PARADOXICAL DECOMPOSITIONS,

2-GENERATOR KLEINIAN GROUPS,

AND VOLUMES OF HYPERBOLIC 3-MANIFOLDS

Marc Culler^{1} and Peter B. Shalen^{2}

Introduction

The ϵ-thin part of a hyperbolic manifold, for an arbitrary positive number ϵ, is defined to consist of all points through which there pass homotopically non-trivial curves of length at most ϵ. For small enough ϵ, the ϵ-thin part is geometrically very simple: it is a disjoint union of standard neighborhoods of closed geodesics and cusps. (Explicit descriptions of these standard neighborhoods are given in Section 1.) If ϵ is small enough so that the ϵ-thin part of M has this structure then ϵ is called a Margulis number of M. There is a positive number, called a 3-dimensional Margulis constant, which serves as a Margulis number for every hyperbolic 3-manifold.

The results of this paper provide surprisingly large Margulis numbers for a wide class of hyperbolic 3-manifolds. In particular we obtain the following result, which is stated as Theorem 10.3:

Let M be a closed orientable hyperbolic 3-manifold whose first Betti number is at least 3. Then log 3 = 1.09 … is a Margulis number for M.

Knowing a Margulis number for a given manifold provides important geometric information. For example, one can give an estimate of the volume of a hyperbolic manifold M in terms of a Margulis number ϵ for M. One first observes that M must contain a hyperbolic ball of radius ϵ/2; indeed, any point in the thick part of M is the center of such a ball. The volume of this ball is a lower bound for the volume of M. This bound can be improved by using an observation of Meyerhoff’s [Me]: the ratio of the volume of M to that of the ball is at least d(ϵ/2), where d(r) is an explicitly given function which, by a theorem of Böröczky, is a bound for the local density of a radius r sphere-packing in hyperbolic space. This technique gives the following corollary (10.4) to Theorem 10.3:

Let M be a closed orientable hyperbolic 3-manifold whose first Betti number is at least 3. Then M contains a hyperbolic ball of radius \frac{1}{2} log 3 = .54 … and the volume of M is greater than .92.

Typeset by AMS-TEX

1. Partially supported by the National Science Foundation, the Sloan Foundation and the Mathematical Sciences Research Institute

2.Partially supported by the National Science Foundation

The estimates given by the above results should be compared with other known estimates and with known examples. The greatest known lower bound for the volume of an arbitrary closed orientable hyperbolic 3-manifold, due to F. Gehring and G. Martin [GeM], is .00115, improving an earlier estimate of .00082 by R. Meyerhoff. The smallest known orientable hyperbolic 3-manifold in terms of volume was discovered by J. Weeks; it has Betti number 0 and has a volume of approximately .94. A result of Shalen and Wagreich [ShW] implies, for a 3-manifold M satisfying somewhat weaker hypotheses than those of Theorem 10.3, that \frac{1}{2} log 3 is a Margulis number for M (so that M contains a hyperbolic ball of radius \frac{1}{4} log 3).

It can be proved that log 3 is not a Margulis number for the complement of the figure 8 knot, and hence that it is not a Margulis constant. Furthermore, there is overwhelming evidence, based on computer experiments by C. Hodgson and J. Weeks, that the largest hyperbolic ball in the Weeks manifold has radius .51 … < \frac{1}{2} log 3. Thus Theorem 10.3, and — according to the experimental evidence — Corollary 10.4 as well, become false if one drops the topological restrictions on the manifold.

Theorem 10.3 is an application of a theorem on 2-generator Kleinian groups. We may regard the orientable hyperbolic 3-manifold M as the quotient of the hyperbolic 3-space H^{3} by a discrete group Γ of orientation-preserving isometries; from this point of view, ϵ is a Margulis number for M if and only if, for any two non-commuting elements ξ and η of Γ, every point of H^{3} is moved a distance at least ϵ by either g or h.

A 3-manifold is said to be topologically tame if it is homeomorphic to the interior of a compact 3-manifold; a torsion-free Kleinian group Γ is topologically tame if H^{3}/Γ is topologically tame. We shall prove the following result, which is a stated as Theorem 9.1:

Let ξ and η be non-commuting isometries of H^{3}. Suppose that ξ and η generate a torsion-free discrete group which is topologically tame, is not co-compact and contains no parabolics. Then every point of hyperbolic space is moved a distance at least log 3 by either ξ or η; that is, we have

max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3

for any z ∈ H^{3}.

In Section 10 we show (Proposition 10.2) that if M = H^{3}/Γ has first Betti number at least 3 then every two-generator subgroup of Γ is non-co-compact and topologically tame. Thus Theorem 9.1 on two-generator groups implies Theorem 10.3 about Margulis numbers.

It is conjectured that any hyperbolic 3-manifold with finitely generated fundamental group is topologically tame. If this were true it would allow one to generalize the above result about Margulis numbers to any hyperbolic 3-manifold whose fundamental group has no 2-generator subgroup of finite index.

We now describe the ingredients of the proof of our main theorem. The proof begins with the observation (Propositon 9.2) that if ξ and η are non-commuting isometries of H^{3} and if the group which they generate is torsion-free, is not co-compact and contains no parabolics then it is free of rank 2.

One novel feature of the proof is the use of a construction, due to Patterson and studied extensively by Sullivan, of a geometrically natural measure on the limit set of a Kleinian group. We use such a measure to relate the combinatorial structure of the free group of rank 2 to its action on its limit set. This part of the argument bears an intriguing resemblance to the construction of a decomposition of the sphere in the Hausdorff-Banach-Tarski paradox [Wag]. By viewing the free group of rank 2 as a set of reduced words in the alphabet {x, y, x^{−1}, y^{−1}}, one obtains a decomposition of the group into four infinite sets and a singleton {1}, where each of the infinite sets consists of all words beginning with a given letter. This decomposition has the peculiar property that each of the infinite sets is mapped onto the complement of another of the infinite sets under left multiplication by a certain letter. For example, left multiplication by x^{−1} sends a word which begins with x to a word which does not begin with x^{−1}.

In the Hausdorff-Banach-Tarski construction one has a rank-2 free group acting by isometries on the sphere. A choice of a point in a given free orbit determines an identification of the group with the orbit and hence induces a decomposition of the orbit. By choosing a point from each orbit one obtains a decomposition of the sphere (ignoring the countable subset of points with non-trivial stabilizer.) Since in this situation the group action preserves the area measure of the sphere, one obtains a paradoxical conclusion.

In our situation we have a free group of rank 2 acting by isometries on hyperbolic space, and hence acting by Möbius transformations on the sphere at infinity. Using a generalization of Patterson’s construction we are able to construct a measure on the limit set of our group which decomposes as a sum of four measures, each of which is transformed to the complement of another by a generator or the inverse of a generator. The key to this construction is the fact that the Patterson measures are obtained as limits of measures which are supported on an orbit of the group, and hence reflect the combinatorial structure of the group itself. It is necessary here to identify the group with an orbit by making a choice of a point z in hyperbolic space.

This construction is especially interesting in the case where the Patterson measure that admits the decomposition is equal to the area measure on the sphere at infinity. (Here the sphere is given the round metric obtained by identifying H^{3} with a ball so that the given point z is the center, and the area measure is normalized so as to have total mass 1.) In this case the “paradoxical” decomposition leads to an elementary proof of the inequality max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3 By symmetry considerations together with a measure-theoretic lemma one obtains a subset of the sphere which has area at most 1/4 but whose image under one of the generators has area at least 3/4. A direct computation then shows that this element moves the center a distance at least log 3.

Given a Patterson measure associated to a discrete group Γ, one can construct a positive, Γ-invariant (generalized) eigenfunction of the hyperbolic Laplacian by integrating the hyperbolic Poisson kernel against the measure. This eigenfunction has a positive eigenvalue and therefore determines a superharmonic function on H^{3}/Γ (which is a hyperbolic manifold if Γ is torsion-free). This allows one to prove that if every superharmonic function on H^{3}/Γ is constant then the area measure is the unique Patterson measure associated to Γ.

An argument due to Thurston shows that certain geometric conditions on the ends of a complete hyperbolic manifold M = H^{3}/Γ imply that all positive superharmonic functions on M are constant. In our setting the appropriate condition is that there exist a sequence of singular surfaces having metrics of intrinsic curvature ≤ −1 and bounded genera, such that every compact subset of M is “enclosed” in a homological sense by a surface in the sequence. By a theorem of Canary’s, such a sequence exists if Γ is topologically tame, contains no parabolics and has limit set equal to the entire sphere at infinity of hyperbolic space. Thus we need only to establish the inequality in the conclusion of Theorem 9.1 in the case that the limit set of Γ is a proper subset of the sphere at infinity, i.e. when Γ has non-empty set of discontinuity.

Notice, however, that if the set of discontinuity is non-empty then the area measure cannot be a Patterson measure. Our strategy is to reduce the case where Γ has non-empty set of discontinuity to the more exotic case where the area measure is a Patterson measure by considering limits of groups with non-empty set of discontinuity. There is an open subset 𝔊𝔉 of the variety PSL_{2}(C) × PSL_{2}(C) consisting of all pairs of elements (ξ, η) such that ⟨ξ, η⟩ is a free group of rank 2 having a non-empty set of discontinuity and containing no parabolics. If (ξ, η) is any point in the closure \overline{𝔊𝔉} (in the complex topology) of 𝔊𝔉, then ⟨ξ, η⟩ is still free and discrete. It is easy to show that for any point z ∈ H^{3} the function (ξ, η) ↦ max ( dist(z, ξ · z), dist(z, η · z)) is a proper continuous function on \overline{𝔊𝔉} with no local minimum on 𝔊𝔉. Hence it takes a minimum value at some point (ξ, η) in the frontier 𝔅 of 𝔊𝔉. This reduces the proof of the inequality max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3 for the case (ξ, η) ∈ 𝔊𝔉 to the proof in the case (ξ, η) ∈ 𝔅. We then complete the proof by showing that there is a dense G_{δ} in 𝔅 consisting of points for which the area measure is indeed a Patterson measure. We use Thurston’s criterion here as well; we show that, for (ξ, η) in a dense G_{δ} subset of 𝔅, there exists a sequence of negatively curved singular surfaces of bounded genera enclosing every compact set in H^{3}/⟨ξ, η⟩. The proof of the existence of this dense G_{δ} contains much of the technical work in the paper.

Each of the singular surfaces that we construct is a map of a closed surface into M = H^{3}/⟨ξ, η⟩. The domain surface is divided into “pairs of pants” by a family of “waist” curves which are mapped to closed geodesics in M. Furthermore, the surfaces in the sequence are girded in the sense that the lengths of the waist geodesics tend to 0 through the sequence. The existence of a sequence of singular surfaces of this type is proved by means of a recent result of Curt McMullen’s asserting the density of maximal cusps in 𝔅. The singular surfaces are chosen to be invariant with respect to a canonical involution of M; such an involution exists because π_{1}(M) is a 2-generator group. By combining this invariance property with the fact that the waist lengths tend to 0, we show that any compact subset of M is enclosed by some surface in the family.

Our construction of girded surfaces works only in the 2-generator case because it uses the involution. However, we believe that a similar picture holds more generally. Let Γ be any geometrically finite Kleinian group without parabolics, and let M be the manifold defined by a generic point in the frontier of the set of quasi-conformal deformations of Γ. We conjecture that M is girded, meaning that there exists a sequence of girded surfaces in M such that every compact subset of M is enclosed by a surface in the sequence. For a precise definition of a girded manifold, see 7.4.

It is worth noting that the singular surfaces which appear in the proof of Canary’s theorem are not required to contain short geodesics. In general a topologically tame manifold with geometrically infinite ends need not contain arbitrarily short geodesics, and thus need not be girded. Conversely, it is not clear in general whether a girded manifold with finitely generated fundamental group is topologically tame.

The paper is organized as follows. In the first section we establish notation and collect the basic facts about hyperbolic manifolds which will be needed in the paper. The next two sections discuss the correspondence, via the Poisson kernel, between measures on the sphere at infinity and eigenfunctions of the hyperbolic Laplacian in a Γ-equivariant setting. In essence this material is classical potential theory, with the Poincaré ball replacing the Euclidean ball.

In Section 4 we present Patterson’s ideas in a generalized setting to give a construction of a Γ-invariant conformal density which respects a given decomposition of a discrete group Γ. Section 5 contains the argument which is used in proving the main estimate in the case where the area measure is the unique Patterson measure. In Section 6 we describe Thurston’s geometric criterion which implies that every positive superharmonic function is constant. We introduce the notion of a Bonahon surface and formulate the criterion in terms of these surfaces.

In Section 7 we develop the notions that we use in constructing our girded surfaces. Section 8 contains the proof of the existence of a dense G_{δ} in 𝔅 consisting of girded groups. In Section 9 we prove the main theorem and in Section 10 we give the applications to closed manifolds.

The preprint of this paper contains a more self-contained and detailed discussion of the topics in Sections 4 and 6 of the current version. In addition the preprint contains a full development of the geometry of ultra-hyperbolic surfaces. The point of view taken in the preprint is somewhat different from the one taken in [Bo], being based on “packing” arguments similar to some that have been used by Thurston and Gromov.

During the course of this research we have benefited greatly from conversations with Robert Brooks, Richard Canary, Craig Hodgson, Howard Masur, Curt McMullen, Robert Meyerhoff, John Smillie and Jeff Weeks. We are indebted to Mohan Ramachandran for guiding us through the maze of potential theory, and to Francis Bonahon for his patient explanation of his ideas.

§1. Hyperbolic space and discrete groups

We establish some notation and conventions that will be used throughout the paper.

1.1. Hyperbolic space. Whenever we consider a metric space, we shall write dist(x, y) to denote the distance function, provided that no confusion can result. In a metric space we also write dist(x, S) to denote the distance from a point x to a closed set S. If S is a subset of a metric space and r is a positive number, we shall denote the closed r-neighborhood of S by nbhd_{r}(S).

By a model of hyperbolic n-space, for n ≥ 2, we shall mean a complete, 1-connected Riemannian n-manifold of constant sectional curvature −1. Throughout this paper we shall fix, abstractly, a model H^{n} of hyperbolic space. Any two models of hyperbolic n-space are isometric, and the isometry group of any model acts transitively on its orthonormal frame bundle. It will frequently be useful to identify H^{n}, via a suitable isometry, with one of the standard concrete models. The concrete models which will be used most often are the upper half-space model, which has underlying space R^{n−1} × [0, ∞), and the Poincaré model, for which has underlying space is the open unit ball B^{n} ∈ R^{n}. The geodesics in these models are intersections with lines or circles in the ambient euclidean space which are orthogonal, respectively, to the plane x_{n+1} = 0 or the unit sphere. In studying convex sets in H^{n} it is useful to consider the projective model, which also has underlying space B^{n}, but with a metric for which the geodesics are intersections of euclidean geodesics with B^{n}. For detailed descriptions of these spaces, and for general information on hyperbolic geometry, the reader is referred to [F].

The sphere at infinity of H^{n} will be denoted by S_{∞}^{n−1} or simply S_{∞}. The canonical compactification of H^{n}, of which S_{∞} is the boundary, will be denoted \overline{H}^{n}. Any selfisometry γ of H^{n} extends to a conformal automorphism of \overline{H}^{n}, which we denote γ̄. We denote by γ_{∞} the conformal automorphism of S_{∞} obtained by restricting γ̄.

Given any point z ∈ H^{n}, there exists an isometry h from H^{n} onto the Poincaré model B^{n} which maps z to 0. Furthermore, h is unique modulo composition with orthogonal linear transformations of R^{n}. It follows that if we pull back the metric of S^{n−1} = ∂B^{n} via h̄, we obtain a metric on S_{∞} which is uniquely determined by the point z; we call it the round metric centered at z.

Any isometry h of the upper half-space model onto H^{n} extends uniquely to a conformal diffeomorphism of R^{n−1} × [0, ∞) onto \overline{H}^{n} − \{ζ\}, where ζ ∈ S_{∞} is a point determined by h. We shall say that h maps ∞ to ζ (or that h^{−1} maps ζ to ∞). Given any point ζ ∈ S_{∞}, there exists an isometry of R^{n−1} × (0, ∞) onto H^{n} which maps ∞ to ζ.

If A_{1} and A_{2} are points of \overline{H}^{n} and B is a point of H^{n}, we denote by ∠A_{1}BA_{2} the angle between the ray from B to A_{1} and the ray from B to A_{2}; here by the ray from B to A_{i} we mean the (hyperbolic) ray starting at B and passing through A_{i} (if A_{i} ∈ H^{n}) or having A_{i} as an endpoint (if A_{i} ∈ S_{∞}).

1.2. Convexity. A subset K of \overline{H}^{n} is termed convex if K ≠ ∅ and if every line segment in H^{n} with endpoints in K is itself contained in K; here a “line segment” may be an entire line or a half-infinite ray. The convex hull of a non-empty subset X of \overline{H}^{n} is defined to be the intersection of all convex subsets of \overline{H}^{n} containing X, and will be denoted \overline{hull}(X). We set hull(X) = \overline{hull}(X) ∩ H^{n}. Clearly \overline{hull}(X) and hull(X) are convex.

Any isometry of H^{n} onto the projective model B^{n} extends to a diffeomorphism of \overline{H}^{n} onto \overline{B}^{n} which maps convex subsets of \overline{H}^{n} onto convex subsets (in the usual Euclidean sense) of \overline{B}^{n}. Hence any compact convex subset of \overline{H}^{n} which has non-empty interior is homeomorphic to a closed n-ball. It follows that any closed convex subset of H^{n} is the intersection of H^{n} with a topological closed n-ball in \overline{H}^{n}, and is therefore a contractible n-manifold-with-boundary.

Proposition. If K is a convex subset of H^{n}, then for any positive number R the set nbhd_{R}(K) is convex.

Proof. It suffices to show that if α ⊂ H^{n} is a compact line segment then nbhd_{R}(α) is convex. This is in turn equivalent to showing that for any line ℓ ⊂ H^{n} the set ℓ∩nbhd_{R}(α) is connected. If φ : H^{n} → R is the continuous function that assigns to each point of R^{n} its minimum distance from α, we have ℓ ∩ nbhd_{R}(α) = (φ | ℓ)^{−1}([0, R)). It therefore suffices to prove that the function φ | ℓ has no local maximum. This is an exercise in hyperbolic geometry which we leave to the reader. □

1.3. Let γ be any non-trivial orientation-preserving isometry of H^{3}. For any ϵ > 0 let us write

C_{ϵ}(γ) = {z ∈ H^{3} : dist(z, γ · z) ≤ ϵ}.

If γ is parabolic, i.e. if γ_{∞} has a unique fixed point ζ ∈ S_{∞}, then C_{ϵ}(γ) is a closed horoball based at ζ.

If γ is loxodromic then γ_{∞} has exactly two fixed points, say ζ_{−} and ζ_{+}, in S_{∞}. The line A joining ζ_{−} and ζ_{+} is by definition the axis of γ, and γ | A is a translation through some distance length(γ). The distance from a point to its image under γ depends on the twist angle of γ as well as on length(γ) and the distance of the point from the axis A. However, the following result is sufficient to permit us to describe C_{ϵ}(γ) in this case.

Proposition. Let γ be a loxodromic isometry of H^{3} with axis A. Then there is a monotonically increasing continuous function f = f_{γ} : [0, ∞) → [0, ∞) such that dist(z, γ · z) = f( dist(z, A)) for every point z ∈ H^{n}.

Proof. Left to the reader. □

It follows from the above proposition that if γ is loxodromic with axis A then we have C_{ϵ}(γ) = ∅ for any ϵ < length(γ), and for any ϵ ≥ length(γ) we have C_{ϵ}(γ) = nbhd_{δ}(A_{γ}), where δ is a non-negative number depending on γ and ϵ. (In the notation of the proposition we have δ = f_{γ}(ϵ).)

1.4. By a Kleinian group we shall mean a discrete group of orientation-preserving isometries of H^{3} which is non-elementary in the sense that it has no abelian subgroup of finite index. If Γ is a Kleinian group we shall write Λ_{Γ} for its limit set.

When Γ is a torsion-free Kleinian group we shall write M(Γ) for the complete hyperbolic 3-manifold H^{3}/Γ. The set hull(Λ_{Γ}) is closed since Λ_{Γ} is compact, and non-empty since Γ is non-elementary. Thus the set nbhd_{1} ( hull(Λ_{Γ})) ⊂ H^{3} is closed and has non-empty interior. It is convex by Proposition 1.2, and is clearly Γ-invariant. Hence by 1.2, nbhd_{1} ( hull(Λ_{Γ})) is a contractible, Γ-invariant 3-manifold-with-boundary. We shall write

N(Γ) = ( nbhd_{1} ( hull(Λ_{Γ})))/Γ ⊂ M(Γ).

By the discussion above, N(Γ) is a 3-manifold-with-boundary and a deformation retract of M(Γ).

1.5. Margulis numbers. The group of all orientation-preserving isometries of H^{3} is isomorphic to PSL_{2}(C). In PSL_{2}(C), every element of order > 2 has an abelian centralizer. Hence if Γ is a torsion-free Kleinian group, every non-trivial element of Γ lies in a unique maximal abelian subgroup. If H is any maximal abelian subgroup of Γ then either H is cyclic and H − {1} consists of loxodromic elements with a common axis, or H is free abelian of rank 1 or 2 and H − {1} consists of parabolic elements with a common fixed point in S_{∞}. In the latter case H will be called a cuspidal subgroup of Γ.

For any point P ∈ M(Γ) we shall denote by short(P) the infimum of the lengths of all homotopically non-trivial loops based at P. Thus for any z ∈ H^{3} lying in the fiber above P we have

short(P) = inf_{1≠γ∈Γ} dist(z, γ · z).

We have 0 < short(P) ≤ ∞, and short(P) = ∞ if and only if Γ = {1}. When Γ ≠ {1}, the infimum in the definition of short(P) is realized. For any interval I ⊂ (0, ∞) we denote by M_{I}(Γ) the set of all points P ∈ M(Γ) such that short(P) ∈ I. Thus for any ϵ > 0, the set M_{(0,ϵ]} consists of all points of M through which there pass homotopically non-trivial closed curves of length ≤ ϵ. We call M_{(0,ϵ]} the ϵ-thin part of M.

Let M̃_{(0,ϵ]} denote the pre-image of M_{(0,ϵ]} in H^{n}. Then by definition, M̃_{(0,ϵ]} consists of all points z ∈ H^{3} such that dist(z, γ · z) ≤ ϵ for some γ ∈ Γ − {1}. Hence we have

M̃_{(0,ϵ]} = ⋃_{1≠γ∈Γ} C_{ϵ}(γ).

