}(C)) is a special case of Thurston’s Double Limit Theorem (see Thurston [85] or Otal [77]), though it will follow from our analysis as well. Our real interest here is in the fact that the limiting end invariants have the expected values.

Proof. Suppose first that ν_{−}^{∞}, ν_{+}^{∞} ∈ D. Then eventually ν_{±}^{i} ∈ D as well, and we have convergence in the space of quasi-Fuchsian groups. In this case the result is already known, by the Ahlfors-Bers theory [10, 12].

More generally, suppose that one invariant, say ν_{+}^{∞}, is in D. This is essentially the case of convergence in a Bers slice. Let σ be a metric representing ν_{+}^{∞} on S, and f_{i} : (S, σ) → Ω_{+}(ρ_{i})/ρ_{i}(π_{1}(S)) be the extremal quasiconformal map inducing ρ_{i} on the fundamental group. Since ν_{+}^{i} → ν_{+}^{∞}, the dilatation K_{i} of f_{i} converges to 1. Thus the lifts f̃_{i} : S̃ → Ω_{+}(ρ_{i}), after appropriate normalization, converge on compact sets, by general compactness theorems for quasiconformal maps (see Lehto-Virtanen [57]), to a conformal embedding f_{∞} : S̃ → Ĉ, and this map induces a limit representation ρ on the fundamental groups. It follows that f_{∞}(S̃) = Ω_{+}(ρ), and hence the top invariant of ρ is ν_{+}.

We now consider the limiting behavior of invariants that escape the interior. Following the discussion in Section 4, we have for each i = 1, …, ∞ a pair of points α_{±}^{i} ∈ ∂D, an edge set E^{i} = E(α_{−}^{i}, α_{+}^{i}), and a pivot sequence with internal pivot subsequence P^{i} and P_{0}^{i}, respectively.

Let us assume for the moment that, if one of ν_{±}^{∞} lies in the interior then it is generic in the sense that α_{±}^{∞} is uniquely determined. Hence the same will eventually be true for ν_{±}^{i}. We’ll return to this point at the end.

We claim that every pivot in P^{∞} is eventually in P^{i}. Consider an edge e ∈ E^{∞}. It separates α_{−}^{∞} from α_{+}^{∞}. If ν_{+}^{∞} ∈ ∂D, we have ν_{+}^{∞} = α_{+}^{∞} and hence ν_{+}^{i} → α_{+}^{∞}. It then follows that α_{+}^{i} → α_{+}^{∞}. If ν_{+}^{∞} lies in the interior then (by the assumption of genericity) eventually ν_{+}^{i} is sufficiently close that α_{+}^{i} = α_{+}^{∞}. The same argument applies to α_{−}^{∞}.

Thus in any case we find that eventually e separates α_{−}^{i} from α_{+}^{i}, and hence e ∈ E^{i}. It follows that every internal pivot of P^{∞} eventually lies in P^{i}.

Suppose that α_{+}^{∞} is in C, and hence is the last, non-internal, pivot of P^{∞}. If ν_{+}^{∞} ∈ D then as above, eventually α_{+}^{i} = α_{+}^{∞}, and in particular α_{+}^{∞} is the last vertex of P^{i}. If ν_{+}^{∞} = α_{+}^{∞} something interesting can happen: Normalize D as H^{2} where α_{+}^{∞} = ∞. In this normalization, |ν_{+}^{i}| → ∞. If Im ν_{+}^{i} → ∞ then eventually α_{+}^{i} = ∞ = α_{+}^{∞}. If not, then | Re ν_{+}^{i}| → ∞, and since ν_{−}^{i} are converging to ν_{−}^{∞} ≠ ∞, eventually an arbitrary number of vertical Farey edges separate α_{−}^{i} from α_{+}^{i}, so that α_{+}^{∞} is an internal vertex of P^{i}.

The same argument applies to α_{−}^{∞}, so our claim is proved.

Suppose now that P^{∞} contains at least two pivots (equivalently that α_{−}^{∞} ≠ α_{+}^{∞}). Let us number them α_{0} and α_{1}, and let B_{0}^{i}, B_{1}^{i} be the corresponding blocks in the model M_{i} for large enough i. The surface S^{i}(1) = ∂_{0}B_{1}^{i} = ∂_{1}B_{0}^{i} has a metric where both α_{0} and α_{1} have shortest length, so we may identify these surfaces, by the identity on the underlying S, for all i. The map f_{i}|_{S(1)} induces the representation ρ_{i}. Thus, after conjugating the image groups appropriately, or equivalently after choosing a lift of f_{i}|_{S(1)} to H^{3} that maps a fixed basepoint to the origin, we find that the generators of the group associated to α_{0} and α_{1} have bounded translation lengths, and hence (by Chuckrow’s theorem) there is a convergent subsequence with some limit representation ρ_{∞}. Let us reindex this as if it were the whole sequence.