For any maximal abelian subgroup H of Γ and for any fixed ϵ > 0, the family of sets {C_{ϵ}(γ) : γ ∈ Γ} is totally ordered by inclusion. It follows from the discreteness of Γ that there is always a maximal set C_{ϵ}(H) in this family. Thus for any ϵ > 0 we may write

M̃_{(0,ϵ]} = ⋃_{H} C_{ϵ}(H),

where H ranges over all maximal abelian subgroups of Γ. The discreteness of Γ also implies that the sets C_{ϵ}(H) form a locally finite family. A positive number ϵ will be called a Margulis number for M(Γ) if the sets C_{ϵ}(H), where ϵ ranges over the distinct maximal abelian subgroups of Γ, are pairwise disjoint. Equivalently, ϵ is a Margulis number for M if for every point z ∈ H^{3} and every pair of non-commuting elements ξ, η of Γ, we have max ( dist(z, ξ · z) , dist(z, η · z)) ≥ ϵ. There exists a positive number which is a 3dimensional Margulis constant in the sense that it is a Margulis number for every complete hyperbolic 3-manifold. The existence of such a constant follows, for example, from [Be, Theorem 5.4.5].

If ϵ is a Margulis number for M then each component of M_{(0,ϵ]} has the form C_{ϵ}(H)/H for some maximal abelian subgroup H of Γ. When H is cuspidal, so that C_{ϵ}(H) is a horoball, we call C_{ϵ}(H)/H a standard cusp neighborhood. When H consists of loxodromic elements, it is cyclic. In this case it follows from Proposition 1.3 that C_{ϵ}(H) = ∅ when ϵ is less than the length of a generator of H, and that otherwise C_{ϵ}(H) = nbhd_{δ}(A), where A is the axis of a generator of H and δ is a non-negative number depending on ϵ and H. In particular, A/H is a simple closed geodesic of length ≤ ϵ and T = nbhd_{δ}(A/H). Topologically, T is a solid torus unless A/H has length ϵ, in which case T = A/H. When H consists of loxodromic elements and C_{ϵ}(H) ≠ ∅, we call T = C_{ϵ}(H)/H a tube.

1.6. If Γ is any Kleinian group and ϵ is any Margulis number for M(Γ), we shall denote by M_{(0,ϵ]}^{c}(Γ) the union of all components of M_{(0,ϵ]}(Γ) which are standard cusp neighborhoods. We shall write

M_{{ϵ}}^{c}(Γ) = ∂M_{(0,ϵ]}^{c}(Γ) ⊂ M_{{ϵ}}(Γ),

and

M_{[ϵ,∞)}^{c}(Γ) = M(Γ) − int M_{(0,ϵ]}^{c}(Γ) ⊃ M_{[ϵ,∞)}(Γ).

Furthermore, if I is any of the intervals (0, ϵ], {ϵ} or [ϵ, ∞), we set

N_{I}^{c}(Γ) = N(Γ) ∩ M_{I}^{c}(Γ).

The pre-image of N_{I}^{c}(Γ) in H^{3} will be denoted Ñ_{I}^{c}(Γ).

For any Margulis number ϵ, the set N_{(0,ϵ]}(Γ) is a 3-manifold-with-boundary and a deformation retract of M_{(0,ϵ]}(Γ). Indeed, each component of M_{(0,ϵ]}^{c}(Γ) has the form C_{ϵ}(H)/H for some maximal abelian subgroup H of Γ consisting of parabolic elements. The fixed point ζ ∈ S_{∞} belongs to Λ_{Γ}, and we have Λ_{Γ} ≠ {ζ} since Γ is non-elementary; hence hull(Λ_{Γ}) ∩ C_{ϵ}(H) ≠ ∅. Since hull(Λ_{Γ}) and C_{ϵ}(H) are compact and convex, it follows that nbhd_{1} ( hull(Λ_{Γ})) ∩ C_{ϵ}(H) is a closed convex subset of H^{3} with non-empty interior, and by 1.2 is therefore a contractible 3-manifold-with-boundary. Hence the quotient of nbhd_{1} ( hull(Λ_{Γ})) ∩ C_{ϵ}(H) by the action of H is a 3-manifold-with-boundary and a deformation retract of C_{ϵ}(H)/H; our assertion follows.

1.7. Geometric finiteness. The Kleinian group Γ is said to be geometrically finite if N_{[ϵ,∞)}^{c}(Γ) is compact for some Margulis number ϵ.

If Γ is geometrically finite then it has only finitely many conjugacy classes of cuspidal subgroups. Indeed, if ϵ is a Margulis constant, the sets C_{ϵ}(H), where H ranges over the cuspidal subgroups of Γ, form a locally finite family (1.5). Hence the corresponding cusp neighborhoods C_{ϵ}(H)/H form a locally finite family of subsets of M(Γ). For each H the fixed point ζ of H belongs to Λ_{Γ}, and Λ_{Γ} ≠ {ζ} since Γ is non-elementary; hence hull(Λ_{Γ}) contains a geodesic with one endpoint at ζ. It follows that C_{ϵ}(H)/H ∩ N_{[ϵ,∞)}^{c}(Γ) ≠ ∅ for every H. Hence if N_{[ϵ,∞)}^{c}(Γ) is compact then the family of cusp neighborhoods C_{ϵ}(H)/H is finite, establishing our assertion.

Lemma. Let H be a discrete group of parabolic isometries of the upper half-space model of hyperbolic space, with common fixed point ∞. Let Λ be a closed non-empty H-invariant subset of R^{2} × {0} such that Λ/H is compact.

(i) If H is free abelian of rank 2 then R^{2} × {h} ⊂ hull(Λ ∪ {∞}) for all sufficiently large h.

(ii) If H is infinite cyclic then there is an H-invariant strip S bounded by two (possibly equal) parallel lines in R^{2} such that Λ ⊂ S and R^{2}×{h}∩hull(Λ∪{∞}) = S ×{h} for all sufficiently large h.

Proof. By convexity, if a point P of upper-half space is contained in hull(Λ ∪ {∞}) then so is the entire vertical ray beginning at P. Now suppose that α and β are points of Λ and that the Euclidean distance from α to β is d. Let σ be the Euclidean line segment from α to β. Since the hyperbolic geodesic from α to β is a Euclidean circle of radius d/2, it follows that for all h > d/2 we have σ × {h} ⊂ hull(Λ ∪ {∞}).

Let γ ∈ H and α ∈ Λ and let ℓ be the line in R^{2} which contains the points γ̄^{n}(α) for n ∈ Z. Let d be the Euclidean distance from γ̄^{n}(α) to γ̄^{n+1}(α). By the observation above, applied to each of the pairs {γ̄^{n}(α), γ̄^{n+1}(α)} we have ℓ × {h} ⊂ hull(Λ ∪ {∞}) for all h > d/2.

Suppose that H is free abelian of rank 2 with generators γ_{1} and γ_{2}. Let d_{1} and d_{2} be the respective translation lengths of γ̄_{1} and γ̄_{2} acting on R^{2} × {0}. Let α be a point of Λ. Let ℓ_{n} denote the γ̄_{1}-invariant line through γ̄_{2}^{n}(α). For all h > d_{1}/2 we have that ℓ_{n} × {h} ⊂ R^{2} × {h} ∩ hull(Λ ∪ {∞}). Every point of R^{2} lies on a line segment of length d_{2} with endpoints in ℓ_{n} and ℓ_{n+1} for some n ∈ Z. It follows that for h > d_{1}/2 + d_{2}/2 we have R^{2} × {h} ⊂ hull(Λ ∪ {∞}).

Suppose that H is infinite cyclic with generator γ and that γ̄ acts on R^{2} × {0} as a Euclidean translation through a distance d. The assumption that Λ/H is compact implies that Λ is contained in a strip S bounded by γ̄-invariant lines ℓ_{1} and ℓ_{2}. Since Λ is closed, the lines ℓ_{1} and ℓ_{2} may be taken to contain points of Λ. If the Euclidean distance from ℓ_{1} to ℓ_{2} is w then for all h > d/2+w/2 we have S×{h} ⊂ hull(Λ∪{∞}). On the other hand, S×[0, ∞) is a convex set containing Λ ∪ {∞}. Therefore R^{2} × {h} ∩ hull(Λ ∪ {∞}) = S × {h}. □

Recall that a topological 3-manifold N is said to be irreducible if every locally flat 2-sphere in N is the boundary of a 3-ball in N.

Proposition. Let Γ be a geometrically finite, torsion-free Kleinian group; set M = M(Γ) and N = N(Γ). Then N_{[ϵ,∞)}^{c} is compact whenever ϵ is a Margulis number for M. Furthermore, for every sufficiently small Margulis number ϵ_{0} the following conclusions hold:

(i) N_{[ϵ_{0},∞)}^{c} is an orientable, irreducible 3-manifold-with-boundary and is a deformation retract of N and hence of M; and

(ii) N_{{ϵ_{0}}}^{c} is a compact 2-manifold-with-boundary, properly embedded in N, and is a deformation retract of N_{(0,ϵ_{0}]} and hence of M_{(0,ϵ_{0}]}.

Proof. Since Γ is geometrically finite, there is a Margulis number ϵ_{1} for M such that N_{[ϵ_{1},∞)}^{c} is compact.

Let ϵ be any Margulis number. In order to show that N_{[ϵ,∞)}^{c} is compact, we observe that for any sufficiently small ϵ′ > 0 we have

Ñ_{[ϵ,∞)}^{c} ⊂ nbhd_{1} ( hull(Λ_{Γ}) ∩ M̃_{[ϵ′,∞)}^{c}).

Hence we need only show that ( hull(Λ_{Γ}) ∩ M̃_{[ϵ′,∞)}^{c})/Γ is compact for sufficiently small ϵ′.

In particular we may suppose that ϵ′ < ϵ_{1}. Since ( hull(Λ_{Γ}) ∩ M̃_{[ϵ_{1},∞)}^{c})/Γ is compact, and since Γ has only finitely many conjugacy classes of cuspidal subgroups by 1.7, we need only show that for any cuspidal subgroup H of Γ the set ( hull(Λ_{Γ}) ∩ (C_{ϵ_{1}}(H) − C_{ϵ}(H)))/H

has compact closure in M. For this we identify H^{3} with the upper half-space model by an isometry mapping the fixed point of H to ∞ and set Λ = Λ_{Γ} − {∞}. Let h and h_{1} be real numbers so that the boundaries of the horoballs C_{ϵ}(H) and C_{ϵ_{1}}(H) are identified with the horizontal planes R^{2} × {h} and R^{2} × {h_{1}} respectively. If H has rank 2 then

(C_{ϵ_{1}}(H) − C_{ϵ}(H))/H ⊂ (R^{2} × [h_{1}, h])/H,

the set on the right being compact. If H has rank 1 then

( hull(Λ_{Γ}) ∩ (C_{ϵ_{1}}(H) − C_{ϵ}(H)))/H ⊂ (S × [h_{1}, h])/H,

where S is the strip provided by the lemma. Again, the set on the right is compact.

Next we prove that for a sufficiently small Margulis number ϵ_{0} the set N_{{ϵ_{0}}}^{c} is a compact 2-manifold-with-boundary properly embedded in N (which by 1.4 is a 3-manifold-with-boundary), and that N_{{ϵ_{0}}}^{c} is a deformation retract of N_{(0,ϵ_{0}]}. This will establish assertion (ii) of the proposition, since N_{(0,ϵ_{0}]} is a deformation retract of M_{(0,ϵ_{0}]} according to 1.6. Since Γ has only finitely many conjugacy classes of cuspidal subgroups, it suffices to show that for any cuspidal subgroup H of Γ there is a Margulis number ϵ_{H} such that for any ϵ ≤ ϵ_{H} the set N ∩ (∂C_{ϵ}(H)/H) is a properly embedded 2-manifold-with-boundary in N and is a deformation retract of C_{ϵ}(H)/H. Equivalently, we must show that for small enough ϵ the set nbhd_{1} ( hull(Λ_{Γ}))∩∂C_{ϵ}(H) is a contractible, properly embedded 2-manifold-with-boundary in nbhd_{1} ( hull(Λ_{Γ})).

We prove this by applying the lemma as before. The statement follows immediately in the case where H is free abelian of rank 2. In the case that H is isomorphic to Z let C denote the horoball R^{2} × [e^{c}, ∞), and D denote the region S × (0, ∞) where S is the strip provided by the conclusion of the lemma. We have hull(Λ_{Γ}) ∩ C = D ∩ C. Hence if C′ denotes the horoball R^{2} × [e^{c+1}, ∞) ⊂ C, so that the minimum distance between ∂C and ∂C′ is 1, we have nbhd_{1} ( hull(Λ_{Γ})) ∩ C′ = nbhd_{1}(D) ∩ C′. But nbhd_{1}(D) is bounded by two equidistant surfaces [F, p. 39] with the two boundary components of D as axial planes; these equidistant surfaces are intersections of the upper half-space with non-horizontal Euclidean planes E_{1} and E_{2} in R^{3}.

We have C′ = C_{ϵ}(H) for some Margulis number ϵ_{H}; and for any ϵ ≤ ϵ_{H}, the set nbhd_{1} ( hull(Λ_{Γ})) ∩ ∂C_{ϵ}(H) is the closed strip in the horizontal Euclidean plane ∂C_{ϵ}(H) bounded by the lines ∂C_{ϵ}(H) ∩ E_{1} and ∂C_{ϵ}(H) ∩ E_{2}. Hence nbhd_{1} ( hull(Λ_{Γ})) ∩ ∂C_{ϵ}(H) is a contractible 2-manifold-with-boundary. By inspection we have that nbhd_{1} ( hull(Λ_{Γ})) ∩ ∂C_{ϵ}(H) is properly embedded in nbhd_{1} ( hull(Λ_{Γ})). This proves (ii) for small ϵ_{0}.

We shall complete the proof by showing that (ii) implies (i). If (ii) holds for a given ϵ_{0} then in particular N_{{ϵ_{0}}}^{c} is a deformation retract of N_{(0,ϵ_{0}]}^{c}; thus N_{[ϵ_{0},∞)}^{c} is a deformation retract of N, and hence (by 1.4) of M. In particular N_{[ϵ_{0},∞)}^{c} is aspherical. Since M is covered by H^{3}, every locally flat 2-sphere S in M(Γ) bounds a unique ball B ⊂ M; if S ⊂ N_{[ϵ_{0},∞)}^{c} then B ⊂ N_{[ϵ_{0},∞)}^{c} by asphericity. Hence N_{[ϵ_{0},∞)}^{c} is irreducible. Since M is orientable, so is N_{[ϵ_{0},∞]}^{c}. This establishes (i). □

1.8. The following proposition about 2-generator groups of isometries of H^{3} follows easily from results which can be found in [Jør] and in [F, V.1].

If ℓ is any (geodesic) line in H^{3}, we denote by τ_{ℓ} the 180^{∘} rotation about ℓ. Thus τ_{ℓ} is the unique element of order 2 in the group Isom (H^{3}) whose fixed point set is ℓ.

Proposition. Let ξ and η be two orientation-preserving isometries of H^{3}. Suppose that ξ_{∞} and η_{∞} have no common fixed point in S_{∞}. Then there is a unique line ℓ ⊂ H^{3} such that τ = τ_{ℓ} satisfies τξτ = ξ^{−1} and τητ = η^{−1}.

If ξ and η satisfy the hypotheses of the above proposition, the line ℓ given by the proposition will be denoted ℓ(ξ, η). This line is the common perpendicular to the axes of ξ and η, suitably interpreted in the degenerate cases where one or both of these isometries are parabolic [Fe, III.3 and V.1].

1.9. The line ℓ(ξ, η) is easily seen to depend continuously on ξ and η in the following sense. Suppose that ξ_{∞} and η_{∞} satisfy the hypotheses of Proposition 1.11, and let (ξ_{i}) and (η_{i}) be sequences of isometries of H^{3} converging to ξ_{∞} and η_{∞} respectively, so that ξ_{i} and η_{i} also satisfy the hypotheses of the proposition for large i. Let ζ_{∞}^{+} and ζ_{∞}^{−} denote the endpoints of ℓ(ξ_{∞}, η_{∞}). Then we may label the endpoints of ℓ(ξ_{i}, η_{i}) (for large i) as ζ_{i}^{+} and ζ_{i}^{−} in such a way that ζ_{i}^{+} → ζ_{∞}^{+} and ζ_{i}^{−} → ζ_{∞}^{−} as i → ∞.

§2. The Poisson kernel

We shall need some elementary facts about the Poisson kernel for hyperbolic space. In this section and the next we present the relevant material from a geometric point of view which is natural for the applications in this paper.

2.1. We define a continuous function Π : H^{n} × H^{n} × H^{n} → R by

Π(z, z′, w) = exp ( dist(z, w) − dist(z′, w)).

From the hyperbolic law of cosines [F, p. 91] we have

\frac{cosh dist(z, w)}{cosh dist(z′, w)} = ( cosh dist(z, z′) − tanh dist(z, w) sinh dist(z, z′) cos ∠z′zw)^{−1}.

It follows that the function Π has a unique positive continuous extension \overline{Π}: H^{n} × H^{n} × \overline{H}^{n} → R. The restriction of \overline{Π} to a function on H^{n} × H^{n} × S_{∞} will be denoted P. This restriction is given explicitly by

(2.1.1)  P(z, z′, ζ) = ( cosh dist(z, z′) − sinh dist(z, z′) cos ∠z′zζ)^{−1}

for any z ≠ z′ ∈ H^{n} and any ζ ∈ S_{∞}^{n−1}; and by P(z, z, ζ) = 1 for any z ∈ H^{n} and any ζ ∈ S_{∞}^{n−1}.

2.2. It is immediate from the definition of the function Π that for any points z, z′, z′′, w ∈ H^{n} we have Π(z, z′′, w) = Π(z, z′, w)Π(z′, z′′, w). Hence in view of the continuity of \overline{Π} we have

P(z, z′′, ζ) = P(z, z′, ζ)P(z′, z′′, ζ)

for any z, z′, z′′ ∈ H^{n} and any ζ ∈ S_{∞}.

2.3. There is a simple formula for the function P in terms of the upper half-space model. Given any point ζ ∈ S_{∞}^{n−1}, let us fix an isometry J of H^{n} onto the upper half-space model R^{n−1} × (0, ∞) which maps ζ to ∞ (see 1.1). The n-th coordinate function on R^{n−1} × (0, ∞) pulls back via J to a function Im : H^{n} → (0, ∞). The effect of replacing J by another isometry mapping ζ to ∞ would be to multiply the function Im by a constant; this is because a self-isometry of the upper half-space model that fixes ∞ is of the form x ↦ ασ(x), where α is a positive constant and σ is a Euclidean isometry σ of R^{n} which fixes the n-th standard basis vector. In particular, for any two points z, z′ ∈ H^{n}, the value of the expression Im z/ Im z′ depends only on the point ζ and not on the choice of J.

Proposition. With the above conventions we have

P(z, z′, ζ) = Im z′/ Im z

for any points z, z′ ∈ H^{n}.

Proof. We identify H^{n} with R^{n−1} × (0, ∞) via the diffeomorphism J, and write z = (x, t) and z′ = (x′, t′), where x, x′ ∈ R^{n−1} and t, t′ > 0. We define two sequences (w_{j})_{j≥1} and (w′_{j})_{j≥1} of points of H^{n} by w_{j} = (x, j) and w′_{j} = (x′, j). For each j, the vertical line segments zw_{j} and z′w′_{j} are hyperbolic geodesic arcs; hence dist(z, w_{j}) = log |j − z_{n}| and dist(z′, w′_{j}) = log |j − z′_{n}|. On the other hand, the horizontal line segment from w_{j} to w′_{j} is an arc of hyperbolic length 1/j, and hence lim_{j→∞} dist(w_{j}, w′_{j}) = 0. From the definition of P we then have

P(z, z′, ζ) = lim_{j→∞} exp ( dist(z, w_{j}) − dist(z′, w_{j}))
= lim_{j→∞} exp ( dist(z, w_{j}) − dist(z′, w′_{j}))
= t′/t = Im z′/ Im z.

□

The function P is our version of the Poisson kernel. If H^{n} is identified with the Poincaré model via an isometry which maps a given point z_{0} to the origin, then the hyperbolic Poisson kernel P: B^{n} × S_{∞} → R discussed, for example, in [N] or [P] is given by

P (z, ζ) = P(z_{0}, z, ζ).

This follows from the proposition above together with the calculation [N, Theorem 5.1.3] that if V is the isometry from the Poincaré model to the upper half-space model which sends the origin to the point (0, 1) ∈ R^{n} × (0, ∞) and maps ζ to ∞ then P (V^{−1}(z), ζ) = Im z.

2.4. Conformal expansion factors. Consider a point z ∈ H^{n} and an isometry γ : H^{n} → H^{n}. By 1.1, γ induces a conformal diffeomorphism γ_{∞} of S_{∞}. If we equip S_{∞} with the round metric centered at z, then at each point ζ ∈ S_{∞}, the map γ_{∞} has a well-defined conformal expansion factor λ > 0. (This means that the tangent map dγ_{∞} : T_{ξ}(S_{∞}) → T_{γ_{∞}(ξ)}(S_{∞}) satisfies |dγ_{∞}(v)| = λ|v| for every tangent vector v at ζ, where | · | denotes length in the round metric.) We shall denote the conformal expansion factor of γ_{∞} at ζ by λ_{γ,z}(ζ). Thus λ_{γ,z} is a smooth positive-valued function on S_{∞}.

Note that if ζ is a fixed point of γ then dγ_{∞}(ζ) is a linear automorphism of T_{ξ}(S_{∞}) and we have λ_{γ,z}(ζ) = | det dγ_{∞}(ζ)|^{1/(n−1)}. Thus in this case λ_{γ,z}(ζ) may be calculated without reference to the round metric or the point z.

Proposition. Let z be any point of H^{n} and let γ be any isometry of H^{n}. Then we have λ_{γ,z}(ζ) = P(z, γ^{−1}z, ζ) for every point ζ ∈ S_{∞}.

Proof. The corresponding statement for the Poincaré model is given in [N, Lemma 3.4.2]. This together with 2.2 gives the proposition. □

2.5. The Laplacian. For any Riemannian n-manifold M we have a Laplacian operator

Δ = ∗d ∗ d : C^{∞}(M) → C^{∞}(M),

where * is the Hodge star and d denotes exterior differentiation.

Using Proposition 2.3, one shows the following by a straightforward calculation in the upper half-space model. (See [N, Theorem 5.1.3]).

Proposition. For any z ∈ H^{n}, and ζ ∈ S_{∞} and any real number r, the function ϕ_{ζ} : H^{n} → R defined by ϕ_{ζ}(z) = P(z_{0}, z, ζ)^{r} satisfies the equation Δϕ_{ζ} = −r(n − r − 1)ϕ_{ζ}.

Proposition 2.5 is the starting point for the use of the Poisson kernel in constructing solutions to the equation Δϕ_{ζ} = −r(n − r − 1)ϕ_{ζ} (“generalized eigenfunctions” for the Laplacian) with prescribed “boundary values.” From the point of view of the present paper this construction is most naturally described in terms of the notion, first formalized by Sullivan, of a “conformal density.” We shall discuss this in the next section.

§3. Conformal densities

In this section we discuss a Γ-equivariant potential theory in hyperbolic space, where Γ is a Kleinian group. Our treatment, which roughly parallels the one given in Nicholls’s recent book [N], is formulated in terms of Sullivan’s notion of a conformal density. This makes the equivariant theory work out quite neatly. Our approach to the uniqueness of a Γ-invariant conformal density when all Γ-invariant super-harmonic functions are constant is somewhat different from the approach used in [N] and [Su1]. We show (Proposition 3.7) that an arbitrary (n − 1)-conformal density on H^{n}, not necessarily Γ-invariant, is determined by the associated harmonic funtion. This works only when the degree of the conformal density is n − 1, but is simpler than the arguments in [N] and [Su1] which depend on the ergodicity of the action of Γ on its limit set with respect to the measure class determined by the conformal density.

3.1. By a Borel measure on a locally compact metrizable space X we shall mean a measure on the Borel subsets of X which is finite on the compact subsets of X. Measures will always be understood to be non-negative: we shall have no occasion to consider signed measures. If μ is a finite measure on a space X, the number μ(X) will be called the total mass of μ. If μ is a measure, and f a measurable function, on a measurable space (X, B), and if ν is the measure defined by ν(E) = ∫_{E} fdμ for every E ∈ B, then we write dν = fdμ.

If x is a point of a locally compact metrizable space X, we denote by δ_{x} the Dirac measure supported at x: by definition, for any Borel set E ⊂ X we have δ_{x}(E) = 1 if x ∈ E and δ_{x}(E) = 0 otherwise.

If X and Y are locally compact metrizable spaces and f : X → Y is a homeomorphism, then any Borel measure μ on Y gives rise to a measure on X, called the pull-back of μ and denoted f^{∗}μ. It is defined by (f^{∗}μ)(E) = μ(f(E)) for any Borel set E ⊂ X.

3.2. Conformal densities. Let n be an integer ≥ 2, and let D be a number in [0, n−1]. By a D-conformal density for the sphere at infinity S_{∞}^{n−1} ⊂ \overline{H}^{n} we mean a family M = (μ_{z})_{z∈H^{n}} of finite Borel measures on S_{∞} = S_{∞}^{n−1}, indexed by the points of H^{n}, such that for any two points z, z′ ∈ H^{n} we have dμ_{z′} = P(z, z′, ·)^{D}dμ_{z}.

A family M = (μ_{z})_{z∈H^{n}} of finite Borel measures on S_{∞} will be called a conformal density if it is a D-conformal density for some D ∈ [0, n − 1].

The trivial conformal density is (0)_{z∈H^{n}}; it will be denoted by 0. Note that 0 is a Dconformal density for every D ∈ [0, n − 1]. On the other hand, it is clear that a non-trivial conformal density M can be a D-conformal density for only one value of D; we call D the degree of M.

If z_{0} is a point of H^{n} and μ is a finite Borel measure on S_{∞}, then for any D ≥ 0 there is a unique D-conformal density M = (μ_{z}) such that μ_{z_{0}} = μ. Indeed, uniqueness is clear since for any z ∈ H^{n} we must have dμ_{z} = P(z_{0}, z, ·)^{D}dμ. To prove existence, we define μ_{z}, for each z ∈ H^{n}, to be the unique Borel measure such that dμ_{z} = P(z_{0}, z, ·)^{D}dμ; using 2.3 we find that for any points z, z′ ∈ H^{n} we have

P(z_{0}, z′, ·)^{D} = P(z_{0}, z, ·)^{D}P(z, z′, ·)^{D},

and hence

dμ_{z′} = P(z_{0}, z′, ·)^{D}dμ = P(z_{0}, z, ·)^{D}P(z, z′, ·)^{D}dμ = P(z, z′, ·)^{D}dμ_{z},

so that (μ_{z}) is the required conformal density.

If M = (μ_{z}) is a conformal density then all the measures μ_{z}, z ∈ H^{n}, have the same support; this follows from the definition, since the function P(z, z′, ·) is strictly positive on S_{∞} for all z, z′ ∈ H^{n}. The common support of the μ_{z} will be called the support of M and will be denoted supp M.

If M = (μ_{z})_{z∈H^{n}} and M′ = (μ′_{z})_{z∈H^{n}} are two D-conformal densities for a given D, it is clear that (μ_{z} + μ′_{z})_{z∈H^{n}} is also a conformal density; it will be denoted M + M′. More generally we can define the sum ∑_{i=}^{n} M_{i} of any finite family (M_{i})_{1≤i≤n} of D-conformal densities.

Now let M = (μ_{z})_{z∈H^{n}} be a D-conformal density and f a continuous positive-valued function on S_{∞}. For each z ∈ H^{n} let μ′_{z} be the Borel measure defined by dμ′_{z} = fdμ_{z}. It is clear that N = (μ′_{z})_{z∈H^{n}} is a D-conformal density; we shall indicate that N is defined in this way from f and M by writing dN = fdM.

3.3. The area density. For any point z ∈ H^{n}, the round metric centered at z determines an area measure on S_{∞}. We let A_{z} denote such an area measure normalized so as to have total mass 1. It follows from Proposition 2.4 and the change-of-variable formula for the Lebesgue integral that (A_{z})_{z∈H^{n}} is an (n − 1)-conformal density. This will be called the area density and will be denoted A.

3.4. Invariant Conformal Densities. Suppose that M = (μ_{z}) is a D-conformal density and that γ: H^{n} → H^{n} is an isometry. Then the family of Borel measures (γ_{∞}^{∗}μ_{γz})_{z∈H^{n}} is also a D-conformal density. Indeed, for any points z, z′ ∈ H^{n} and any continuous function f on S_{∞}, we have

∫ fd(γ_{∞}^{∗}μ_{γz′}) = ∫ (f ∘ γ_{∞}^{−1})dμ_{γz′}
= ∫ f ∘ γ_{∞}^{−1}(w)P(γz, γz′, w)^{D}dμ_{γz}(w)
= ∫ f ∘ γ_{∞}^{−1}(w)P(z, z′, γ^{−1}w)^{D}dμ_{γz}(w)
= ∫ fP(z, z′, ·)^{D}d(γ_{∞}^{∗}μ_{γz}).

We call (γ_{∞}^{∗}μ_{γz}) the pull-back of the conformal density M, and we shall denote it by γ^{∗}M.

Now let Γ be any group of isometries of H^{n}. A conformal density M will be termed Γ-invariant if we have γ^{∗}M = M for every γ ∈ Γ. The area density is clearly invariant under the full group of isometries of H^{n}.

If a D-conformal density M = (μ_{z}) is Γ-invariant then for any z ∈ H^{n} and any γ ∈ Γ we have d(γ^{∗}μ_{z}) = dμ_{γ^{−1}z} = P(z, γ^{−1}z, ·)^{D} · dμ_{z}; hence in view of 2.4 we have

(3.4.1)  d(γ^{∗}μ_{z}) = λ_{γ,z}^{D}dμ_{z}.

3.5. Eigenfunctions of the Laplacian. Let n ≥ 2 be an integer, and let D be a number in the interval [0, n − 1]. Let M = (μ_{z})_{z∈H^{n}} be a D-conformal density. Let us consider the non-negative-valued function u = u_{M} on H^{n} defined by u(z) = μ_{z}(S_{∞}). If we fix any point z_{0} ∈ H^{n}, and set μ = μ_{z_{0}}, then according to the definition of a conformal density, u may be written as

(3.5.1)  u(z) = ∫_{S_{∞}} P(z_{0}, z, ζ)^{D}dμ(ζ).

For the area density A we clearly have u_{A} ≡ 1.

According to Proposition 2.5, for any ζ ∈ S_{∞} the function ϕ_{ζ} = P(z_{0}, ·, ζ)^{D} is C^{∞} and satisfies Δϕ_{ζ} = −D(n − D − 1)ϕ_{ζ}. By differentiating under the integral sign in (3.5.1) it follows that for any D-conformal density M the function u = u_{M} is C^{∞} and satisfies the equation

(3.5.2)  Δu = −D(n − D − 1)u.

In particular, if D = n − 1 then (3.5.2) says that u is harmonic.

If M is invariant (3.4) under a group Γ of isometries of H^{n}, then for any γ ∈ Γ and any z ∈ H^{n} we have

u(γz) = μ_{γz}(S_{∞}) = (γ^{∗}μ_{z})(S_{∞}) = μ_{z}(S_{∞}) = u(z).

Thus u is Γ-invariant (i.e. constant on the orbits of Γ.)

3.6. Proposition. Let f be a continuous function on S_{∞} and let N be the conformal density defined by dN = fdA, where A is area density. Then the function ū : \overline{H}^{n} → R, defined by ū | H^{n} = u_{N} and ū | S_{∞} = f, is continuous.

Proof. We identify H^{n} via a conformal diffeomorphism with the Poincaré model. Thus S_{∞} is identified with the unit sphere S^{n−1}. We write A = A_{0}. Since f is continuous and u = u_{N} is C^{∞}, we need only show that if (z_{i}) is a sequence of points in H^{n} converging to a point ζ_{0} ∈ S_{∞}, then u(z_{i}) converges to f(ζ_{0}). We set C = f(ζ_{0}). For each i we have

u(z_{i}) = ∫_{S_{∞}} P(0, z_{i}, ζ)^{n−1}f(ζ)dA(ζ).

On the other hand, the integral ∫_{S_{∞}} P(0, z_{i}, ζ)^{n−1}dA(ζ) represents the total mass of A_{z_{i}}, which is 1. Hence

u(z_{i}) − C = ∫_{S_{∞}} P(0, z_{i}, ζ)^{n−1}(f(ζ) − C)dA(ζ).

To show that u(z_{i}) → C, it suffices to show that lim sup |u(z_{i}) − C| < ϵ for any prescribed ϵ > 0. Given ϵ, let U be a neighborhood of ζ_{0} in S_{∞} such that |f(ζ) − C| < ϵ for every ζ ∈ U. We may take U to be the δ-neighborhood for some δ > 0 in the standard metric on S^{n−1}; thus a point ζ lies in U if and only if ∠(ζ_{0}, 0, ζ) < δ. We may write u(z_{i}) − C as

∫_{U} P(0, z_{i}, ζ)^{n−1}(f(ζ) − C)dA(ζ) + ∫_{S_{∞}−U} P(0, z_{i}, ζ)^{n−1}(f(ζ) − C)dA(ζ).

The first term in the above expression is bounded above in absolute value by ϵ. We shall complete the proof by showing that the second term tends to 0 as i → ∞.

Since f is continuous on S_{∞}, it is enough to show that sup_{ζ∈S_{∞}−U} P(0, z_{i}, ζ) tends to 0 as i → ∞. This follows easily from the formula (2.1.1) for P. □

3.7. Proposition. If M and M′ are (n − 1)-conformal densities for H^{n} such that u_{M} = u_{M′}, then M = M′.

Proof. For the purpose of this argument we identify H^{n} with the Poincaré model B^{n}, so that \overline{H}^{n} is identified with \overline{B}^{n}. Then any point in H^{n} may be written in the form tθ, where θ ∈ S^{n−1} and 0 ≤ t < 1.

We will use the following fundamental symmetry of the function P which follows directly from the formula (2.1.1):

(3.7.1)  P(0, tζ, θ) = P(0, tθ, ζ)

for any t ∈ [0, 1) and any ζ, θ ∈ S^{n−1}.

Now let M = (μ_{z}) be any (n−1)-conformal density and let f be any continuous function on S_{∞}. Let N be the conformal density defined by dN = fdA, where A is the area density. By Proposition 3.6 the function ū : \overline{H}^{n} → R, defined by ū | H^{n} = u_{N} and ū | S_{∞} = f, is continuous. Hence

(3.7.2)  ∫ fdμ_{0} = lim_{t→1^{−}} ∫ u_{N} (tζ)dμ_{0}(ζ).

For any t < 1 we have, by (3.7.1) and the Fubini Theorem,

∫ u_{N} (tζ)dμ_{0}(ζ) = ∫ ∫ f(θ)P(0, tζ, θ)^{n−1}dA(θ)dμ_{0}(ζ)
= ∫ ∫ P(0, tθ, ζ)^{n−1}dμ_{0}(ζ)f(θ)dA(θ) = ∫ u_{M}(tθ)f(θ)dA(θ).

If M′ = (μ′_{z}) is a second conformal density such that u_{M} = u_{M′}, then for any t < 1 we have

∫ u_{N} (tζ)dμ′_{0}(ζ) = ∫ u_{M′}(tθ)f(θ)dA(θ) = ∫ u_{M}(tθ)f(θ)dA(θ) = ∫ u_{N} (tζ)dμ_{0}(ζ).

Taking limits as t → 1, it follows by (3.7.2) that

∫ fdμ_{0} = ∫ fdμ′_{0}

for every continuous function f on S_{∞}. Hence μ_{0} = μ′_{0}. By 3.2 it follows that M = M′. □

3.8. We shall need the following elementary result concerning the degree of a Γ-invariant conformal density, when Γ is a discrete group (cf. [N, Corollary 3.4.5]).

Proposition. Let Γ be a non-elementary discrete group of isometries of H^{n}. Then every non-trivial Γ-invariant conformal density for H^{n} has strictly positive degree.

Proof. Suppose that there exists a non-trivial Γ-invariant conformal density of degree 0. By definition, this means that there is a non-zero Γ-invariant finite Borel measure μ on S_{∞}.

Since Γ is non-elementary, some element γ of Γ is loxodromic; let P and Q be the fixed points of γ_{∞}. Then any point ζ ∈ S_{∞} − {P, Q} has a neighborhood U in S_{∞} such that γ_{∞}^{k}(U) ∩ U = ∅ for all k > 0. Since μ is finite and countably additive, it follows that μ(U) = 0. This shows that supp μ ⊂ {P, Q}. But supp μ is non-empty and invariant under Γ; thus S_{∞} contains a Γ-invariant subset of cardinality 1 or 2. Since Γ is discrete, it follows that Γ has an abelian subgroup of finite index. This contradicts the hypothesis that Γ is non-elementary, and the proposition is proved. □

3.9. A C^{∞} function f on a Riemannian manifold is termed superharmonic if it satisfies the inequality Δf ≤ 0.

Proposition. Let Γ be a non-elementary discrete group of isometries of H^{n}. Suppose that every Γ-invariant positive-valued superharmonic function on H^{n} is constant. Then any Γ-invariant conformal density for H^{n} is a constant multiple of the area density.

Proof. Suppose that M = (μ_{z}) is a Γ-invariant D-conformal density for some D ∈ [0, n−1]. Since Γ is non-elementary, it follows from Proposition 3.9 that D ≠ 0. We may assume that M ≠ 0, so that the function u = u_{M} is positive-valued. By 3.5 u is Γ-invariant and satisfies the equation Δu = −D(n − D − 1)u; in particular u is superharmonic. Hence by the hypothesis u is a constant C.

Thus 0 = Δu ≡ D(n − D − 1)u. Since D ≠ 0 it follows that D = n − 1. Since u_{M} = C = u_{CA}, Proposition 3.7 then guarantees that M = CA. □

Note that our approach to this uniqueness question is based on Proposition 3.7, which applies to an arbitrary (n − 1)-conformal density. The support of the conformal density M plays no role in the argument.

Another, somewhat less elementary, approach is found in the literature. Suppose that M = (μ_{z}) is a Γ-invariant D-conformal density with supp M = Λ_{Γ}. (A construction, due to Patterson, of conformal densities with these properties will be discussed in the next section.) Then any Γ-invariant D-conformal density supported on Λ_{Γ} is a constant multiple of M provided that Γ acts ergodically on S_{∞} with respect to the measure class determined by M [N, Theorem 4.2.1]. Suppose that A is a Γ-invariant subset of S_{∞} such that A and S_{∞} − A are sets of positive measure with respect to this measure class. Let M′ be given by dM′ = χ_{A}dM. Using points of density one can show that the ratio u_{M′}/u_{M} tends to χ_{A} along almost all (μ_{0}) rays from the origin in the Poincaré model. Given this, the ergodicity follows if one knows that the ratio of any two positive Γ-invariant eigenfunctions of Δ, with eigenvalue −D(n − D − 1), is bounded above and below by positive constants. In [Su3] Sullivan uses this method to give examples of Kleinian groups Γ, for which there exist unique Γ-invariant conformal densities of degree 2 supported on Λ_{Γ}, but where these conformal densities are not constant multiples of the area density. (In fact, for these groups Λ_{Γ} has area 0, although it follows from the existence of such conformal densities that Λ_{Γ} has Hausdorff dimension 2.)

§4. The Patterson construction

In this section we describe a construction first introduced by Patterson [P], and extensively studied by Sullivan ([Su1], [Su2], [Su3]), for associating to a Kleinian group Γ a Γ-invariant conformal density supported on the limit set of Γ. By taking a slightly more general point of view than Patterson’s, we are able to construct decompositions of a Patterson density corresponding to decompositions of the group. In Section 5, by specializing to the case of a free group, we shall obtain the paradoxical decomposition described in the introduction.

4.1. Uniformly discrete sets. A subset W of H^{n} will be called uniformly discrete if there is a number ϵ > 0 such that dist(z, w) > ϵ for any two distinct points z, w ∈ W. Such a number ϵ will be called a modulus of discreteness for W.

If Γ is any discrete group of isometries of H^{n}, then for any w ∈ H^{n}, the orbit W = Γw is a uniformly discrete set. (Indeed, since Γ is discrete and w has compact stabilizer in the isometry group of H^{n}, the orbit Γw is discrete. Thus there is a number ϵ > 0 such that nbhd_{ϵ}(w) ∩ Γw = {w}. Since Γ consists of isometries it follows that ϵ is a modulus of discreteness.)

If W ⊂ H^{n} is any uniformly discrete set, the closure of W in \overline{H}^{n} has the form W ∪ Λ, where Λ is a closed subset of S_{∞}. We call Λ the limit set of W. Note that Λ is empty if and only if W is finite.

If Γ is a discrete group of isometries, the limit set of any orbit of Γ is the limit set of Γ.

4.2. The main result of this section is:

Proposition. Let W be an infinite, uniformly discrete subset of H^{n}, and let 𝔙 be a countable collection of subsets of W. Suppose that W ∈ 𝔙. Then there exist a number D ∈ [0, n − 1] and a family (M_{V})_{V ∈𝔙} of D-conformal densities for H^{n}, indexed by the collection 𝔙, satisfying the following conditions.

(i) M_{W} ≠ 0.

(ii) For any finite family (V_{i})_{1≤i≤m} of disjoint sets in 𝔙 such that V = ∐_{i=1}^{m} V_{i} ∈ 𝔙,

we have M_{V} = ∑_{i=1}^{m} M_{V_{i}}. (iii) For any V ∈ 𝔙 and any isometry γ: H^{n} → H^{n} such that γV ∈ 𝔙, we have

γ_{∞}^{∗}(M_{γV}) = M_{V}. (iv) For any V ∈ 𝔙, the support of M_{V} is contained in the limit set of M_{V}. In

particular, for any finite set V ∈ 𝔙 we have M_{V} = 0.

Before turning to the proof of the Proposition, let us point out one of its consequences. If Γ is any non-elementary discrete group of isometries of H^{n}, then any orbit W of Γ is uniformly discrete by 4.1 and infinite because Γ is non-elementary. Thus we may apply Proposition 4.2 to W, taking 𝔙 to be any countable collection of subsets of W with W ∈ 𝔙 (for example 𝔙 = {W }). This gives a conformal density M = M_{W}. It follows from 4.2(iii) that M is Γ- invariant. In particular the support of M is Γ- invariant. On the other hand, we have supp M ≠ ∅ by 4.2(i); and by 4.2(iv) we have supp M ⊂ Λ_{Γ}. Since the action of Γ on Λ_{Γ} is topologically minimal, it follows that supp M = Λ_{Γ}.

In particular, taking 𝔙 = {W }, we obtain the following.

Corollary. (Patterson [P]) If Γ is any non-elementary discrete subgroup of isometries of H^{n}, there is a Γ-invariant conformal density for H^{n} whose support is Λ_{Γ}. □

The rest of this section is devoted to the proof of the proposition. We are given an infinite, uniformly discrete set W ⊂ H^{n} and a countable collection 𝔙 of subsets of W, with W ∈ 𝔙.

4.3. The Poincaré Series. For any point z ∈ H^{n} and any s ≥ 0 we denote by Σ(z, s) the sum of the “Poincaré series” for the uniformly discrete set W. That is,

Σ(z, s) = ∑_{w∈W} e^{−s dist(w,z)}

where we interpret the sum as a non-negative number or +∞.

The following properties of Σ(z, s) are immediate from the definition.

4.3.1. For any z ∈ H^{n} and any s ≥ s′ ≥ 0 we have Σ(z, s′) ≥ Σ(z, s).

4.3.2. For any z, z′ ∈ H^{n} and any s ≥ 0 we have

Σ(z′, s) ≤ e^{s dist(z,z′)}Σ(z, s).

Lemma. There is a unique non-negative number D ≤ n − 1 such that, for every point z ∈ H^{n}, the sum Σ(z, s) is finite for all s > D and infinite whenever 0 < s < D.

Proof. First we note that Σ(z, s) is finite for s > n−1. This is proved, in the case that W is an orbit of a discrete group, in [N, Theorem 1.6.1]. The proof applies as well to an arbitrary uniformly discrete set. Briefly, the idea is that the Poincaré series converges as long as s is larger than the exponential growth rate of the number N_{R} of points of W in the ball of radius R about z. The uniform discreteness of W implies that N_{R} cannot grow faster than the volume V (R) of a ball of radius R. However, we have V (R) = A_{n} ∫_{0}^{r} sinh^{n−1} x dx, where A_{n} is a number depending only on the dimension n. Hence for all R we have V (R) ≤ B_{n}e^{(n−1)R}, where B_{n} is again a number depending only on n.

It now follows from (4.3.1) and the discussion in 4.3 that for each z ∈ H^{n} there is a unique number D_{z} ≤ n−1 such that Σ(z, s) < ∞ for all s > D_{z} and Σ(z, s) = ∞ whenever 0 < s < D_{z}. It follows from 4.3.2 that D_{z} is independent of z. □

The number D given by the lemma is called the critical exponent of W.

4.4. The critically divergent case. The construction of the conformal densities in the statement of Proposition 4.2 is simplest in the case where Σ(z, D) is infinite for every z ∈ H^{n}. We sketch the construction in this case before discussing the general case.

First we define a family of measures μ_{V,z,s} on \overline{H}^{n} by

μ_{V,z,s} = \frac{1}{Σ(z, s)} ∑_{w∈V} e^{−s dist(w,z)}δ_{w},

for z ∈ H^{n} and s > D.

Now fix an arbitrary point z_{0} ∈ H^{n} and let (s_{j})_{j≥0} be a decreasing sequence of real numbers converging to D such that the sequence of measures (μ_{V,s_{j},z_{0}}) is a weakly convergent sequence. Let μ_{V,z_{0}} denote the limit. The condition Σ(z, D) = ∞ implies that the support of μ_{V,z_{0}} is contained in S_{∞}.