Taking a further subsequence, we may also assume by Theorem 11.2 that there is a geometric limit f_{∞} : M_{∞} → N_{∞}, so that S(1) still embeds in M_{∞} between the first two blocks, and f_{∞}|_{S(1)} induces the representation ρ_{∞}. We now wish to determine the ending invariants of ρ_{∞}.

The width sequence for P^{∞} is, by the above discussion, exactly reproduced as a subset of the width sequence for M_{∞}, and the last pivot α_{+} of P^{∞} (if there is one) appears as the first (lowest-numbered) cusp in M_{∞} above S(1). Similarly the first pivot α_{−} of P^{∞}, if there is one, appears as the last cusp below S(1).

Lift M_{∞} to the cover M̃_{∞} corresponding to S(1), and lift f_{∞} to a map from M̃_{∞} to the corresponding cover Ñ_{∞} of N_{∞}. The map induced from π_{1}(S) = π_{1}(M̃_{∞}) to π_{1}(Ñ_{∞}) is just the limit representation ρ_{∞}. If there is a cusp of M_{∞} above S(1) corresponding to α_{+}, it lifts to a rank 1 cusp T̃ in the top half of M̃_{∞}, where we use our orientation convention to order the unbounded components of M̃_{∞} \ S(1). Since f̃_{∞} is now a degree 1 proper map between manifolds homeomorphic to S ×R, it follows that f̃_{∞}(T̃) is in the unbounded component of Ñ_{∞} \ f̃_{∞}(S(1)) which is a neighborhood of its e_{+} end. Thus the e_{+} ending invariant of ρ_{∞} is exactly α_{+}^{∞} = ν_{+}^{∞} in this case.

If in fact ν_{+}^{∞} is irrational then P^{∞} is infinite in the forward direction, and so is M_{∞}. The pivots in M^{∞} therefore converge (in the normalization determined by α_{0} and α_{1}) to ν_{+}^{∞}. In this case the portion of M_{∞} above S(1) lifts homeomorphically to M̃_{∞}, and maps to the e_{+} end of Ñ_{∞}. Thus the e_{+} invariant of ρ_{∞} is again ν_{+}^{∞}.

The same argument applies to e_{−} when ν_{−}^{∞} is a boundary point.

We have shown that some subsequence of the original {ρ_{i}} converges, and that for every limit ρ that can arise, the invariants ν(ρ) are exactly (ν_{±}^{∞}). The Ending Lamination Theorem (Theorem A) now implies that all of these limits are equal, and hence in fact the entire sequence converges. (We remark, however, that the set of possible geometric limits can be quite large, as we have seen).

We have left out the possibility that α_{−}^{∞} = α_{+}^{∞}. Since ν_{+}^{∞} and ν_{−}^{∞} are different by assumption, this can only occur if at least one of them, say ν_{+}^{∞}, is in the interior. By the Bers-type argument in the beginning of the proof we therefore have a convergent subsequence, and the rest of the argument goes through in the same way.

Finally, let us return to the minor assumption that ν_{+}^{∞} (similarly ν_{−}^{∞}) is generic, in the sense that if it lies in D it is closest to a unique α_{+}^{∞} ∈ C. If this should be false then there are two or three possible choices, and any one can be made. However in the sequence ν_{+}^{i} → ν_{+}, perhaps for each i a unique α_{+}^{i} can be chosen, and not the same for each i. This is easily resolved by separating the sequence into a finite number of subsequences in each of which α_{±}^{i} are eventually constant. The limits for each subsequence will agree, as above.

The fact that the map ν is surjective is already known: Bers’ Simultaneous Uniformization theorem implies that every pair in D × D is obtained and in fact that ν is a homeomorphism from the quasi-Fuchsian groups to D×D. The boundary points of \overline{D} × \overline{D} \ Δ can be reached using work of Maskit [61] and Keen-Maskit-Series [54] for rational points and Thurston [85, 77] for irrational points. See also Ohshika [74] for generalizations. Our Lemma 12.1, which reproduces Thurston’s theorem in our setting, also accomplishes this.