It is immediate from the definition that for all s > D

dμ_{V,z′,s} = Π(z, z′, ·)^{s} dμ_{V,z,s}.

This implies first of all that the sequence (μ_{V,z,s_{j}}) is weakly convergent for each z ∈ H^{n}. Furthermore, letting μ_{V,z} denote the limit of the sequence (μ_{V,z,s_{j}}), and viewing these measures as measures on S_{∞}, we obtain the formula

dμ_{V,z′} = P(z, z′, ·)^{D}dμ_{V,z}.

In particular, M_{V} = μ_{V,z} is a D-conformal density.

4.5. Adjustment functions. The above construction must be modified in the case where Σ(z, D) < ∞; the idea, invented by Patterson, is to perturb the terms in the sum Σ(z, s) to obtain an expression Σ′(z, s) that is always finite when s > D and infinite when 0 ≤ s ≤ D. In our formulation, we show that for a suitably chosen real-valued function α the construction sketched above can be carried out with dist(z, w) replaced by α(dist(z, w)) throughout.

If α is any real-valued function on [0, ∞), then for any point z ∈ H^{n} and any s > 0 we define

Σ(α, z, s) = ∑_{w∈W} e^{−sα( dist(w,z))}

where we again interpret the sum as a non-negative number or +∞. Note that if id denotes the identity function on [0, ∞) then Σ(id, z, s) = Σ(z, s) for any z and s.

We define an adjustment function to be a C^{1} real-valued function α on [0, ∞) such that (i) α(0) = 0, (ii) 0 ≤ α′(t) ≤ 1 for all t > 0, and (iii) lim_{t→∞} α′(t) = 1. In particular an adjustment function α must satisfy 0 ≤ α(t) ≤ t for all t ≥ 0. The identity function on [0, ∞) is obviously an adjustment function.

The following generalizations of 4.3.1 and 4.3.2 hold for any adjustment function α.

4.5.1. For any z ∈ H^{n} and any s ≥ s′ ≥ 0 we have Σ(α, z, s′) ≥ Σ(α, z, s).

4.5.2. For any z, z′ ∈ H^{n} and any s ≥ 0 we have

Σ(α, z′, s) ≤ e^{s dist(z,z′)}Σ(α, z, s).

Indeed, 4.5.1 is immediate; and 4.5.2 follows from the observation that, by virtue of condition (ii) in the definition of an adjustment function, we have

|α( dist(w, z)) − α( dist(w, z′))| ≤ | dist(w, z) − dist(w, z′)| ≤ dist(z, z′)

for any w ∈ W.

We also have the following extension of 2.1.

Lemma. For any adjustment function α, the function Π̄_{α} : H^{n} × H^{n} × \overline{H}^{n} → R defined by

Π̄_{α}(z, z′, w) = exp(α(dist(z, w)) − α(dist(z′, w)))

for (z, z′, w) ∈ H^{n} × H^{n} × H^{n}, and

Π̄_{α}(z, z′, ζ) = P(z, z′, ζ)

for (z, z′, ζ) ∈ H^{n} × H^{n} × S_{∞}, is continuous.

Proof. It is enough to prove that if (z_{i}), (z′_{i}) and (w_{i}) are sequences of points in H^{n} which converge respectively to points z ∈ H^{n}, z′ ∈ H^{n} and ζ ∈ S_{∞}, then

lim_{i→∞} exp (α( dist(z_{i}, w_{i})) − α( dist(z′_{i}, w_{i}))) = P(z, z′, ζ).

By definition, we have

lim_{i→∞} exp ( dist(z_{i}, w_{i})) − dist(z′_{i}, w_{i})) = P(z, z′, ζ).

In particular, dist(z_{i}, w_{i}) − dist(z′_{i}, w_{i}) converges. Since α′(t) → 1 as t → ∞ we have

lim_{i→∞} \frac{α( dist(z_{i}, w_{i})) − α( dist(z′_{i}, w_{i}))}{dist(z_{i}, w_{i}) − dist(z′_{i}, w_{i})} = 1.

The result follows. □

4.6. Lemma. There exists an adjustment function α such that, for every z ∈ H^{n}, we have Σ(α, z, s) < ∞ for all s > D, and Σ(α, z, s) = ∞ whenever 0 ≤ s ≤ D.

Proof. First consider an arbitrary adjustment function α. Since α(t) ≤ t for all t ≥ 0, we have Σ(α, z, s) ≥ Σ(z, s) for all s; hence Σ(α, z, s) = ∞ when 0 ≤ s < D. Now suppose that s > D and choose a positive number k < 1 such that sk < D. Since lim_{t→∞} α′(t) = 1, there is a constant C such that α(t) ≥ kt − C for all t ≥ 0. Hence, for s > D, we have

Σ(α, z, s) ≤ e^{Cs}Σ(z, sk) < ∞.

Thus we need only to construct α so that Σ(α, z, D) = ∞. We first dispose of the special case D = 0. Since W is infinite, it is clear that Σ(z, 0) = +∞ for every z. Hence in the case D = 0 we may take α to be the identity function.

Now suppose that D > 0. If α is any adjustment function, it follows from 4.5.2 that Σ(α, z, D) is either finite for all z ∈ H^{n} or infinite for all z ∈ H^{n}. Hence we need only show that for a given point z ∈ H^{n} there is an adjustment function α such that Σ(α, z, D) = +∞.

Let (θ_{m})_{m≥1} be a strictly increasing sequence of positive numbers which converge to 1. For each m ≥ 1 we set s_{m} = θ_{m}D. Since 0 < s_{m} < D, the sum

Σ(z, s_{m}) = ∑_{w∈W} e^{−s_{m} dist(w,z)}

is infinite. Hence there is a finite subset X_{m} of W such that

∑_{w∈X_{m}} e^{−s_{m} dist(w,z)} ≥ m.

Let (R_{m})_{m≥1} be a monotone increasing sequence of positive numbers tending to +∞, such that for each m ≥ 1 and each w ∈ X_{m} we have dist(z, w) ≤ R_{m}. Let β be any continuous, monotone increasing real-valued function on [0, ∞) such that β(R_{m}) = θ_{m} for all m > 0 and β(t) < 1 for all t ≥ 0. Then α(t) = ∫_{0}^{t} β(u) du is an adjustment function and α(t) ≤ θ_{m}t whenever 0 ≤ t ≤ R_{m}. For each m > 0 and each w ∈ X_{m} we have dist(w, z) < R_{m}, and hence

α( dist(w, z))D ≤ θ_{m}D dist(w, z) = s_{m} dist(w, z).

Thus

Σ(α, z, D) ≥ ∑_{w∈X_{m}} e^{−Dα(dist(w,z))} ≥ ∑_{w∈X_{m}} e^{−s_{m} dist(w,z)} ≥ m.

Since this holds for all m ≥ 1, we must have Σ(α, z, D) = +∞. □

For the rest of this section, we fix an adjustment function α constructed as in the lemma. We shall write Σ′(z, s) = Σ(α, z, s) for all z and s.

4.7. For any s > D and any V ∈ 𝔙 we have

∑_{w∈V} e^{−sα(dist(w,z))} ≤ Σ′(z, s) < ∞

by Lemma 4.6. Hence we may define a Borel measure

∑_{w∈V} e^{−sα(dist(w,z))}δ_{w}

on \overline{H}^{n}. This measure will be denoted μ_{V,z,s}. Its total mass is ≤ Σ′(z, s), with equality when V = W.

4.8. For the rest of this section we fix, arbitrarily, a point z_{0} ∈ H^{n}.

Lemma. There exists a sequence (s_{j})_{j≥0} of real numbers with the following properties: (i) s_{j} > D for all j, and lim_{j→∞} s_{j} = D; (ii) for every V ∈ 𝔙, the sequence (Σ′(z_{0}, s_{j})^{−1}μV, z_{0}, s_{j})_{j≥0} is a weakly convergent

sequence of Borel measures on \overline{H}^{n}.

Proof. Let (t_{i}) be any sequence of numbers > D such that lim_{i→∞} t_{i} = D. For any V ∈ 𝔙, we have a sequence of measures (Σ′(z_{0}, t_{i})^{−1}μ_{V,z_{0},t_{i}}) which by 4.7 all have total mass ≤ 1. Given any V ∈ 𝔙, there is a subsequence (t_{i_{j}}) such that the sequence (Σ′(z_{0}, t_{i_{j}})^{−1}μ_{V,z_{0},t_{i_{j}}}) is weakly convergent. Since 𝔙 is countable, the conclusion of the lemma follows via diagonalization. □

For the rest of the section we fix a sequence (s_{j}) having the above properties (i) and (ii). For each j ≥ 0 we set C_{j} = Σ′(z_{0}, s_{j}). For each V ∈ 𝔙 we let μ_{V}^{0} denote the weak limit of the sequence (\frac{1}{C_{j}} μ_{V,z_{0},s_{j}}); by definition, μ_{V}^{0} is a Borel measure on \overline{H}^{n}.

4.9. Lemma. We have C_{j} → +∞ as j → ∞.

Proof. By 4.6 we have Σ′(z_{0}, D) = +∞. This implies the lemma since the terms of the sum Σ′(z, s) = ∑_{i∈I} e^{−sα(dist(w_{i},z))} are positive-valued continuous functions of s. □

4.10. Lemma. For each V ∈ 𝔙, the measure μ_{V}^{0} has support contained in S_{∞}^{n−1}.

Proof. It is enough to show that μ_{V}^{0} (C) = 0 for any compact set C ⊂ H^{n}. The set V ∩ C is finite since V is a subset of the uniformly discrete set W. We have

μ_{V,z_{0},s_{j}}(C) = ∑_{w∈V} exp(−s_{j}α(dist(w, z))) ≤ card (V ∩ C)

for each j ≥ 0. Thus \frac{1}{C_{j}} μ_{V}^{0} (C) is at most \frac{1}{C_{j}} card (V ∩ C), which tends to 0 as j → ∞ by virtue of Lemma 4.9. Hence μ_{V}^{0} (C) = 0. □

4.11. Lemma. For any V ∈ 𝔙 and any point z ∈ H^{n}, the sequence (\frac{1}{C_{j}} μ_{V,z,s_{j}}) converges weakly to μ_{V,z}.

Proof. It is immediate from the definition that dμ_{V,z,s_{j}} = \overline{Π}_{α}(z_{0}, z, ·)^{s_{j}}dμ_{V,z_{0},s_{j}}, where Π̄_{α} is the function given by Lemma 4.5. Setting μ_{j}^{0} = \frac{1}{C_{j}} μ_{V,z_{0},s_{j}} and μ_{j} = \frac{1}{C_{j}} μ_{V,z,s_{j}}, we therefore obtain, for any continuaous function f on \overline{H}^{n} and any j ≥ 0,

∫ f dμ_{j} = ∫ \overline{Π}_{α}(z_{0}, z, ·)^{s_{j}}f dμ_{j}^{0}.

Now since lim_{j→∞} s_{j} = D, and since f and \overline{Π} are continuous on the compact space \overline{H}^{n}, we have Π̄_{α}(z_{0}, z, ·)^{s_{j}}f → Π̄_{α}(z_{0}, z, ·)^{D}f uniformly. Since the μ_{j}^{0} converge weakly to μ_{V}^{0} , it follows that

lim_{j→∞} ∫ \overline{Π}_{α}(z_{0}, z, ·)^{s_{j}}f dμ_{j}^{0} = ∫ \overline{Π}_{α}(z_{0}, z, ·)^{D}f dμ_{V}^{0} = ∫ P^{D}f dμ_{V}^{0} .

The right-hand side of the last equation is equal to ∫ fdμ_{V,z} since M_{V} = (μ_{V,z}) is a D-conformal density. So we have

lim_{j→∞} ∫ fdμ_{j} = ∫ fdμ_{V,z}.

This proves the lemma. □

Proof of Proposition 4.2.. In light of Lemma 4.10, for each V ∈ 𝔙, the measure μ_{V}^{0} may be regarded as a measure on S_{∞}. According to 3.3, there exists for each V ∈ 𝔙 a unique D-conformal density M_{V} = (μ_{V,z}) such that μ_{V,z_{0}} = μ_{V}^{0} . We shall prove Proposition 4.2 by showing that the family (M_{V}) of D-conformal densities satisfies conditions (i) - (iv) of the proposition.

To prove (i) it is enough to show that μ_{W,z_{0}} ≠ 0. But as we observed in 4.7, for each s > D and each z ∈ H^{n}, the measure μ_{W,z,s} has total mass Σ′(z, s); in particular, for each j ≥ 0 the total mass of μ_{W,z_{0},s_{j}} is C_{j}. Thus μ_{W,z_{0}} is the weak limit of a sequence of measures of total mass 1, and therefore itself has total mass 1.

To prove (ii) we observe that if V = ∐_{i=1}^{m} V_{i}, where V and the V_{i} belong to 𝔙, then for each s > D and each z ∈ H^{n} we have μ_{V,z,s} = ∑_{i=1}^{m} μ_{V_{i},z,s} by the definitions of μ_{V,z,s} and μ_{V_{i},z,s}. Hence for each j ≥ 0 we have

\frac{1}{C_{j}} μ_{V,z,s} = \frac{1}{C_{j}} ∑_{i=1}^{m} μ_{V_{i},z,s}.

Using Lemma 4.11 we can take weak limits of both sides to obtain

μ_{V,z} = ∑_{i=1}^{m} μ_{V_{i},z}.

This proves (ii).

To prove (iii) we must show that if V is a set in 𝔙 and γ is a hyperbolic isometry such that γV ∈ 𝔙, then for each z ∈ H^{n} we have γ_{∞}^{∗}μ_{γV,γz} = μ_{V,z}. If we think of μ_{γV,γz} and μ_{V,z} as measures on \overline{H}^{n}, then γ_{∞}^{∗}μ_{γV,γz} may be rewritten as γ̄^{∗}μ_{γV,γz}. For each s > D we have

γ̄^{∗}μ_{γV,γz,s} = ∑_{w∈V} e^{−sα(dist(γw,γz))}γ̄^{∗}(δ_{γw}) = ∑_{w∈V} e^{−sα(dist(w,γz))}δ_{w} = μ_{V,z,s}.

Hence for each j ≥ 0 we have

γ̄^{∗}(\frac{1}{C_{j}} μ_{γV,γz,s_{j}}) = \frac{1}{C_{j}} μ_{V,z,s_{j}}.

Using Lemma 4.11 we can take weak limits of both sides to obtain γ̄^{∗}μ_{γV,γz} = μ_{V,z}, as required.

To prove (iv) we consider an arbitrary set V ∈ 𝔙. For any s > D, it follows from the definition of the measure μ_{V,z_{0},s} that the support of μ_{V,z_{0},s} is exactly V ∪ Λ_{V}, the closure of V in \overline{H}^{n}. Hence supp \frac{1}{C_{j}} μ_{V,z_{0},s_{j}} = V ∪ Λ_{V} for each j ≥ 0. By Lemma 4.11 it follows that supp μ_{V}^{0} ⊂ V ∪ Λ_{V}. But supp μ_{V}^{0} ⊂ S_{∞} by 4.10, and hence supp μ_{V}^{0} ⊂ Λ_{V}. This implies (iv). □

§5. The paradoxical decomposition

This section is devoted to the proof of the following two results.

5.1. Proposition. Let Γ be a Kleinian group which is free on a generating set Ξ with k elements, where 2 ≤ k < ∞. Suppose that every Γ-invariant conformal density for H^{3} is a constant multiple of the area density. Then for any z ∈ H^{3} we have

max_{ξ∈Ξ} dist(z, ξ · z) ≥ \frac{1}{2} log((k − 1)(2k − 1)).

5.2. Proposition. Let ξ and η be two orientation-preserving isometries of H^{3}. Suppose that the group Γ generated by ξ and η is discrete and is free on the generators ξ and η. Suppose in addition that every Γ-invariant conformal density for H^{3} is a constant multiple of the area density. Then for any z ∈ H^{3} we have

max(dist(z, ξ · z), dist(z, η · z)) ≥ log 3.

5.3. The key step in the proofs of Propositions 5.1 and 5.2 is the following lemma. When the hypotheses of 5.1 or 5.2 are saøisfied, the lemma will give the “paradoxical” decomposition of the normalized area measure on S_{∞} which was described in the introduction.

Lemma. Let Γ be a Kleinian group which is free on a generating set Ξ. Set Ψ = Ξ⨿Ξ^{−1} ⊂ Γ. Let z_{0} be any point of H^{3}. Then there exist a number D ∈ [0, 2], a Γ-invariant Dconformal density M = (μ_{z}) for H^{3} and a family (ν_{ψ})_{ψ∈Ψ} of Borel measures on S_{∞} such that

(i) μ_{z_{0}}(S_{∞}) = 1; (ii) μ_{z_{0}} = ∑_{ψ∈Ψ} ν_{ψ}; and (iii) for each ψ ∈ Ψ we have

∫ (λ_{ψ,z_{0}})^{D}dν_{ψ^{−1}} = 1 − ∫ dν_{ψ}.

If Ξ is a two-element set {ξ, η} and if z_{0} ∈ ℓ(ξ, η) (see 1.8) then we also have (iv)

∫ dν_{ξ^{−1}} = ∫ dν_{ξ}  and  ∫ dν_{η^{−1}} = ∫ dν_{η}.

Proof. Every element γ of Γ can be written uniquely as a reduced word ψ_{1} ⋯ ψ_{m}, where m ≥ 0, each ψ_{i} is an element of Ψ, and ψ_{i+1} ≠ ψ_{i}^{−1} for i = 1,…, m − 1. If γ ≠ 1, i.e. if m > 0, we shall call ψ_{1} the initial letter of γ. For each ψ ∈ Ψ, let J_{ψ} denote the set of non-trivial elements of Γ that have initial letter ψ. Then Γ is a finite disjoint union

{1} ⨿ ∐_{ψ∈Ψ} J_{ψ}.

Since Γ is discrete and torsion-free, it acts freely on H^{3}. Hence the orbit W = Γz_{0} is a disjoint union

(5.3.1)  W = {z_{0}} ⨿ ∐_{ψ∈Ψ} V_{ψ}

where V_{ψ} = {γz : γ ∈ J_{ψ}}. Let 𝔙 denote the finite collection of all subsets of W which are sub-unions of the above disjoint union; that is, W consists of all sets of the form ∐_{ψ∈Ψ′} V_{ψ} or {z_{0}} ⨿ ∐_{ψ∈Ψ′} V_{ψ} for Ψ′ ⊂ Ψ. We apply Proposition 4.2 with these choices of W and 𝔙. We take D to be a number, and (M_{V})_{V ∈𝔙} a family of conformal densities, for which conditions (i)–(iv) of 4.2 are satisfied. We write M_{V} = (μ_{V,z})_{z∈H^{n}}. We set M = M_{W}, and ν_{ψ} = μ_{V_{ψ},z_{0}} for each ψ ∈ Ψ. According to 4.2, M is Γ-invariant. It follows from condition (i) of 4.2 and the definition of a conformal density that μ_{z_{0}} = μ_{W,z_{0}} ≠ 0; hence after multiplying by a constant we may assume that μ_{z_{0}} has total mass 1. This is conclusion (i) of the present lemma. We shall show that the other conclusions of the lemma also hold with these definitions of M and of the ν_{ψ}.

By (5.3.1) above and condition (ii) of 4.2, we have

μ_{z_{0}} = μ_{W,z_{0}} = μ_{{z_{0}},z_{0}} + ∑_{ψ∈Ψ} μ_{V_{ψ},z_{0}}.

But μ_{{z_{0}},z_{0}} = 0 by condition (iv) of 4.2. Hence

μ_{z_{0}} = ∑_{ψ∈Ψ} μ_{V_{ψ},z_{0}} = ∑_{ψ∈Ψ} ν_{ψ}

which is conclusion (ii) of the lemma.

It follows immediately from the definition of the sets J_{ψ} that for each ψ ∈ Ψ we have

ψJ_{ψ^{−1}} = Γ − J_{ψ},

so that

ψV_{ψ^{−1}} = W − V_{ψ}.

Since W − V_{ψ} = {z_{0}} ⨿ ∐ ψ′ ≠ ψV_{ψ′} ∈ 𝔙, condition (iii) of 4.2 gives

(5.3.2)  M_{V_{ψ}−1} = ψ_{∞}^{∗}(M_{W −V_{ψ}}).

On the other hand, by 4.2(ii) we have

M = M_{W −V_{ψ}} + M_{V_{ψ}}.

Thus (5.3.2) can be rewritten in the form M_{V_{ψ^{−1}}} = ψ_{∞}^{∗}(M − M_{V_{ψ}}); and in particular we have

(5.3.3)  μ_{V_{ψ^{−1},ψ(z_{0})}} = ψ_{∞}^{∗}(μ_{z_{0}} − μ_{V_{ψ},z_{0}}) = ψ_{∞}^{∗}(μ_{z_{0}} − ν_{ψ}).

Since M_{V_{ψ^{−1}}} is a D-conformal density, we have

dμ_{V_{ψ^{−1}} ,ψ(z_{0})} = P(z_{0}, ψ^{−1}z_{0}, ·)^{D}dμ_{V_{ψ^{−1}}} = λ_{ψ,z_{0}}^{D}dμ_{V_{ψ^{−1}}} ,

where the last equality follows from Proposition 2.4. Hence, equating the total masses of the two sides of (5.3.3), we obtain

∫ λ_{ψ,z_{0}}^{D}dμ_{V_{ψ^{−1}}} = ∫ d(ψ_{∞}^{∗}(μ_{z_{0}} − ν_{ψ})) = ∫ d(μ_{z_{0}} − ν_{ψ}) = 1 − ∫ dν_{ψ},

and conclusion (iii) of the lemma is established.

It remains to show that conclusion (iv) holds if Ξ = {ξ, η} and z_{0} ∈ ℓ(ξ, η). Since the involution τ = τ_{ξ,η} satisfies τξτ = ξ^{−1} and τητ = η^{−1}, we have τΓτ = Γ; furthermore, τJ_{ξ}τ = J_{ξ^{−1}} and τJ_{η}τ = J_{η^{−1}}. On the other hand, since z_{0} ∈ ℓ(ξ, η) we have τz_{0} = z_{0}. Hence V_{ξ^{−1}} = {γz_{0} : γ ∈ J_{ξ^{−1}}} = {τγτz_{0} : γ ∈ J_{ξ}} = τV_{ξ}. By condition (iii) of 4.2 it follows that τ_{∞}^{∗}M_{V_{ξ}} = M_{V_{ξ^{−1}}} . In particular, τ_{∞}^{∗}νξ = ν_{ξ^{−1}}. Taking total masses of both sides gives ∫ dν_{ξ^{−1}} = ∫ dν_{ξ}. The same argument shows that ∫ dν_{η^{−1}} = ∫ dν_{η}. This establishes (iv) and completes the proof of the lemma. □

The proof of Propositions 5.1 and 5.2 will require combining Lemma 5.3.3 with the elementary considerations covered by the following two lemmas.

5.4. Lemma. Let μ and μ_{0} be measures of finite total mass on a measurable space (X, B). Suppose that 0 ≤ μ_{0} ≤ μ. Let C be a Borel set such that μ(C) ≥ μ_{0}(X). Let f be a measurable, non-negative real-valued function on X such that inf f(C) ≥ sup f(X − C). Then

∫_{X} fdμ_{0} ≤ ∫_{C} fdμ.

Proof. We write μ = μ_{0} + μ_{1}, where μ_{1} is again a measure on (X, B). We have

μ_{0}(X − C) = μ_{0}(X) − μ_{0}(C) ≤ μ(C) − μ_{0}(C) = μ_{1}(C),

so that

∫_{X−C} fdμ_{0} ≤ (sup f(X − C)) · μ_{0}(X − C) ≤ (inf f(C)) · μ_{1}(C) ≤ ∫_{C} fdμ_{1}.

Hence

∫_{X} f dμ_{0} = ∫_{C} f dμ_{0} + ∫_{X−C} f dμ_{0} ≤ ∫_{C} f dμ_{0} + ∫_{C} f dμ_{1} = ∫_{C} f dμ.

□

5.5. Lemma. Let a and b be numbers such that 0 ≤ a ≤ \frac{1}{2} and 0 ≤ b ≤ 1, let γ be an isometry of H^{3} and let z be a point in H^{3}. Suppose that ν is a measure on S_{∞} such that

(i) ν ≤ A_{z}, (ii) ν(S_{∞}) ≤ a, and (iii) ∫_{S_{∞}} λ_{γ,z}^{2} dν ≥ b.

Then

dist(z, γ · z) ≥ \frac{1}{2} log \frac{a(1 − b)}{b(1 − a)}.

Proof. We let h denote the constant dist(z, γ · z), and set c = cosh h and s = sinh h. We let λ denote the function λ_{γ,z}. We identify \overline{H}^{3} conformally with the unit ball in R^{3} in such a way that z is the origin (so that S_{∞} has the round metric centered at z) and γ^{−1} · z is on the positive vertical axis.

According to 2.4, we have λ(ζ) = P(z, γ^{−1} · z, ζ) for all ζ ∈ S_{∞}. Hence by (2.1.1), λ is given by the formula

λ(ζ) = (c − s cos ϕ)^{−1}

where ϕ = ϕ(ζ) is the angle between the positive vertical axis and the ray from the origin through ζ; thus ϕ is the polar angle of ζ in spherical coordinates.

Set A = A_{z}. Since S_{∞} has the round metric centered at z, the measure A is obtained by dividing the area measure on the unit sphere by the area 4π of the sphere. In spherical coordinates θ and ϕ on the unit sphere we have dA = (1/4π) sin ϕ dϕdθ.

Set ϕ_{0} = arccos(1 − 2a), and let C ⊂ S_{∞} denote the spherical cap defined by the inequality ϕ ≥ ϕ_{0}. Then we have

A(C) = \frac{1}{4π} ∫_{0}^{2π} ∫_{0}^{ϕ_{0}} sin ϕ dϕdθ = \frac{1}{2}(1 − cos ϕ_{0}) = a.

Thus by hypothesis (ii) we have A(C) ≥ ν(S_{∞}). Observe also that since λ is given by the function (c − s cos ϕ)^{−1}, which is positive and monotone decreasing for 0 ≤ ϕ ≤ π, we have inf λ(C) ≥ sup λ(Ξ − C). Since we also have ν ≤ A by hypothesis (i), we may apply Lemma 5.4 with f = λ^{2} to obtain

∫_{S_{∞}} λ^{2}dν ≤ ∫_{C} λ^{2}dA = \frac{1}{4π} ∫_{0}^{2π} ∫_{0}^{ϕ_{0}} \frac{sin ϕ}{(c − s cos ϕ)^{2}} dϕdθ

= \frac{1}{2} ∫_{0}^{ϕ_{0}} \frac{sin ϕ}{(c − s cos ϕ)^{2}} dϕ = \frac{1}{2s}(\frac{1}{c − s} − \frac{1}{c − cos ϕ_{0}} )

where the last step follows from the substitution u = c − s cos ϕ. Recalling that cos ϕ_{0} = 1 − 2a and using hypothesis (iii), we find that

b ≤ ∫_{S_{∞}} λ^{2}dν ≤ \frac{a}{(c − s)(c − s + 2as)}.

After rewriting c and s respectively as \frac{1}{2}(e^{h} + e^{−h}) and \frac{1}{2}(e^{h} − e^{−h}), and simplifying, we conclude that

e^{−2h} ≤ \frac{a(1 − b)}{b(1 − a)}.

□

Proof of Proposition 5.1. Suppose that Γ and Ξ satisfy the hypotheses of the proposition, and let z_{0} ∈ H^{n} be an arbitrary point.

We wish to show that

max_{ξ∈Ξ} dist(z_{0}, ξ · z_{0}) ≥ \frac{1}{2} log((k − 1)(2k − 1)).

In particular Γ and Ξ satisfy the satisfy the hypotheses of Lemma 5.3, and so there exist a number D ∈ [0, 2], a conformal density M = (μ_{z}) and a family of Borel measures (ν_{ψ})_{ψ∈Ψ} (where Ψ = Ξ ⨿ Ξ^{−1}), for which conclusions (i)–(iii) of 5.3 hold. But the hypotheses of 5.1 imply that M is a constant multiple of the area density A; hence D = 2. Since μ_{z_{0}} has total mass 1 by 5.3(i), we must in fact have M = A.

By 5.3(ii) we have

1 = ∑_{ψ∈Ψ} ν_{ψ}(S_{∞}) = ∑_{ξ∈Ξ}(ν_{ξ}(S_{∞}) + ν_{ξ^{−1}}(S_{∞})).

Hence for some ξ_{0} ∈ Ξ we have ν_{ξ_{0}}(S_{∞}) + ν_{ξ_{0}^{−1}}(S_{∞}) ≤ \frac{1}{k}. It follows that for some ψ_{0} ∈ \{ξ_{0}, ξ_{0}^{−1}\} we have ν_{ψ_{0}}(S_{∞}) ≤ \frac{1}{2k}.

We set λ = λ_{ψ_{0},z} and ν = ν_{ψ_{0}} − 1 . By 5.3(iii) we have

∫_{S} λ^{2}dν = 1 − ν_{ψ_{0}}(S_{∞}) ≥ 1 − (ν_{ξ_{0}}(S_{∞}) + ν_{ξ_{0}} − 1(S_{∞})) ≥ 1 − \frac{1}{k} .

Furthermore, it follows from 5.3(ii) that ν ≤ μ_{z_{0}} = A_{z_{0}}. Hence we may apply Lemma 5.5 with z = z_{0}, γ = ψ_{0}, a = \frac{1}{2k} and b = 1 − \frac{1}{k} . This gives

dist(z, ψ_{0} · z) ≥ \frac{1}{2} log((k − 1)(2k − 1)).

But since ψ_{0} = ξ_{0}^{±1} we have dist(z, ψ_{0} · z) = dist(z, ξ_{0} · z), and the conclusion follows. □

Proof of Proposition 5.2. Suppose that Γ, ξ and η satisfy the hypotheses of the proposition. Let z_{0} ∈ H^{n} be any point; we wish to show that max(dist(z_{0}, ξ ·z_{0}), dist(z_{0}, η ·z_{0})) ≥ log 3. Set ℓ = ℓ(ξ, η). It follows from Proposition 1.15 that

max(dist(z_{0}, ξ · z_{0}), dist(z_{0}, η · z_{0})) ≥ max(dist(z_{1}, ξ · z_{1}), dist(z_{1}, η · z_{1})),

where z_{1} denotes the orthogonal projection of z_{0} onto ℓ. Hence we may assume without loss of generality that z_{0} ∈ ℓ.

In particular Γ and Ξ = {ξ, η} satisfy the hypotheses of Lemma 5.3, and so there exist a number D ∈ [0, 2], a conformal density M = (μ_{z}) and a family of Borel measures (ν_{ψ})_{ψ∈Ψ} (where Ψ = {ξ, ξ^{−1}, η, η^{−1}}), for which conclusions (i)–(iv) of 5.3 hold. But the hypotheses of 5.2 imply that M is a constant multiple of the area density A; hence D = 2. Since μ_{z_{0}} has total mass 1 by 5.3(i), we must in fact have M = A.

By conclusions (ii) and (iv) of 5.3 we have

1 = 2ν_{ξ}(S_{∞}) + 2ν_{η}(S_{∞})

so that either ν_{ξ}(S_{∞}) or ν_{η}(S_{∞}) is ≤ 1/4. By symmetry we may assume that ν_{ξ}(S_{∞}) ≤ 1/4. We set λ = λ_{ξ^{−1},z_{0}}. By 5.3(iii) and (iv) we have

∫_{S} λ^{2}dν_{ξ} = 1 − ν_{ξ^{−1}}(S_{∞}) = 1 − ν_{ξ}(S_{∞}) ≥ 3/4.

Furthermore, it follows from 5.3(i) that ν_{ξ} ≤ A. Hence we may apply Lemma 5.5 with ν = ν_{ξ}, γ = ψ_{0}, a = \frac{1}{4} and b = \frac{3}{4} . This gives

dist(z_{0}, ξ^{−1} · z_{0}) ≥ log 3.

Since dist(z_{0}, ξ · z_{0}) = dist(z_{0}, ξ^{−1} · z_{0}), the conclusion follows. □

§6. Bonahon surfaces and superharmonic functions

The results of Section 5 depend on the hypothesis that every Γ-invariant conformal density is a constant multiple of the area density, where Γ is the given free Kleinian group. According to Proposition 3.9, this is always true if every positive, Γ -invariant superharmonic function on H^{3} is constant. Thus in order to apply the results of Section 5 one needs a geometric condition that guarantees that the hyperbolic manifold M(Γ) admits no non-constant positive superharmonic functions. In this section we describe such a condition. Very similar conditions have been used in Thurston [Th 8.12] and, later, in Sullivan [Su3], Bohanon [Bo] and Canary [Ca1].

For our applications we need only consider Kleinian groups Γ which contain no parabolics and satisfy M(Γ) = N(Γ). We will restrict attention to these groups whenever convenient. While the complications which arise from the existence of parabolics can be dealt with, it would unnecessarily complicate the exposition to do so here.

6.1. Let Σ be a closed 2-manifold, not necessarily connected. Let u ∈ H_{2}(Σ; Z/2) denote the fundamental class of Σ, i.e. the sum of the images under inclusion of the fundamental classes of the components of Σ. A map f: Σ → M will be said to be null-homologous if the class f_{∗}(u) ∈ H_{2}(M; Z/2) is trivial. Suppose that f: Σ → M is a null-homologous map and that P is a point of M − f(P). We will say that P is strictly enclosed by f if the class f_{∗}(u) ∈ H_{2}(M − {P }; Z/2) is non-trivial, where we are regarding f as a map from Σ to M − {P }. We will say that a point P ∈ M is enclosed by the null-homologous map f if either P ∈ f(Σ) or P is strictly enclosed by f. A compact set K will be said to be enclosed by f if every point of K is enclosed by f.

Proposition. Let Γ be a Kleinian group without parabolics such that M(Γ) = N(Γ). Suppose that f_{j}: Σ_{j} → M is a sequence of maps from closed 2-manifolds into M such that every compact subset of M(Γ) is enclosed by some f_{j}. If there exists a constant V such that the volume of nbhd_{1} f_{j}(Σ_{j}) is less than V for all j then every positive superharmonic function on M is constant.

Proof. This statement is proved by an argument given in [Th, 8.12] and [Ca, Theorem 7.1]. In that argument it is assumed that M has k simply degenerate ends with neighborhoods homeomorphic to S_{1} × [0, ∞),…, S_{k} × [0, ∞). The argument is applied to a sequence of maps f_{j}: Σ → M where Σ = Σ_{j} is the disjoint union of S_{1},…, S_{k} and where the restriction of f_{j} to S_{i} is homotopic to the inclusion of S_{j} × {0} into M. The argument uses two properties of these maps. First that there is a bound on the volume of the sets nbhd_{1} (f_{j}(Σ_{j})) and second that a flow line which exits an end of M must pass through infinitely many of these sets. The second property follows immediately from our hypothesis that every compact set in M is enclosed by one of the maps f_{j}. □

The rest of this section is devoted to a description of a geometric construction, due to Bonahon, of maps that satisfy the hypotheses of Proposition 6.1. The construction is a simplification of Thurston’s construction of “pleated surfaces.”

6.2. Virtual triangulations. A mapping φ of an affine k-simplex Δ^{k} into a topological space X will be said to define a virtual k-simplex in X if φ is one-to-one on the interior of each face of Δ^{k}. Two maps φ : Δ → X and φ′ : Δ′ → X will be said to define the same virtual simplex if there is an affine isomorphism I : Δ → Δ′ such that φ ∘ I = ψ. By the interior of the virtual simplex (defined by) φ : Δ → X we mean the set φ(int Δ). The dimension of φ is the dimension of Δ. A virtual simplex φ_{1} : Δ_{1} → X is called a face of a virtual simplex φ if there is an affine isomorphism J of Δ_{1} onto a face of Δ such that φ ∘ J = φ_{1}.

Let Σ be a topological 2-manifold-with-boundary. (We may have ∂Σ = ∅, and Σ may be disconnected.) A virtual triangulation of Σ is a collection Φ of virtual simplices in Σ such that

(i) Φ contains the faces of each simplex in Φ, and (ii) as a set, Σ is the disjoint union of the interiors of the virtual simplices in Φ.

(Note that two faces of a simplex may coincide.)

For any integer i, the i-skeleton of Σ in a given virtual triangulation is defined to be the union of the interiors of all the virtual simplices of dimension ≤ i; we denote it by skel_{i}(K). The virtual 0-simplices of Σ will be called vertices.

A virtual 1-simplex φ in the 2-manifold-with-boundary Σ may be regarded as a (nonoriented) path in Σ. If φ is a homotopically trivial loop we shall say that it is degenerate. A virtual triangulation will be termed non-degenerate if it has no degenerate 1-simplices.

6.3. Piecewise hyperbolic structures. Let φ : Δ → Σ be a virtual 2-simplex in a topological surface Σ. A homeomorphism h of Δ = domain φ onto a triangle in H^{2} which maps vertices onto vertices will be said to define a hyperbolic structure on φ. Two such homeomorphisms h and h′ will be said to define the same hyperbolic structure on φ if h′h^{−1} is an isometry of hyperbolic triangles. If ψ : Δ_{1} → Σ is a 1-dimensional face of φ, any hyperbolic structure on φ induces a linear metric on Δ_{1}.

A virtual 1-simplex in the interior of a virtually triangulated surface Σ either occurs as a face of two distinct 2-simplices or as two coincident faces of a single 2-simplex. Thus if each 2-simplex in Σ is given a hyperbolic structure then for each interior 1-simplex there are two linear metrics induced on Δ_{1}. A piecewise-hyperbolic (PH) structure on Σ is a family of hyperbolic structures on the virtual 2-simplices of Σ such that, for every interior 1-simplex of Σ, the two induced linear metrics on Δ_{1} agree. A PH-structure on Σ determines, in a natural way, the structure of a (necessarily incomplete) hyperbolic 2-manifold with geodesic boundary on Σ − skel_{0}(Σ),

If φ: Δ → Σ is a virtual 2-simplex and w is a vertex of Δ, we shall denote by θ_{w}(φ) the angle in the hyperbolic triangle Δ at the vertex w. For any vertex x of Σ we define the angle sum at x in a given triangulation to be ∑_{φ,w} θ_{w}(φ), where φ ranges over all virtual 2-simplices of the given virtual triangulation, and w ranges over all the vertices of dom φ which are mapped by φ to x. (Note that a given virtual 2-simplex φ can contribute as many as three terms to this sum.) If x is a point of Σ which is not a vertex, we define the angle sum at x to be 2π if x ∈ int Σ and π if x ∈ ∂Σ.

Let x be any point of Σ and let a denote the angle sum at x. If x ∈ int Σ we set excess(x) = a − 2π. If x ∈ ∂Σ we set excess(x) = a − π. We call x a singular point if excess(x) ≠ 0. If x ∈ ∂Σ and excess(x) < 0, we call x a corner. We shall denote by sing Σ the set of all singular points of Σ. It is clear that the natural hyperbolic structure on Σ − skel_{0}(Σ) admits a unique extension to a hyperbolic structure on Σ − sing Σ.

Given a virtual 2-simplex φ: Δ → Σ in the PH surface Σ, a path in φ(Δ) will be called geodesic if it has the form φ ∘ α for some geodesic path α (parametrized proportionally to arc length) in Δ. A path in Σ will be termed piecewise geodesic (PG) if it is a composition γ = (φ_{1} ∘ α_{1}) ∗ ⋯ ∗ (φ_{n} ∘ α_{n}) of finitely many geodesic paths supported in 2-simplices of Σ. The length of γ is the sum of the lengths of the α_{i}. Any two points in the same component of Σ can be joined by a PG path, and it is not hard to show that a metric on each component of Σ can be constructed by defining dist(x, y) to be the infimum of the lengths of PG paths joining x and y. It is also not difficult to verify that this metric agrees with the hyperbolic metric on the components of Σ − sing Σ.

6.4. Ultrahyperbolic surfaces. We define an ultrahyperbolic structure on a virtually triangulated surface Σ to be a PH structure such that excess(x) ≥ 0 for every x ∈ int Σ. The ultrahyperbolic surfaces form a rich class of geometric objects which are natural generalizations of hyperbolic surfaces. For example, by a generalization of the Gauss-Bonnet theorem which applies to PH surfaces (c.f. [Bo, Lemme 1.9]), each component of a surface with an ultrahyperbolic structure must have negative Euler characteristic. The fundamental geometric property of these surfaces which we will need here is the version of Bonahon’s Bounded Diameter Lemma stated below.

Given an ultrahyperbolic surface Σ and a point P ∈ Σ we define short(P) to be the infinum of the lengths of homotopically non-trivial PG loops based at P. If I is an interval contained in (0, ∞) we define Σ_{I} as in 1.5. In particular, for ϵ > 0, the ϵ-thin part of Σ is the subset Σ_{(0,ϵ]}. Given a PG path γ in Σ we define the length of γ modulo Σ_{(0,ϵ]} to be the length, i.e. the Lebesgue measure, of γ^{−1}(Σ_{(ϵ,∞)}). If P and Q are points of Σ then the distance from P to Q modulo Σ_{(0,ϵ]} is the infimum of the set of lengths modulo Σ_{(0,ϵ]} of PG paths from P to Q. The diameter of Σ modulo Σ_{(0,ϵ]} is the supremum over all P, Q ∈ Σ of the distance modulo Σ_{(0,ϵ]} from P to Q.

The following result is due to Bonahon [Bo, Lemme 1.10].

Proposition. For each positive integer g and each positive number ϵ there exists a constant D(g, ϵ) such that if Σ is a closed ultrahyperbolic surface of total genus g then the diameter of σ modulo Σ_{(0,ϵ]} is less than D(g, ϵ).

6.5. Hyperbolically simplicial surfaces. Let us consider a complete hyperbolic manifold M = H^{n}/Γ. A map f of a virtually triangulated (not necessarily connected) surface Σ into M will be called hyperbolically simplicial if for every virtual simplex φ : Δ → Σ, the map f_{φ} = f ∘ φ admits a continuous lift f̃_{φ} : Δ → H^{n} such that f̃_{φ}(Δ) is a hyperbolic simplex and either (i) dim f̃_{φ}(Δ) < dim Δ or (ii) f̃_{φ} maps Δ homeomorphically onto f̃_{φ}(Δ). If (ii) holds for every simplex of Σ we shall say that f is non-degenerate. If f is non-degenerate then for each virtual 2-simplex φ the map f̃_{φ} induces a hyperbolic structure on the 2-simplex Δ which is obviously independent of the choice of lift. The linear metric induced on the domain of a 1-simplex ψ by an adjacent 2-simplex agrees with the linear metric determined by f̃_{ψ}. Thus a non-degenerate hyperbolically simplicial map from Σ to M induces a PH structure on Σ. If we equip Σ with the PH structure induced by the non-degenerate hyperbolically simplicial map f, then for every PG path γ in Σ, the path f ∘ γ has the same length as γ. Hence if we give each component of Σ the metric defined by its PH structure, the restriction of f to each component of Σ is distance-decreasing.

By a hyperbolically simplicial surface in M we mean a pair (Σ, f), where Σ is a closed, orientable surface equipped with a virtual triangulation, and f : Σ → M is a hyperbolically simplicial map. We shall say that (Σ, f) is non-degenerate if the virtual triangulation of Σ and the map f are non-degenerate. The Euler characteristic and the total genus of a hyperbolically simplicial surface (Σ, f) are defined respectively to be the Euler characteristic and the total genus of the surface Σ.

Suppose we are given a hyperbolically simplicial surface (Σ, f). Then any small perturbation of f |skel_{0} Σ can be extended over Σ to give a hyperbolically simplicial surface (Σ, f′) such that f′ is homotopic to f. One first modifies f in a neighborhood of the 0-skeleton to make it agree with f′ on the vertices. The restrictions to skel_{1} Σ and skel_{2} Σ can then be successively modified to make them geodesic. The map f′ can be made arbitrarily close to f in the uniform topology by choosing the initial perturbation to be sufficiently small.

We will say that a hyperbolically simplicial surface (Σ, f) encloses a compact set K ⊂ M provided that the map f: Σ → M is null-homologous and encloses K.

6.6. Bohanon surfaces. A continuous map f of a topological 1-manifold S into a hyperbolic 3-manifold M will be called geodesic if for every component C of S there is a covering map p : R → C such that f ∘ p is a geodesic. In the case where S is a simple closed curve, i.e. is homeomorphic to S^{1}, a geodesic map f : S → M is a re-parametrization of a closed geodesic in M, and thus has a well-defined length.

A non-degenerate hyperbolically simplicial surface (Σ, f) in M will be called a Bonahon surface if there is a closed 1-manifold S ⊂ Σ such that skel_{0}(Σ) ⊂ S ⊂ skel_{1}(Σ) and f | S is geodesic. The following observation is contained in [Bo, Lemme 1.8].

Proposition. If (Σ, f) is a Bonahon surface in a complete hyperbolic manifold M, then the PH structure induced on Σ by f is ultrahyperbolic.

Proof. We must show that the angle sum at any vertex x of Σ is at least 2π. Let e_{1},…, e_{n} be the distinct oriented virtual 1-simplices of Σ having x as initial point. Then f maps each e_{i} to a geodesic path γ_{i} : [0, 1] → Σ with initial point x. Let U denote the unit sphere in the tangent space to the three manifold M at f(x). Let u_{i} ∈ U be the normalized tangent vector to γ_{i} at 0. Then the angle sum at x is equal to the length of a piecewise-geodesic curve γ in U whose vertices are u_{1},…, u_{n}. Now according to the definition of a Bonahon surface, the vertex x lies on a simple closed curve C ⊂ skel_{1}(Σ) such that f | C is geodesic. This implies that the curve γ passes through two antipodal points of U, and therefore has length ≥ 2π. □

6.7. Neighborhoods of Bonahon surfaces. Let (Σ, f) be a Bonahon surface in M, and let Σ be given the PH structure induced by f. Let ϵ be a positive number. We shall say that (Σ, f) is ϵ-incompressible if for every homotopically non-trivial PG closed curve γ : S^{1} → Σ of length < ϵ, the closed curve f ∘ γ is homotopically non-trivial in M.