The ending lamination theorem serves to show that ν is injective, and hence

ν^{−1} : (\overline{D} × \overline{D}) \ Δ → D(π_{1}(S))

is well defined. Although ν is not a homeomorphism (it is discontinuous, as above), Lemma 12.1 implies exactly that ν^{−1} is continuous.

This gives the first statement of Theorem B, and the fact that every representation in D is the limit of quasi-Fuchsian representations. Now let us consider Bers and Maskit slices.

A Bers slice, in our terminology, is the image by ν^{−1} of a slice \{x_{0}\} × \overline{D}, where x_{0} ∈ D. Bers defined these slices for a general surface S (see Gardiner [33]), as a subset of D(π_{1}(S)) where the “bottom” invariant component of Ω has quotient equal to a fixed point in the Teichmüller space of S. He showed that such a slice is compact and conjectured that it was the closure of its interior, identified with the Teichmüller space of S.

In our case, since ν^{−1} is continuous and \{x_{0}\}×\overline{D} is compact, the image must be compact and the restriction of ν^{−1} is a homeomorphism. Hence Bers’ conjecture holds for the punctured torus.

A Maskit slice is a similar construction, where x_{0} is taken to be a rational point on the boundary. In this case the point (x_{0}, x_{0}) must be omitted so the slice is ν^{−1}(\{x_{0}\} × (\overline{D} \ \{x_{0}\})). (Again, the original definition of Maskit applies for the higher genus case as well, and does not depend on the existence of the irrational end invariants.)

Since the Maskit slice is not compact, we must be more careful in showing that ν^{−1} restricts to a homeomorphism. It suffices to show that ν^{−1}|_{\{x_{0}\}×(\overline{D}\\{x_{0}\})} is proper, or in other words given a sequence (x_{0}, y_{i}) with y_{i} → x_{0}, show that the corresponding representations ρ_{i} must diverge. Consider again our pivot sequence for (x_{0}, y_{i}).

Recall that y_{i}(x_{0}) is the image of y_{i} in a fixed normalization of H^{2} sending x_{0} to ∞. Suppose first that Im y_{i}(x_{0}) → ∞. Then x_{0} is represented by a very short curve on both sides of the convex core for each ρ_{i}. An annulus between these two curves therefore cuts the convex core into a thrice-punctured sphere times an interval, and every curve in S intersecting x_{0} must have geodesic representative crossing this annulus, so its length will go to infinity as i → ∞. It follows that the translation lengths diverge for at least one of any generator pair, so the sequence of representations diverges. This is in fact the exceptional geometric limit we have mentioned before, in which there is one double-boundary block in the model and the geometric limit is a thrice-punctured sphere group.

The remaining case is that Im y_{i}(x_{0}) remain bounded and hence Re y_{i}(x_{0}) → ±∞.

In this case we can apply the pivot sequence analysis to show that there is a sequence of integers β_{i} → ±∞ representing neighbors of x_{0} in C, with ℓ_{ρ_{i}}(β_{i}) bounded. Namely, if y_{i} is in a triangle T_{i} with vertex x_{0}, take β_{i} to be one of the other vertices. Otherwise take β_{i} to be the first internal pivot in the pivot sequence for ρ_{i}. (More concretely, β_{i} is the integer part of Re y_{i}(x_{0}), plus or minus 1).

Now letting tr_{n}^{i} denote the trace in the representation ρ_{i} of the neighbor of x_{0} indexed by n, we may apply the trace identity (7.2) together with the parabolic commutator condition, as in Section 7, to conclude that tr_{n}^{i} = tr_{0}^{i} +cn where c = ±2\sqrt{−1}. Since tr_{β_{i}}^{i} stays bounded we conclude that tr_{0}^{i} → ∞. Thus again there is no fixed choice of generators that converge.

This proves that the Maskit slice is a homeomorphic image of a disk minus a boundary point.

Boundaries of embeddings. Both Maskit and Bers slices have well-known embeddings into finite-dimensional vector spaces. In the case of the punctured torus these have one complex dimension (see Wright [89] and Keen-Series [53] for a thorough development and good pictures of this embedding for a Maskit slice). Our results imply in particular that the the boundaries of these embeddings are Jordan curves.

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State University of New York, Stony Brook, NY 11794

   E-mail address: yair@math.sunysb.edu