The following proposition is proved in [Ca, Lemma 7.2]. (The pleated surface version is found in [Th, 8.12.1].)

Proposition. For each positive integer g and each positive number ϵ there exists a constant V (g, ϵ) such that if (f, Σ) is any ϵ-incompressible Bonahon surface of genus g in a complete hyperbolic 3-manifold M then the volume of nbhd_{1}(f(Σ)) is at most V (g, ϵ).

6.8. The following Proposition, which is an immediate consequence of Propositions 6.1 and 6.7, gives a basic criterion for every positive superharmonic function on M(Γ) to be constant.

Proposition. Let Γ be a Kleinian group without parabolics such that M(Γ) = N(Γ). Suppose that there exist positive integers g_{1},…, g_{k} and a real number ϵ such that every compact subset of M(Γ) is enclosed by a system (Σ_{1}, f_{1}),…, (Σ_{k}, f_{k}) of ϵ-incompressible Bonahon surfaces such that the genus of Σ_{i} is less than g_{i}. Then every positive superharmonic function on M(Γ) is constant.

6.9. Topological tameness. A 3-manifold is said to be topologically tame if it is homeomorphic to the interior of a compact 3-manifold. A Kleinian group Γ is topologically tame if M(Γ) is topologically tame.

A Theorem due to R. Canary [Ca1] implies that if Γ is a Kleinian group without parabolics such that M(Γ) = N(Γ) and if Γ is topologically tame then the hypotheses of Proposition 6.8 hold. In particular we have the following special case of [Ca, Theorem 7.2].

Proposition. Let Γ be a Kleinian group without parabolics such that M(Γ) = N(Γ). If Γ is topologically tame then every positive super-harmonic function on M(Γ) is constant.

In Section 8 we show, by quite different methods, that the hypotheses of Proposition 6.8 hold for most rank-2 free Kleinian groups which have no parabolics and lie on the boundary of the space of rank-2 free geometrically finite groups. The proof of our main theorem in Section 9 breaks into two cases which are handled respectively by the results of Section 8 and the proposition above.

6.10. We conclude this section with a result that will be needed in Section 8.

Proposition. Let M be a 3-manifold with an involution T such that every connected component of Fix T is homeomorphic to R. Let (Σ, f) be a null-homologous hyperbolically simplicial surface in M, and let T_{Σ} be an orientation-preserving involution of Σ such that f ∘ T_{Σ} = T ∘ f. Suppose that for some component L of Fix T there are exactly two fixed points v_{+} and v_{−} of T_{Σ} which are mapped into L by f; let s denote the compact arc in L bounded by f(v_{+}) and f(v_{−}). Then (Σ, f) encloses s.

Proof. It is enough to prove that every point of s ∩ (M − f(Σ)) is strictly enclosed by (Σ, f). Let P be any point of s∩(M −f(Σ)). By a small perturbation, as described in 6.5, one can approximate the map f arbitrarily well, in the uniform topology on the space of maps of Σ into M, by a hyperbolically simplicial map f_{1} which is topologically transverse to L and satisfies f_{1} ∘ T_{Σ} = T ∘ f_{1}. In particular, f_{1} maps the finite set Fix T_{Σ} into the 1-manifold Fix T. Hence if f_{1} is close enough to f then f_{1} maps v_{+} and v_{−} into distinct components of L − {P } and maps the remaining fixed points of T_{Σ} into (Fix T) − L. Thus the hypotheses of the theorem hold when f is replaced by f_{1}. Furthermore, if f_{1} is close enough to f then f_{1} maps Σ into M − {P } and is homotopic in M − {P } to f. Hence f_{1} strictly encloses P if and only if f does. Thus we may assume without loss of generality that f is topologically transverse to L.

In this case the set f^{−1}(L) is finite, and has even cardinality since (Σ, f) is nullhomologous. Let L_{+} and L_{−} denote the components of L − {P } containing f(v_{+}) and f(v_{−}) respectively. To say that (Σ, f) encloses P is equivalent to saying that f^{−1}(L_{+}) (and hence also f^{−1}(L_{−})) has odd cardinality. But f^{−1}(L_{+}) is invariant under the involution T_{Σ} (since f ∘ T_{Σ} = T ∘ f) and contains only one fixed point of T_{Σ}, namely a. Hence card f^{−1}(L_{+}) is indeed odd. □

§7. Girded surfaces and girded 3-manifolds

In this section we introduce a special class of Bonahon surfaces which will be used in the proof of our main theorem.

7.1. Let ϵ be a positive number. By an ϵ-girded surface in M we mean a triple (Σ, f, S), where (Σ, f) is a non-degenerate hyperbolically simplicial surface in M and S ⊂ Σ is a closed 1-manifold, such that

(i) skel_{0}(Σ) ⊂ S ⊂ skel_{1}(Σ); (ii) each component of Σ − S is homeomorphic to a sphere with three punctures; (iii) f | S is geodesic, and f | C has length < ϵ/3 for each component C of S; and (iv) for each component R of Σ − S, the map f | R induces an injective homomorphism

from π_{1}(R) to π_{1}(M). The maps f | C, where C ranges over the components of S, are re-parametrizations of

closed geodesics. These closed geodesics will be said to be carried by (Σ, f, S). It follows from condition (ii) above that S has 3g −3 components, where g is the genus of Σ. Hence if k denotes the number of distinct closed geodesics carried by Σ, we have 1 ≤ k ≤ 3g − 3. It follows from the definitions that if (Σ, f, S) is an ϵ-girded surface then (Σ, f) is a Bonahon surface. Note also that since no component of Σ is a sphere, condition (iv) of the above definition implies that the closed geodesics carried by (Σ, f, S) are non-trivial; in particular the components of S are homotopically non-trivial in Σ.

7.2. Proposition. Let ϵ be a Margulis number for M, and let (Σ, f, S) be an ϵ/3-girded surface in M. Then (Σ, f, S) is ϵ/3-incompressible.

Proof. Let γ : S^{1} → Σ be any homotopically non-trivial PG closed curve of length < ϵ/3. If γ(S^{1}) is disjoint from S, then by 7.1(iv), f ∘ γ is homotopically non-trivial in M.

Now suppose that γ(S^{1}) meets a component C of S. Then f(γ(S^{1})) meets the set f(C), which by 7.1(iii) is the support of a closed geodesic of length ϵ/3 in M. In particular f(γ(S^{1})) meets M_{(0,ϵ/3]}. But since γ has length < ϵ/3 and f is distance-decreasing, f(γ(S^{1})) has diameter < ϵ/3 and hence f(γ(S^{1})) ⊂ nbhd_{ϵ/3}(M_{(0,ϵ/3]}) ⊂ M_{(0,ϵ)}. Since ϵ is a Margulis number for M, the component T of M_{(0,ϵ)} containing f(γ(S^{1})) is a Margulis tube, and the core of T is the only closed geodesic meeting T . Hence if S′ ⊃ C denotes the union of all components of S that meet γ(S^{1}), then f(S′ ∪ γ(S^{1})) ⊂ T .

Let Z denote the connected set S′ ∪γ(S^{1}). Since Z is a finite union of geodesic arcs, it is a subcomplex of Σ in some triangulation. Hence there is a connected neighborhood U of Z in Σ such that Z carries π_{1}(U). We may take U to be a compact 2-manifold-with-boundary contained in f^{−1}(T ). Now we let U′ denote the 2-manifold-with-boundary obtained by adjoining to U all disk components of Σ − U, and we let U′′ denote the complement in U′ of a regular neighborhood of S′ in int U′. Then π_{1}(U′) is carried by Z ⊂ f^{−1}(T ), and hence (f | U′)_{#}(π_{1}(U′)) ⊂ π_{1}(M) is cyclic. On the other hand, it follows from 7.1(iv) and the construction of U′′ that f induces an injection from the fundamental group of each component of U′′ into π_{1}(M). Hence each component of U′′ has cyclic fundamental group and must therefore be an annulus.

This implies that U′ has Euler characteristic zero. Since Σ has genus > 1, U′ must be an annulus. Since C is a homotopically non-trivial simple closed curve in U′, it is a topological core of U′. But γ is a homotopically non-trivial closed curve in U′ and is therefore homotopic in U′ to a power of C. As f | C is a non-trivial closed geodesic, and as the complete hyperbolic manifold M necessarily has torsion-free fundamental group, we conclude that f ∘ γ is homotopically non-trivial as required. □

7.3. Proposition. Let M be a hyperbolic 3-manifold and let ϵ be a Margulis number for M. Let (Σ, f, S) be an (ϵ/3)-girded surface of total genus g in M. Let α_{1},…, α_{k} be the distinct geodesics carried by (Σ, f, S), and let T_{i} denote the component of M_{(0,ϵ]} containing α_{i}. Then f(Σ) ⊂ nbhd_{D}(T_{1} ∪ ⋯ ∪ T_{k}), where D = D(g, ϵ/3) in the notation of Proposition 6.4.

Proof. In the metric defined by the PH structure of Σ induced by f, we have by Proposition 6.4 that the diameter of Σ modulo Σ_{(0,ϵ/3]} is less than D = D(g, ϵ/3).

Thus Σ = nbhd_{D}(Σ_{(0,ϵ/3]}). Since f is distance-decreasing it follows that f(Σ) ⊂ nbhd_{D}(f(Σ_{(0,ϵ/3]})). In order to prove the proposition, it therefore suffices to show that f(Σ_{(0,ϵ/3]}) ⊂ T_{1} ∪ ⋯ ∪ T_{k}.

Let x be any point of Σ_{(0,ϵ/3]}. Then there is a closed curve γ : S^{1} → Σ which passes through x, is homotopically non-trivial in Σ and has length < ϵ/3. If γ(S^{1}) intersects S then x is in the ϵ/3-neighborhood of S; since f is distance-decreasing it follows that f(x) ∈ nbhd_{ϵ/3}(α_{i}) for some i. Since the closed geodesic α_{i} has length < ϵ/3 it is clear that nbhd_{ϵ/3}(α_{i}) is contained in M_{(0,ϵ]} and hence in T_{i}; thus f(x) ∈ T_{i} in this case.

Finally, suppose that γ(S^{1}) is disjoint from S. Then γ is contained in some component R of Σ−S. Since γ is homotopically non-trivial in Σ, the graph G = γ(S^{1}) is non-contractible in R (i.e. the inclusion G → R is not homotopic to a constant). Hence G contains a simple closed curve C′ which is non-contractible in R. By condition (ii) in the definition of an ϵ-girded surface, R is homeomorphic to a sphere with three punctures, and hence C′ is parallel in Σ to a component C of S. Since γ has length < ϵ/3, the curve C′ has length < ϵ/3, and x lies in the ϵ/3-neighborhood of C′. Hence there is a loop γ′ based at x which is freely homotopic to the simple closed curve C and has length < ϵ. Now f ∘ γ′ has length < ϵ and is freely homotopic to α_{i}. Hence f(γ′(S^{1})) ⊂ T_{i}. In particular f(x) ∈ T_{i}. □

7.4. Girded manifolds. An ϵ-girded surface is said to enclose a set if its underlying hyperbolically simplicial surface is null-homologous and encloses the set. Let M be a complete hyperbolic 3-manifold. We shall say that M is girded if there exist an integer g and a point P_{0} ∈ M such that for every ϵ > 0 there is an ϵ-girded surface of total genus at most g which encloses P_{0}. A Kleinian group Γ will be termed girded if the hyperbolic 3-manifold M(Γ) is girded.

Proposition. If a complete hyperbolic 3-manifold M is girded then every positive superharmonic function on M is constant.

Proof. Let M be a complete hyperbolic 3-manifold and let ϵ be a Margulis constant for M.

We will show that every compact set in M is enclosed by an ϵ/3- girded surface of total genus at most g. Any ϵ/3-girded surface is ϵ/3-incompressible by Proposition 7.2 and is by definition a Bonahon surface. The proposition then follows immediately from Proposition 6.8.

Let K be a compact set in M. By enlarging K if necessary we may assume that K is connected and that P_{0} ∈ K. Set D = D(g, ϵ/3), in the notation of Proposition 6.4. Since nbhd_{D}(K) is compact there are only finitely many components of M_{(0,ϵ]} which meet nbhd_{D}(K). Each of these is a tube around a geodesic of length less than ϵ. Let ϵ′ < ϵ/3 be a positive number which is less than the length of the core geodesic of any of the components of M_{(0,ϵ]} which meets nbhd_{D}(K). Thus if α is a geodesic of length less than ϵ′ then the component T of M_{(0,ϵ]} which contains α is disjoint from nbhd_{D}(K); equivalently, nbhd_{D}(T ) is disjoint from K.

Now it follows from Proposition 7.3 that any ϵ′-girded surface is disjoint from K. Let (Σ, f, S) be an ϵ′-girded surface of total genus at most g that encloses the point P_{0}. Since the set of points of M which are enclosed by (Σ, f, S) is open and closed in M − f(Σ) and since K is connected and contains P_{0} it follows that (Σ, f, S) encloses K. An ϵ′-girded surface is also ϵ/3-girded. Thus we have constructed an ϵ/3-girded surface of total genus at most g that encloses K, as desired. □

§8. Girded groups are dense

8.1. We shall let F denote the abstract free group on two generators x and y. We fix an identification of the group of orientation-preserving isometries of H^{3} with PSL_{2}(C). The representations of F in PSL_{2}(C) are in 1-1 correspondence with points of the complex affine variety V = PSL_{2}(C) × PSL_{2}(C); the representation ρ = ρ_{ξ,η} corresponding to a point (ξ, η) of V is defined by ρ(x) = ξ, ρ(y) = η. We shall always understand V to be endowed with the complex topology. We shall denote by 𝔇 the set of all points (ξ, η) ∈ V such that ρ_{ξ,η} is a faithful representation with discrete image, i.e. such that the group ⟨ξ, η⟩ generated by ξ and η is a Kleinian group which is free of rank 2. According to [Ch], 𝔇 is a closed subset of V. We shall denote by 𝔊𝔉 the set of all points (ξ, η) ∈ 𝔇 such that ⟨ξ, η⟩ is a geometrically finite group (1.7) without parabolic elements. It follows from [Mar, Theorem 8.1] that 𝔊𝔉 is an open subset of V. We shall denote by 𝔅 the frontier of 𝔊𝔉 in V; thus 𝔅 = \overline{𝔊𝔉} − 𝔊𝔉.

8.2. The purpose of this section is to prove the following result.

Theorem. There is a dense G_{δ}-set ℭ ⊂ 𝔅 such that for every (ξ, η) ∈ ℭ, the Kleinian group ⟨ξ, η⟩ is girded.

8.3. Note that for any (ξ, η) ∈ 𝔅, the hyperbolic 3-manifold M(⟨ξ, η⟩) = H^{3}/⟨ξ, η⟩ satisfies H_{2}(M(⟨ξ, η⟩); Z/2) ≅ H_{2}(F; Z/2) = 0 since F is free. Thus every hyperbolically simplicial surface in M(⟨ξ, η⟩) is null-homologous. In order to prove that ⟨ξ, η⟩ is girded for a given (ξ, η) ∈ 𝔅, it suffices to show that for every ϵ > 0, every compact subset of M(⟨ξ, η⟩) is enclosed by some connected ϵ-girded surface of genus 2.

8.4. Since the details of the of the proof of Theorem 8.2 are rather involved, we begin with a rough sketch of the argument. There are two main issues. First one must produce, for a generic point (ξ, η) ∈ 𝔅 and an arbitrary ϵ > 0, a connected ϵ-girded surface of genus 2 in the hyperbolic 3-manifold M = M(⟨ξ, η⟩). Second one must choose this girded surface to enclose a prescribed point in M.

To deal with the first issue we observe that an ϵ-girded surface of genus 2 determines three closed geodesics of length < ϵ in M; since these geodesics are defined by simple closed curves in the domain surface, their homotopy classes are of a restricted type. Conversely, if one can realize a suitable triple of homotopy classes by a triple of geodesics of length < ϵ, one can construct an ϵ-girded genus 2 surface in M. That such a triple of geodesics exists for a generic (ξ, η) will be deduced from a recent result due to Curt McMullen, asserting the density of “maximal cusps” in 𝔅 (see Theorem 8.9 below).

To deal with the second issue, we choose our ϵ-girded surfaces to be equivariant with respect to suitable involutions of M and of the domain surface. We then apply Proposition 6.10 to conclude that our girded surfaces enclose appropriately defined “reference points” in M. The existence of the appropriate involution of M follows from the results presented in 1.8–1.15. This stage of the argument depends strongly on the hypothesis that F is a 2-generator group.

8.5. For the rest of the section we fix, arbitrarily, a base point w_{0} ∈ H^{3}. We shall let p_{ξ,η} denote the covering projection from H^{3} to M(⟨ξ, η⟩).

For any (ξ, η) ∈ 𝔇 the group ⟨ξ, η⟩ is free of rank 2. In particular ⟨ξ, η⟩ is not solvable, and hence ξ and η have no common fixed point. It follows by 1.8 that each (ξ, η) ∈ 𝔇 determines a line ℓ(ξ, η) in H^{3}. We shall write τ_{ξ,η} = τ_{ℓ(ξ,η)}; by 1.8, the involution τ = τ_{ξ,η} satisfies τξτ = ξ^{−1} and τητ = η^{−1}. Since τ_{ξ,η} normalizes ⟨ξ, η⟩, it induces an involution T_{ξ,η} of M(⟨ξ, η⟩).

We shall denote by F̂ the group defined by the presentation

⟨x, y, t | t^{2} = 1, txt = x^{−1}, tyt = y^{−1}⟩.

We may regard F̂ as a semi-direct product of the free group F = ⟨x, y⟩ by the cyclic group ⟨t | t^{2} = 1⟩. Thus F may be identified with an index-2 subgroup of F̂. On the other hand, if we rewrite the presentation for F̂ in terms of the generators t_{1} = t, t_{2} = tx and t_{3} = ty, we obtain ⟨t_{1}, t_{2}, t_{3} | t_{1}^{2} = t_{2}^{2} = t_{3}^{2} = 1⟩, so that F̂ is a free product of three groups of order 2. It follows (for example by the Kurosh subgroup theorem [K, p. 34]) that every element of finite order in F̂ is conjugate to one of the t_{j}.

For every point (ξ, η) ∈ 𝔇 the representation ρ_{ξ,η} : F → PSL_{2}(C) extends to a representation ρ̂ = ρ̂_{ξ,η} : F → PSL_{2}(C) defined by ρ̂(x) = ξ, ρ̂(y) = η, ρ̂(t) = τ_{ξ,η}. The index of ρ_{ξ,η}(F) = ⟨ξ, η⟩ in F̂_{ξ,η} = ρ̂_{ξ,η}(F̂) = ⟨ξ, η, τ_{ξ,η}⟩ is exactly 2; indeed, the index is at most 2 since F̂ has index 2 in F, and we cannot have F_{ξ,η} = F̂_{ξ,η} since F_{ξ,η} is free and F̂_{ξ,η} contains an element of order 2. It follows that ρ̂ is a faithful representation with discrete image.

Thus F̂_{ξ,η} is a Kleinian group having exactly three conjugacy classes of elements of finite order. Hence the set L(ξ, η) of fixed points of elements of F̂_{ξ,η} is the union of a locally finite disjoint family of lines in H^{3}, each of which is the fixed point set of a unique element of F̂_{ξ,η}; and this family consists of exactly three F̂_{ξ,η}-orbits of lines. Hence Fix T_{ξ,η} = p_{ξ,η}(L(ξ, η)) is a disjoint union of three geodesics in M(⟨ξ, η⟩).

Explicitly, the three conjugacy classes of elements of finite order in F̂_{ξ,η} are represented by the elements τ_{ξ,η,j} = ρ̂_{ξ,η}(t_{j}) for j = 1, 2, 3. We have τ_{ξ,η,1} = τ_{ξ,η}, τ_{ξ,η,2} = τ_{ξ,η}ξ and τ_{ξ,η,3} = τ_{ξ,η}η. We set ℓ_{j}(ξ, η) = Fix τ_{ξ,η,j}. We have ℓ_{1}(ξ, η) = ℓ(ξ, η). The components of Fix T_{ξ,η} are p_{ξ,η}(ℓ_{j}(ξ, η)) for j = 1, 2, 3. We shall write L_{j}(ξ, η) = p_{ξ,η}(ℓ_{j}(ξ, η)).

8.6. Proposition. For any (ξ, η) ∈ 𝔇 and for j = 1, 2, 3, the line ℓ_{j}(ξ, η) is mapped homeomorphically onto L_{j}(ξ, η) by p_{ξ,η}.

Proof. Since L_{j}(ξ, η) is a component of Fix T_{ξ,η}, it is a closed subset of M(⟨ξ, η⟩). Hence if the lemma were false, L_{j}(ξ, η) would be a closed geodesic, and ℓ_{j}(ξ, η) would be the axis of a loxodromic element γ of F. Let us set ℓ = ℓ_{j}(ξ, η) and τ = τ_{ξ,η,j}. Then τ fixes ℓ (pointwise) and hence so does the commutator [τ, γ] ∈ F̂. Since F̂ is discrete it follows that [τ, γ] has finite order. But the commutator subgroup of F̂ ≅ Z_{2} ∗ Z_{2} ∗ Z_{2} is torsion-free, and hence γ must commute with τ. But the centralizer of τ in F̂ has order 2 (for example by the Kurosh subgroup theorem) and γ has infinite order; this is a contradiction. □

8.7. We let P̃_{j}(ξ, η) denote the point of ℓ_{j}(ξ, η) which is closest to the base point w_{0}, and we set P_{j}(ξ, η) = p_{ξ,η}(P̃(ξ, η)) ∈ L_{j}(ξ, η). The points P_{j}(ξ, η), j = 1, 2, 3, will be called the reference points of M(⟨ξ, η⟩). Thus M(⟨ξ, η⟩) contains exactly three reference points, and there is one reference point in each component of Fix T_{ξ,η}.

8.8. We shall let 𝔄 denote the set of all points (ξ, η) ∈ 𝔇 such that the group ⟨ξ, η⟩ contains no parabolics. The following result, which will be needed for the proof of Theorem 8.2, was announced in [Ber1]. A version of Bers’s λ-lemma can be used to complete the proof: see [McM, Corollary 1.5].

Proposition. The set 𝔄 ∩ 𝔅 is a dense G_{δ} in 𝔅. □

8.9. The following result, due to Curt McMullen, is the deepest ingredient in the proof of Theorem 8.2. McMullen has informed us that the theorem below is proved by a generalization of the techniques of [McM].

Theorem. There is a dense subset 𝔈 of 𝔅 such that for every (ξ, η) ∈ 𝔈, the free Kleinian group ⟨ξ, η⟩ is geometrically finite and has three distinct conjugacy classes of cuspidal subgroups (1.5).

8.10. The proof of Theorem 8.2 will be proved by combining the results stated above with the following result.

Proposition. Let (α, β) be a point of 𝔇 such that ⟨α, β⟩ is geometrically finite and contains three distinct conjugacy classes of cuspidal subgroups. Let ϵ be any positive number. Then (α, β) has a neighborhood U = U_{ϵ}(α, β) in 𝔅 such that for every (ξ, η) ∈ 𝔄 ∩ U, there is a connected ϵ-girded surface of genus 2 in M(⟨ξ, η⟩) which encloses at least one of the reference points of M(⟨ξ, η⟩).

8.11. Before proving Proposition 8.10, let us give the

Proof that Proposition 8.10 implies Theorem 8.2. According to Theorem 8.9 there is a dense set 𝔈 ⊂ 𝔅 such that for every (α, β) ∈ 𝔈, the group ⟨α, β⟩ is geometrically finite and has three non-conjugate maximal cuspidal subgroups. For each ϵ > 0 and each (α, β) ∈ 𝔈, Proposition 8.10 gives a neighborhood U = U_{ϵ}(α, β) of (α, β) in 𝔅. For every ϵ > 0 the set

W_{ϵ} = ⋃_{(α,β)∈𝔈} U_{ϵ}(α, β)

is open and dense in 𝔅. On the other hand, by Proposition 8.8, 𝔄 ∩ 𝔅 is a dense G_{δ} in 𝔅. Hence the set

ℭ = 𝔄 ∩ ⋂_{0<n∈Z} W_{1/n}

is a dense G_{δ} in 𝔅.

To complete the proof it suffices to show that for every (ξ, η) ∈ ℭ the group ⟨ξ, η⟩ is girded. If (ξ, η) ∈ ℭ then for every positive integer n there is a point (α_{n}, β_{n}) ∈ 𝔈 such that (ξ, η) ∈ 𝔄∩U_{1/n}(α_{n}, β_{n}). According to the defining property of U_{ϵ}(α_{n}, β_{n}) (see Proposition 8.10) this means that for every n > 0 there is a reference point z_{j_{n}} of M(⟨ξ, η⟩) which is enclosed by a connected (1/n)-girded surface of genus 2 in M(⟨ξ, η⟩).

Since M(⟨ξ, η⟩) has only three reference points, it follows that there is a single reference point z of M which is enclosed by a connected (1/n)-girded surface of genus 2 for arbitrarily large values of n; it follows at once that M(⟨ξ, η⟩) is girded. □

8.12. The rest of this section is devoted to the proof of Proposition 8.10. We suppose that we are given a point (α, β) of 𝔇 such that the free Kleinian group ⟨α, β⟩ is geometrically finite and contains three distinct conjugacy classes of cuspidal subgroups. We set ρ̂_{0} = ρ̂_{α,β}, p_{0} = p_{α,β} : H^{3} → M(⟨α, β⟩), and T_{0} = T_{α,β}.

8.13. Lemma. For each component L of Fix T_{0} we have N(⟨α, β⟩)∩L ≠ ∅. Furthermore, there is a constant η > 0 such that for every ϵ < η and every component L of Fix T_{0} we have N_{[ϵ,∞)}(⟨α, β⟩) ∩ L ≠ ∅.

Proof. To prove the first assertion we choose a component ℓ of p_{0}^{−1}(L). Then ℓ is a component of L(α, β), and hence by 8.5 we have ℓ = Fix τ for some order-2 element τ of ρ̂_{0}(F̂). Recall from 1.4 that the set Ñ(⟨α, β⟩) = p_{0}^{−1}(N(⟨α, β⟩)) = nbhd_{1}(hull(Λ_{⟨α,β⟩})) is ⟨α, β⟩-invariant and convex. Since τ normalizes ⟨α, β⟩ (see 8.5), Ñ(⟨α, β⟩) is τ-invariant.

It follows that for any point z ∈ Ñ(⟨α, β⟩), the midpoint of the line segment joining z to τz lies in N(⟨α, β⟩) ∩ L, and the first assertion is proved.

It follows from the first assertion that for any component L of Fix T_{0} there is a number η_{L} > 0 such that N_{[η_{L},∞)}(⟨α, β⟩) ∩ L ≠ ∅. The second assertion now follows if we set η = min_{L} η_{L}, where L ranges over the three components of Fix T_{0}. □

8.14. Let ϵ_{0} be a 3-dimensional Margulis constant such that conclusions (i) and (ii) of Proposition 1.7 hold with Γ = ⟨α, β⟩. According to 1.7 we may take ϵ_{0} to be arbitrarily small. In particular we may suppose ϵ_{0} to be chosen so that N_{[ϵ_{0},∞)}(⟨α, β⟩) ∩ L ≠ ∅ for every component L of Fix T_{0}.

By conclusion (i) of 1.7, J = N_{[ϵ_{0},∞)}(⟨α, β⟩) is a compact, orientable, irreducible 3manifold-with-boundary. Furthermore, J is a deformation retract of M(⟨α, β⟩); in particular J is connected and π_{1}(J) is a free group of rank 2. According to [He, Theorem 5.2], any compact, connected, orientable, irreducible 3-manifold-with-boundary whose fundamental group is free of rank 2 is a genus-2 handlebody.

We set Σ = ∂J. Then Σ is a closed orientable surface of genus 2. Since the fundamental group of a handlebody is carried by its boundary, and since J is a deformation retract of M(⟨α, β⟩), the inclusion homomorphism π_{1}(Σ) → π_{1}(M(⟨α, β⟩)) is surjective.

Hence Σ̃ = p_{0}^{−1}(Σ) is a connected covering surface of Σ. Clearly Σ̃ is invariant under the action of F̂ on H^{3} defined by the representation ρ_{0}. Restricting this action, we obtain a natural action of F̂ on Σ̃.

Since the group ⟨α, β⟩ contains three conjugacy classes of cuspidal subgroups, the set M_{(0,ϵ_{0}]}^{c}(⟨α, β⟩) has three components. Each component of M_{(0,ϵ_{0}]}^{c}(⟨α, β⟩) has the form C/Π, where Π is a cuspidal subgroup of Γ and C is a horoball stabilized by Π. Since ⟨α, β⟩ is free, each of these groups Π is cyclic. Thus M_{(0,ϵ_{0}]}(⟨α, β⟩) has the homotopy type of a disjoint union of three circles. Hence if we set A = N_{{ϵ_{0}}}(⟨α, β⟩), it follows from conclusion (ii) of 1.7 that A is a disjoint union of three annuli in Σ, all homotopically non-trivial in M(⟨α, β⟩).

We set Ã = p_{0}^{−1}(A) ⊂ Σ̃ ⊂ H^{3}. Each component Ã of Ã is contained in a unique horoball in H^{3}, which we denote by C(Ã). We denote the base point of C(Ã) in S_{∞} by ζ(Ã). The stabilizer of Ã in ⟨ξ, η⟩ is an infinite cyclic group consisting of parabolic elements, and this group leaves C(Ã) invariant and fixes ζ(Ã).

Since the three annuli comprising A correspond to different conjugacy classes of (cuspidal) maximal infinite cyclic subgroups of Γ, their cores are non-contractible and (freely) non-homotopic simple closed curves in M(⟨α, β⟩). In particular they are non-contractible and non-homotopic in Σ. Hence each component of \overline{Σ − A} is a planar surface with three boundary curves, and at least two of the components of A are non-separating annuli in Σ.

It is apparent from the definitions of J and A that they are invariant under any selfisometry of M(⟨α, β⟩). Thus in particular J is invariant under T_{0}, so that T_{0} induces an involution T_{Σ} of Σ; and A is invariant under T_{Σ}. (We shall sharpen this assertion in Lemma 8.16 below.)

Observe also that since T_{0} is induced by an orientation-preserving isometry of H^{3}, it is itself orientation-preserving. It follows that T_{0} | J is orientation-preserving, and therefore so is T_{Σ}.

8.15. Lemma. For every component L of Fix T_{0} we have card(L ∩ Σ) = 2.

Proof. Recall from 8.14 that we chose ϵ_{0} in such a way that J ∩L = N_{[ϵ_{0},∞)}(⟨α, β⟩)∩L ≠ ∅ for every component L of Fix T_{0}. Since J ∩ L is a union of components of the fixed set of the involution T_{0} |J of the compact 3-manifold-with-boundary J, it is a properly embedded 1-manifold-with-boundary in J; it has no closed components since L is homeomorphic to R by 8.6. Thus for each component L of Fix T_{0}, the set Σ ∩ L = ∂(J ∩ L) has non-zero, finite, even cardinality. On the other hand, the sum of the cardinalities of Σ ∩ L as L ranges over the three components of Fix T_{0} is equal to card Fix T_{Σ}, the number of fixed points of an orientation-preserving involution of a closed connected surface, which by the Hurwitz branching formula is at most 6. Hence we must have card(Σ ∩ L) = 2 for each component L of Fix T_{0}. □

8.16. Lemma. Each component of A is invariant under T_{Σ}.

Proof. If the assertion is false then two components A_{1} and A_{2} are interchanged by T_{Σ}.

Since A has at least two non-separating components, A_{1} and A_{2} cannot both separate Σ; since they are interchanged by the homeomorphism T_{Σ}, they are both non-separating. Hence Σ′ = \overline{Σ − (A_{1} ∪ A_{2})} is connected. Since neither A_{1} nor A_{2} can contain a fixed point of T_{Σ}, and since T_{Σ} has six fixed points by Lemma 8.15, T_{Σ′} = T_{Σ} | Σ′ is an orientation-preserving involution, with six fixed points, of a connected surface having nonempty boundary and Euler characteristic −2. It then follows from the Hurwitz branching formula that Σ′/T_{Σ′} is a connected surface having non-empty boundary and Euler characteristic 2. But no such surface exists. □

8.17. It follows from Lemma 8.16 that for each component A of A we may choose a T_{Σ}- invariant core curve S_{A}. Having made such a choice, we set S = ⋃_{A} S_{A}, where A ranges over the components of A. It follows from 8.14 that the S_{A} are homotopically non-trivial curves in M(⟨α, β⟩), that at least two of the S_{A} are non-separating curves in Σ, and that each component of Σ − S is a three-punctured sphere. We set S = p_{0}^{−1}(S) ⊂ Σ̃.

Lemma. For each component R of Σ − S, the inclusion homomorphism from π_{1}(R) to π_{1}(M(⟨α, β⟩)) is injective.

Proof. Since J is a deformation retract of M(⟨α, β⟩) by 8.14, it suffices to show that the inclusion homomorphism π_{1}(R) → π_{1}(J) is injective. If it is not, it follows from the loop theorem [He, Theorem 4.2] that some non-trivial element of ker(π_{1}(R) → π_{1}(J)) is represented by a simple closed curve in R. Since R is a three-punctured sphere, every simple closed curve in R is parallel to a component of ∂R ⊂ S. But every component of S is homotopically non-trivial in M(⟨α, β⟩) and hence in J. □

8.18. Lemma. Every non-separating component of S meets Fix T_{Σ} in two points. If C is a separating component of S then C is disjoint from Fix T_{Σ}, and each component of Σ − C is T_{Σ}-invariant.

Proof. It follows from Lemmas 8.15 and 8.16 that T_{Σ} is an orientation-preserving involution of Σ with six fixed points and that each component of S is invariant under T_{Σ}. The Hurwitz branching formula implies that the orbit surface Ψ = Σ/T_{Σ} is a sphere. Let q : Σ → Ψ denote the quotient map.

If C is any component of S, either C contains two fixed points of T_{Σ} and T_{Σ} reverses the orientation of C, or C contains no fixed point of T_{Σ} and T_{Σ} preserves the orientation of C. If C separates Σ then since the other components of S are T_{Σ}-invariant, each component of Σ − C must be T_{Σ}-invariant; hence T_{Σ} preserves the orientation of C. Conversely, if T_{Σ} preserves the orientation of C then q(C) is a simple closed curve and therefore separates Ψ. Hence C = q^{−1}(q(C)) separates Σ. □

8.19. Lemma. There is a T_{Σ}-invariant virtual triangulation of Σ such that (i) skel_{0}(Σ) ⊂ S, (ii) S ∪ Fix T_{Σ} ⊂ skel_{1}(Σ), and (iii) no virtual 2-simplex has more than one edge supported in S.

Proof. Let us label the components of S as C_{1}, C_{2} and C_{3}, where C_{1} and C_{2} are nonseparating curves in Σ, and C_{3} may or may not separate Σ.

Consider the case in which C_{3} is non-separating. Then Σ − S has two components Y_{1} and Y_{2}, each of which is the interior of a compact planar sub-surface of Σ with three boundary curves. Since the C_{i} are non-separating, T_{Σ} reverses the orientation of C_{i}; hence T_{Σ} interchanges Y_{1} and Y_{2}. By Lemma 8.18, each C_{i} contains two fixed points of T_{Σ}. Thus Fix T_{Σ} is a subset of ∂\overline{Y_{1}} containing exactly two points of each component of ∂\overline{Y_{1}}. It follows (see the figure) that \overline{Y_{1}} has a virtual triangulation with Fix T_{Σ} as its 0-skeleton, and in which no virtual 2-simplex has more than one edge supported in ∂\overline{Y_{1}}.

The induced triangulation of ∂\overline{Y_{1}} is automatically T_{Σ}-invariant, and hence the virtual triangulation of \overline{Y_{1}} extends to a unique T_{Σ}-invariant virtual triangulation of Σ; this extended triangulation clearly has the required properties.

Now consider the case in which C_{3} separates Σ. For i = 1, 2, let J_{i} denote the component of Σ − C_{3} containing C_{i}. As in the proof of Lemma 8.18 we observe that Ψ = Σ/T_{Σ} is a sphere and we let q : Σ → Ψ denote the quotient map. Since J_{i} is T_{Σ}-invariant by Lemma 8.16, J_{i} is a 2-fold branched cover of the disk q(J_{i}) ⊂ Ψ. But J_{i} is a genus-1 surface with connected boundary, and it therefore follows from the Hurwitz branching formula that J_{i} contains exactly three fixed points of T_{Σ}. Hence there is a unique fixed point u_{i} of T_{Σ} lying in J_{i} − C_{i}. Now let v be any point of C_{3}. For i = 1, 2, there is a T_{Σ}-invariant, properly embedded arc α_{i} in J_{i} which has v and T_{Σ}(v) as endpoints and u_{i} as an interior point: we may in fact define α_{i} = q^{−1}(a_{i}), where a_{i} is an arc in q(J_{i}) joining q(v) to q(u_{i}). If α_{i} separates J_{i}, then since T_{Σ} clearly reverses the orientation of α_{i}, it must interchange the two components of J_{i} − α_{i}; this implies that J_{i} has even genus, a contradiction. Hence α_{i} is a non-separating arc in J_{i}.

It follows that C′_{3} = α_{1} ∪ α_{2} is a non-separating simple closed curve in Σ. Furthermore, since C′_{3} intersects each J_{i} in an arc, it cannot be parallel to C_{1} or C_{2}. Hence if we set S′ = C_{1} ∪C_{2} ∪C′_{3}, then Σ−S′ has two components Y_{1} and Y_{2}, each of which is the interior of a compact planar sub-surface of Σ with three boundary curves. Since the components of S′ are non-separating, T_{Σ} reverses the orientation of each component of S′; hence T_{Σ} interchanges Y_{1} and Y_{2}. The set

K = ((Fix T_{Σ}) ∩ (C_{1} ∪ C_{2})) ∪ \{v, T_{Σ}(v)\} ⊂ ∂\overline{Y_{1}}

contains exactly two points of each component of ∂\overline{Y_{1}}. It follows that \overline{Y_{1}} has a virtual triangulation with K as its 0-skeleton, and in which no virtual 2-simplex has more than one edge supported in ∂\overline{Y_{1}}. Note also that the arc a = C_{3} ∩ \overline{Y_{1}} is non-boundary parallel in \overline{Y_{1}}; for otherwise a would be parallel to one of the α_{i}, and this α_{i} would separate J_{i}, a contradiction. It follows that we may choose the virtual triangulation of \overline{Y_{1}} in such a way that a is contained in the 1-skeleton. As in the previous case, the induced triangulation of ∂\overline{Y_{1}} is T_{Σ}-invariant so that the virtual triangulation of \overline{Y_{1}} extends to a unique T_{Σ}-invariant virtual triangulation of Σ; and again the extended triangulation clearly has the required properties. □

8.20. For the rest of this section we fix a virtual triangulation of Σ having the properties stated in Lemma 8.19. Since the virtual triangulation of Σ is T_{Σ}-invariant, it induces an F̂-invariant virtual triangulation of Σ̃. We shall always understand Σ̃ to be equipped with this virtual triangulation.

Let (ξ, η) be any point in 𝔇, and let f̃ be a map of Σ̃ into H^{3}. We shall say that f̃ is (ξ, η)-good if (i) f̃ is hyperbolically simplicial, (ii) f̃| S̃ is geodesic and (iii) f̃ is F̂- equivariant with respect to the standard action of F̂ on Σ̃ and the action on H^{3} associated to (ξ, η). Any (ξ, η)-good map f̃ : Σ̃ → H^{3} induces a hyperbolically simplicial map f : Σ → M(⟨ξ, η⟩), and f | S is geodesic. The F̂-equivariance of f̃ implies that f satisfies T_{ξ,η} ∘ f = f ∘ T_{Σ}. If f̃ is non-degenerate (as a hyperbolically simplicial map) then so is f.

8.21. Lemma. For every (ξ, η) ∈ 𝔄 there exists a (ξ, η)-good map f̃ = f̃_{ξ,η}: Σ̃ → H^{3}.

Proof. It follows from Lemma 8.19 that skel_{0}(Σ̃) ⊂ S̃ ⊂ skel_{1}(Σ̃). The first step in the construction of f̃ is to construct a geodesic, F̂-equivariant map f̃_{S}: S̃ → H^{3}.

Let us denote the components of S by C_{1}, C_{2} and C_{3}. For i = 1, 2, 3, we choose a component S̃_{i} of p_{0}^{−1}(C_{i}). We let G_{i} and Ĝ_{i} denote the stabilizers of S̃_{i} in F and F̂ respectively. Then G_{i} is infinite cyclic, and it has index 2 in Ĝ_{i} since C_{i} is T_{Σ}-invariant. Hence Ĝ_{i} is either infinite cyclic or infinite dihedral. If γ_{i} denotes a generator of G_{i} then γ_{i} has infinite order in F, and γ_{i} is not parabolic since (ξ, η) ∈ 𝔄. Hence γ_{i} has a unique axis Z_{i} ⊂ H^{3}. Since Ĝ_{i} normalizes G_{i}, the line Z_{i} is Ĝ_{i}-invariant.

Up to equivariant homeomorphism, each of the groups Z and D_{∞} has a unique properly discontinuous action on R. Hence there is a Ĝ_{i}-equivariant homeomorphism f̃_{i} : S̃_{i} → Z_{i}. Using the maps f̃_{i} we define f̃_{S} as follows. Given any point u ∈ S̃ there exist an index i ∈ {1, 2, 3} and an element γ of F̂ such that γ · u ∈ S̃_{i}; the index i is uniquely determined by u, but the element γ is not. We wish to define f̃_{S}(u) = γ^{−1} · f̃_{i}(γ · u). If γ_{1} is another element of F̂ such that γ_{1}·u ∈ S̃_{i}, then δ = γ_{1}γ^{−1} belongs to Ĝ_{i}, and by the Ĝ_{i}-equivariance of f̃_{i} we have

γ_{1}^{−1} · f̃_{i}(γ_{1} · u) = γ^{−1}δ^{−1} · f̃_{i}(δγ · u) = γ^{−1} · f̃_{i}(γ · u).

Thus f_{S} is well-defined. It follows readily from the definition that it is F̂-equivariant. The next step is to extend f̃_{S} to an F̂-equivariant map of skel_{1}(Σ̃) into H^{3}. Let φ : Δ → Σ̃ be a virtual 1-simplex in the virtual triangulation of Σ̃ which is not carried by S̃. Let f_{φ} : Δ → H^{3} denote the map which agrees with f̃_{S} ∘ (φ | ∂Δ) on the boundary of Δ, and is geodesic in the sense that if we identify Δ with an interval in R by any affine homeomorphism then f_{φ} becomes a geodesic path (parametrized proportionally to arc length). (In particular, if f̃_{S} ∘ (φ | ∂Δ) maps the two boundary points of Δ to the same point, then f_{φ} is constant.) There is a unique map f̃^{(1)} : skel_{1}(Σ̃) → H^{3} such that f̃^{(1)} | S̃ = f̃_{S} and such that for every virtual 1-simplex φ in the virtual triangulation of Σ̃ which is not carried by S̃ we have f̃^{(1)} ∘ φ = f_{φ}. Since f̃_{S} is F̂-equivariant, so is f̃^{(1)}.

To extend f̃^{(1)} to Σ̂ we may use the following construction. Let φ : Δ → Σ̃ be any 2-simplex. Let m denote the baycenter of Δ. Let δ denote the convex hull of f̃^{(1)}(φ(Δ^{(0)}), where Δ^{(0)} denotes the set of vertices of Δ. Then δ is either a hyperbolic triangle, a hyperbolic line segment or a point. If δ is a hyperbolic triangle we let μ denote its centroid [F, p. 125]. If δ is a line segment we define μ to be its midpoint, and if δ consists of a single point we define μ to be this point. Now we define a map f_{φ} : Δ → δ by stipulating that f_{φ} | ∂Δ = f^{(1)} ∘ (φ | ∂Δ), that f(m) = μ, and that for each affine segment σ ⊂ Δ joining m to a point in ∂Δ, the map f_{φ} | σ be geodesic (in the above sense). Now there is a unique map f̃: Σ̃ → H^{3} such that f̃| skel_{1}(Σ) = f̃^{(1)} and such that for every virtual 2-simplex φ in the virtual triangulation of Σ̃ we have f̃^{(1)} ∘ φ = f_{φ}. Since f̃^{(1)} is F̂-equivariant, so is f̃. It is clear from the construction that f̃ is hyperbolically simplicial and that f̃| S̃ is geodesic. □

8.22 Lemma. Let (ξ^{i}, η^{i})_{i≥1} be a sequence of points in 𝔄 which converges in 𝔇 to (α, β). For each i ≥ 1 let f̃^{i}: Σ̃ → H^{3} be a (ξ^{i}, η^{i})-good map. Let C̃ be any component of S̃ and let Ã denote the unique component of Ã containing C̃. Then for any point z ∈ S̃ we have lim_{i→∞} f̃^{i}(z) = ζ(Ã) in \overline{H^{3}}. Furthermore, if z and z′ are distinct points of C̃ and U is any neighborhood of ζ(Ã) in \overline{H}, then we have f̃^{i}(z) ≠ f̃^{i}(z′) for every i, and for sufficiently large i the line joining f̃^{i}(z) to f̃^{i}(z′) is contained in U.

(The definition of ζ(Ã) was given in 8.14.)

Proof. Let FC̃ denote the stabilizer of C̃ under the action of F on H^{3} defined by ρ_{0}. By 8.14 and 8.17, FC̃ is cyclic and is generated by a parabolic element h, and ρ_{0}(h) fixes ζ = ζ(Ã). For each i ≥ 1, since f̃^{i} is a (ξ, η)-good map, f̃^{i}(C̃) is a geodesic which is invariant under ρ_{ξ^{i},η^{i}}(h). Thus f̃^{i}(C̃) is the axis of the isometry ρ_{ξ}i_{,η}i(h) (which is loxodromic since (ξ^{i}, η^{i}) ∈ 𝔇). As i → ∞ we have ρ_{ξ_{,}^{i}η^{i}}(h) → ρ_{0}(h). Hence if U is any

neighborhood of the fixed point ζ of ρ_{0}(h) in \overline{H}, the endpoints of the line f̃^{i}(C̃) lie in U for all sufficiently large i. Hence for large i we have f̃^{i}(C̃) ⊂ U. This implies the first assertion of the lemma.

To prove the second assertion we observe that since the geodesic map f̃^{i} | C̃ is FC̃- equivariant, it is non-constant and is therefore a homeomorphism. This shows that for every i we have f̃^{i}(z) ≠ f̃^{i}(z′), and that f̃^{i}(C̃) is the unique line joining f̃^{i}(z) to f̃^{i}(z′).

Since we have shown that f̃^{i}(C̃) ⊂ U for large i, the second assertion follows. □

8.23. Lemma. Let (ξ^{i}, η^{i})_{i≥1} be a sequence of points in 𝔄 which converges in 𝔇 to (α, β). For each i ≥ 1 let f̃^{i}: Σ → H^{3} be a (ξ^{i}, η^{i})-good map, and let f^{i} : Σ → M(⟨ξ^{i}, η^{i}⟩) be the map induced by f̃^{i}. Let ϵ be any positive number. Then for all sufficiently large i the triple (Σ, f^{i}, S) is an ϵ-girded surface.

Proof. We must show that for sufficiently large i the hyperbolically simplicial surface (Σ, f^{i}) is non-degenerate and conditions (i)—(iv) of the definition 7.1 of an ϵ-girded surface hold. Conditions (i) and (ii) are properties of the virtually triangulated surface Σ and the 1-manifold S ⊂ Σ that have been established in 8.17 and 8.19. We claim that condition (iv) holds for every i. Indeed, the isomorphisms ρ_{0} : F → ⟨α, β⟩ and ρ_{(ξ^{i},η^{i})} : F → ⟨ξ^{i}, η^{i}⟩ give rise to an isomorphism (well-defined modulo inner automorphisms) between ⟨α, β⟩ = π_{1}(M(⟨α, β⟩)) and ⟨ξ^{i}, η^{i}⟩ = π_{1}(M(⟨ξ^{i}, η^{i}⟩)). If we identify π_{1}(M(⟨α, β⟩)) and π_{1}(M(⟨ξ^{i}, η^{i}⟩)) via this isomorphism, then the equivariance condition in the definition of a good map implies that f_{#} : π_{1}(Σ) → π_{1}(M(⟨ξ^{i}, η⟩^{i})) is identified (modulo inner automorphisms) with the inclusion homomorphism π_{1}(Σ) → π_{1}(M(⟨α, β⟩). Our claim therefore follows from Lemma 8.17.

Next we show that condition (iii) in the definition of an ϵ-girded surface holds for large i. It follows from the definition of a (ξ, η)-good map that for every i the map f^{i} | S is geodesic. Now let S be any component of S. Then S is a core curve of some component A of A. Choose any component Ã of p_{0}^{−1}(A), and let FÃ denote the stabilizer of Ã in F. Then FÃ is also the stabilizer of the component C̃ of S̃ contained in Ã. By 8.14, FÃ is cyclic and is generated by an element h such that ρ_{0}(h) is parabolic. Since (ξ^{i}, η^{i}) → (α, β) as i → ∞, it follows that trace ρ_{ξ^{i},η^{i}}(h) → trace ρ_{0}(h) = ±2. Hence the translation length of ρ_{ξ^{i},η^{i}}(h) approaches 0 as i → ∞. But the closed geodesic defined by the map f^{i} | S is C̃/FÃ. Hence the length of f^{i} | S approaches 0 as i → ∞, and is in particular < ϵ for large i.

It remains to show that (Σ, f^{i}) is non-degenerate for large i. Recall from the definition (6.5) that for every virtual simplex φ : Δ → Σ and every i, the map f_{φ}^{i} = f^{i} ∘ φ admits a continuous lift f̃_{φ}^{i} : Δ → H^{n} such that f̃_{φ}^{i}(Δ) is a hyperbolic simplex and either (i) dim f̃_{φ}^{i}(Δ) < dim Δ or (ii) f̃_{φ}^{i} maps Δ homeomorphically onto f̃_{φ}^{i}(Δ). To say that (Σ, f^{i}) is non-degenerate for a given i means that (ii) holds for every virtual simplex Δ of Σ. It is clearly enough to show this for the case of a virtual 2-simplex. Since Δ has only finitely many simplices, it suffices to show that for each virtual 2-simplex φ there is an integer i_{φ} such that (ii) holds for every i ≥ i_{φ}.

Let φ̃ be a virtual 2-simplex in Σ̃ that covers φ. Let v_{j} (j = 1, 2, 3) denote the vertices of φ̃. If (i) holds for a given i then the points f̃^{i}(v_{1}), f̃^{i}(v_{2}) and f̃^{i}(v_{3}) are collinear in H^{3}. Thus we need only show that the f̃^{i}(v_{j}) are non-collinear for large i. According to conclusion (i) of Lemma 8.19, each v_{j} lies in some component of S̃. It follows from conclusion (iii) of Lemma 8.19 that at most two of the v_{j} can lie in any given component of S̃.

Consider first the case in which all the v_{j} lie in three distinct components of S̃. Let C̃_{j} denote the component of S̃ containing v_{j}, and let Ã_{j} denote the component of Ã containing C̃_{j}. Then the horospheres C(Ã_{1}), C(Ã_{2}) and C(Ã_{3}) are distinct components of p_{0}^{−1}(M_{{ϵ_{0}}}(⟨α, β⟩)). This implies that the base points ζ(Ã_{j}) of the C(Ã_{j}) are three mutually distinct points in S_{∞}. But by Lemma 8.22 we have for j = 1, 2, 3 that f̃^{i}(v_{j}) → ζ(Ã_{j}) as i → ∞. It follows that the three points f̃^{i}(v_{j}), j = 1, 2, 3, are non-collinear when i is large.

There remains the case in which two of the v_{j}, say v_{1} and v_{2}, lie in the same component C̃ of S̃, and the third, say v_{3}, lies in a component C̃′ ≠ C̃ of S̃. Let Ã and Ã′ denote the components of Ã containing C̃ and C̃′ respectively. Then ζ(Ã) and ζ(Ã′) are distinct points of S_{∞} and hence have disjoint neighborhoods U and U′ in \overline{H}. By the first assertion of Lemma 8.22 we have f̃^{i}(v_{3}) ∈ U′ for large i. On the other hand, the second assertion of Lemma 8.22 implies that f̃^{i}(v_{1}) ≠ f̃^{i}(v_{2}) for all i, and that for large i the line joining f̃^{i}(v_{1}) to f̃^{i}(v_{2}) is contained in U. Since U ∩ U′ = ∅ it follows that the f̃^{i}(v_{j}) are non-collinear for large i. □

8.24. Lemma. Let (ξ, η) be any point of 𝔄, let f̃ be any (ξ, η)-good map and let f : Σ → M(⟨ξ, η⟩) be the map induced by f̃. Let L be any component of Fix T_{ξ,η}. Then Fix T_{Σ} ⊂ Σ contains exactly two points, say P_{+} and P_{−}, which are mapped into L by f. Furthermore, the arc in L with endpoints f(P_{+}) and f(P_{−}) is enclosed by the singular surface (Σ, f).

Proof. To prove the first assertion we use the notation of 8.5: thus the components of Fix T_{ξ,η} are L_{j}(ξ, η) for j = 1, 2, 3. We must show that for each j there are exactly two of the six fixed points of Fix T_{Σ} that are mapped by f into L_{j}(ξ, η). The equivariance condition in the definition of a good map shows that f̃(ℓ_{j}(α, β)) = f̃(Fix τ_{α,β,j}) ⊂ Fix τ_{ξ,η,j} = ℓ_{j}(ξ, η). Hence f(L_{j}(α, β)) ⊂ L_{j}(ξ, η). In particular f maps L_{j}(α, β) ∩ Σ into L_{j}(ξ, η). But L_{j}(α, β) ∩ Σ consists of fixed points of T_{0}, and has cardinality 2 by Lemma 8.15. Thus the first assertion is proved. Since T_{(ξ,η)} ∘ f = f ∘ T_{Σ} by 8.20, the second assertion follows from the first by virtue of Proposition 6.10. □

Proof of Proposition 8.2. It suffices to prove that if ϵ is a positive number, and (ξ^{i}, η^{i}) is a sequence of points in 𝔄 which converges to (α, β), then for every sufficiently large i there exists an ϵ-girded surface in M(⟨ξ, η⟩) which encloses at least one of the reference points of M(⟨ξ, η⟩). According to Lemma 8.21, for each i there exists a (ξ^{i}, η^{i})-good map f̃^{i} : Σ̃ → H^{3}. Let f^{i} : Σ → M(⟨α, β⟩) denote the map induced by f̃^{i}. By Lemma 8.23, (Σ, f^{i}, S) is an ϵ-girded surface for every sufficiently large i. Thus in order to prove the proposition it suffices to show that for every sufficiently large i the hyperbolically simplicial surface (Σ, f^{i}) encloses some reference point of M(⟨ξ^{i}, η^{i}⟩).

Recall from 8.17 that S has at least two non-separating components; by Lemma 8.18, each such component contains at least two fixed points of T_{Σ}. Thus card(S ∩ Fix T_{Σ}) ≥ 4. On the other hand, we have Fix T_{Σ} ⊂ Fix T_{0}, and T_{0} has three components. Hence there must be two points P_{+}, P_{−} ∈ S ∩ Fix T_{Σ} that lie in the same component L of Fix T_{0}. By Lemma 8.15 we must have L∩Σ̃ = {P_{+}, P_{−}}. In the notation of 8.5, we have L = L_{j_{0}}(ξ, η) for some j_{0} ∈ {1, 2, 3}. Let us set ℓ = ℓ_{j_{0}}(α, β), so that L = p_{0}(ℓ). Then Σ̃ meets ℓ in two points P̃_{+} ∈ p_{0}^{−1}(P_{+}) and P̃_{−} ∈ p_{0}^{−1}(P_{+}).

By 8.5, ℓ is the fixed line of τ_{α,β,j_{0}} = ρ̂_{0}(t_{j_{0}}). For each i ≥ 1, we let ℓ^{i} denote the fixed line of τ_{ξ^{i},η^{i},j_{0}} = ρ_{ξ^{i},η^{i}}(t_{j_{0}}), and set L^{i} = p_{ξ^{i},η^{i}}(ℓ^{i}). Since f̃^{i} is F̂-equivariant for every i, we have f̃^{i}(P̃_{−}), f̃^{i}(P̃_{+}^{i}) ∈ ℓ^{i}. Hence f^{i}(P_{−}), f^{i}(P_{+}) ∈ L^{i}. If α^{i} denotes the arc in L^{i} with endpoints f^{i}(P_{−}), f^{i}(P_{+}), it follows from Lemma 8.24 that (Σ, f^{i}) encloses α^{i}. We shall complete the proof of the proposition by showing that α^{i} contains a reference point of M(⟨ξ^{i}, η^{i}⟩) for sufficiently large i.

We have P̃_{−}, P̃_{+} ∈ S̃ ⊂ Ã. Let C̃_{+} ⊂ Ã_{+} and C̃_{−} ⊂ Ã_{−} denote the components of S̃ and Ã containing P̃_{+} and P̃_{−} respectively. Since Ã = p_{0}^{−1}(N_{{ϵ_{0}}}(⟨α, β⟩)) is invariant under the action of F̂ on H^{3} defined by the representation ρ̂_{0}, and since the involution τ_{α,β,j_{0}} fixes P̃_{−} and P̃_{+}, the sets Ã_{−} and Ã_{+} are invariant under τ_{α,β,j_{0}}. Hence ζ(Ã_{±}), which is the base point of the unique horoball C(Ã_{±}) containing Ã_{±}, is fixed by τ_{α,β,j_{0}}. It follows that each of the points ζ(Ã_{−}) and ζ(Ã_{+}) is an endpoint of the line ℓ. We shall show that ζ(Ã_{−}) ≠ ζ(Ã_{+}), so that ζ(Ã_{−}) and ζ(Ã_{+}) are the two endpoints of ℓ.

Since the horospheres C(Ã_{−}) and C(Ã_{+}) contain the endpoints of ℓ, they meet ℓ orthogonally; hence each of these horospheres meets ℓ in at most a single point. Since P_{−} ≠ P_{+}, it follows that C(Ã_{−}) ≠ C(Ã_{+}). Thus the horospheres C(Ã_{−}) and C(Ã_{+}) are distinct components of p_{0}^{−1}(M_{{ϵ_{0}}}(⟨α, β⟩)). This implies that the base points ζ(Ã_{−}) and ζ(Ã_{+}) of C(Ã_{−}) and C(Ã_{+}) are distinct. Thus ζ(Ã_{−}) and ζ(Ã_{+}) are indeed the two endpoints of ℓ. For each i ≥ 1, let P̃^{i} denote the point of ℓ^{i} that is closest to w_{0}. By definition, P^{i} = p_{ξ}i_{,η}i(P̃^{i}) is a reference point of M(⟨ξ_{i}, η_{i}⟩). Let P̃ denote the point of ℓ that is closest to w_{0}. By 1.9 we have ℓ^{i} → ℓ as i → ∞, so that P̃^{i} → P̃. On the other hand, since P̃_{+} ∈ S̃_{+} and P̃_{−} ∈ S̃_{−}, Lemma 8.22 gives that f̃^{i}(P̃_{+}) → ζ(Ã_{+}) and f̃^{i}(P̃_{−}) → ζ(Ã_{−}) in \overline{H} as i → ∞. Thus we have a sequence of triples of points (f̃^{i}(P̃_{−}), P̃^{i}, f̃^{i}(P̃_{+}))_{i≥1}; the points in the i-th triple all lie in the line ℓ_{i}; and in the limit triple (ζ(Ã_{−}), P̃, ζ(Ã_{+})), the point P̃ lies on the line ℓ ⊂ H^{3}, whereas ζ(Ã_{−}) and ζ(Ã_{+}) are the endpoints of ℓ. It follows that for sufficiently large i the point P̃^{i} lies between f̃^{i}(P̃_{−}) and f̃^{i}(P̃_{+})).

By Proposition 8.6 this means that for sufficiently large i the point P^{i} lies between f^{i}(P_{−}) and f^{i}(P_{+})), so that P^{i} ∈ α^{i} for large i, as required. □

§9. Estimates for geometrically finite groups

9.1. The following theorem is the main result of this paper.

Theorem. Let ξ and η be non-commuting isometries of H^{3}. Suppose that ξ and η generate a torsion-free discrete group which is topologically tame, is not co-compact and contains no parabolics. Then for any z ∈ H^{3} we have

max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3

The following two results are used in the proof of Theorem 9.1, and put the statement in perspective.

9.2. Proposition. Let ξ and η be non-commuting isometries of H^{3} that generate a torsion-free discrete group which is not co-compact and contains no parabolics. Then ⟨ξ, η⟩ is a free group on the generators ξ and η.

Proof. Set Γ = ⟨ξ, η⟩. It follows from the hypotheses that the space M = H^{3}/Γ is a complete, non-compact, orientable hyperbolic 3-manifold without cusps. In particular, the universal cover of M is homeomorphic to R^{3}, so that M is irreducible; and Γ = π_{1}(M) has no free abelian subgroup of rank 2.

It follows from [BaS, Proposition 3 and Theorem A] that if M is any non-compact, irreducible, orientable 3-manifold such that π_{1}(M) has no free abelian subgroup of rank 2, then π_{1}(M) is 2-free in the sense that each of its 2-generator subgroups is free of some rank ≤ 2. (This fact is also implicit in [Tuc] and [JaS, Sect. VI.4].) Thus Γ is 2-free; since Γ is itself generated by two non-commuting elements, it is a free group of rank 2. Hence by [K, p. 59], Γ is in fact free on the given generators ξ and η. □

9.3. Proposition. Suppose that ξ and η satisfy the hypotheses of Theorem 9.1. Then either the group ⟨ξ, η⟩ is geometrically finite, or we have M(⟨ξ, η⟩) = N(⟨ξ, η⟩).

Proof. Since Γ = ⟨ξ, η⟩ is not co-compact, it follows from Proposition 9.2 that Γ is a free group. In particular, H_{2}(Γ; Z) = 0.

According to 1.4, the set N = N(Γ) ⊂ M(Γ) is a 3-manifold-with-boundary and a deformation retract of M(Γ). Thus N is aspherical and π_{1}(N) ≅ Γ, and therefore H_{2}(N; Z) ≅ H_{2}(Γ; Z) = 0. If ∂N has a compact component, it follows that N is compact (and has connected boundary); by the definition (1.7) this implies that Γ is geometrically finite. On the other hand, if ∂N = ∅ then N = M(Γ). Hence to prove the proposition it suffices to show that ∂N is compact.

Let Ω denote the set of discontinuity of Γ. The surface Ω/Γ inherits a conformal structure from Ω. According to the Ahlfors finiteness theorem [A], Ω/Γ is a hyperbolic Riemann surface which has finite area in the Poincaré metric. Since Γ contains no parabolics, it also follows from Ahlfors’s theorem that Ω/Γ has no cusps. Hence Ω/Γ is compact. We shall prove that ∂N is compact by showing that there is a continuous map of Ω/Γ onto ∂N. For this purpose it suffices to produce a Γ-equivariant surjection Θ : Ω → ∂Ñ, where Ñ = nbhd_{1}(hull(Λ_{Γ})) is the pre-image of N in H^{3}. Recall from 1.4 that Ñ is convex.

For any point ζ ∈ Ω let C_{ζ} denote the smallest horoball based at ζ which meets the set Ñ. Then C_{ζ} meets Ñ in a single point; for if z and z′ were distinct points of Ñ ∩ C_{ζ} then, since a horoball is convex, the open line segment in H^{3} joining z to z′ would be contained in Ñ ∩ int C_{ζ}, and the minimality of C_{ζ} would be contradicted. We define Θ(ζ) to be the point of intersection of C_{ζ} with Ñ. It is clear that Θ(ζ) ∈ ∂Ñ ∩ ∂C_{ζ} for each ζ ∈ Ω.

The function Θ: Ω → ∂Ñ is obviously Γ-equivariant. To see that it is surjective, we observe that since Ñ is closed and convex, every point z ∈ ∂Ñ lies on the boundary of a hyperbolic half-space Q ⊂ H^{3} with Ñ ∩ int Q = ∅. (One deduces the existence of such a “supporting half-space” for a closed convex set in H^{3} from the corresponding fact in Euclidean space by using the projective model as in 1.2.) Now let C denote the unique horoball which is contained in Q and tangent to the plane ∂Q, and let ζ denote the base point of C. It is clear that ζ ∈ Ω and that Θ(ζ) = z. Thus Θ is surjective.

It remains to show that Θ is continuous. For this purpose we identify H^{3} with the upper half-space model R^{2} × (0, ∞) by an isometry that maps some point of Λ_{Γ} to ∞. This guarantees that every vertical ray in R^{2} ×(0, ∞) whose lowest point is in N is entirely contained in N. Now let (ζ_{i}) be a convergent sequence in Ω with ζ_{∞} = lim ζ_{i} ∈ Ω. For 1 ≤ i ≤ ∞ we set z_{i} = Θ(ζ_{i}) and C_{i} = C_{ζ_{i}}. The closure \overline{C_{i}} of C_{i} in R^{2} × [0, ∞) is a Euclidean ball of some radius r_{i}, tangent to R^{2} × \{0\} at ζ_{i}, and \overline{C_{i}} ∩ N = \{z_{i}\}. Since the vertical ray with lowest point z_{i} is contained in N, the point z_{i} must lie on the closed upper hemisphere U_{i} of the sphere ∂\overline{C_{i}}.

We claim that r_{i} → r_{∞} as i → ∞. If this is false then after passing to a subsequence we can assume that r_{i} → r for some r ≠ r_{∞} in [0, ∞]. If r > r_{∞} then for large enough i we have U_{∞} ⊂ int C_{i}, and in particular z_{∞} ∈ int C_{i}. This is impossible since \overline{C_{i}} ∩ N = \{z_{i}\} ⊂ ∂\overline{C_{i}}. Similarly, if r < r_{∞} then for large i we have U_{i} ⊂ int C_{∞}, and again we obtain a contradiction. This proves the claim.

Now suppose that (z_{i}) does not converge to z_{∞}. Then after passing to a subsequence we may assume that z_{i} → w for some point z ∈ N − \{z_{∞}\}. Since z_{i} ∈ \overline{C_{i}} for 1 ≤ i < ∞, and since ζ_{i} → ζ and r_{i} → r, we have z ∈ \overline{C_{∞}}. This is impossible since \overline{C_{∞}} ∩N = \{z_{∞}\}. Hence we must have z_{i} → z_{∞}. This shows that Θ is continuous and completes the proof. □

9.4. By Proposition 9.3, the proof of Theorem 9.1 breaks up into two disjoint cases, according to whether the group Γ = ⟨ξ, η⟩—which is free by 9.2—satisfies N(Γ) = M(Γ) or is geometrically finite. The first case is covered by the following result.

Lemma. Let ξ and η be non-commuting isometries of H^{3} that freely generate a rank-2 free Kleinian group Γ which is topologically tame, contains no parabolics and satisfies N(Γ) = M(Γ). Then for any z ∈ H^{3} we have

max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3.

Proof. According to Proposition 6.9 every positive superharmonic function on the manifold M(Γ) is constant. Thus by Proposition 3.9, every Γ-invariant conformal density is a constant multiple of the area density. By Proposition 5.2 we therefore have max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3 for every z ∈ H^{3}. □

9.5. The following result covers the remaining case of Theorem 9.1.

Lemma. Let ξ and η be non-commuting elements in PSL_{2}(C) that freely generate a rank-2 free Kleinian group which is geometrically finite and contains no parabolics. Then for any z ∈ H^{3} we have

max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3.

Proof. We use the notation of Section 8. Let a point z ∈ H^{3} be given. Let us define a continuous non-negative-valued function f on V by

f(ξ, η) = max ( dist(z, ξ · z), dist(z, η · z)).

The proposition asserts that f(ξ, η) ≥ log 3 for all (ξ, η) ∈ 𝔊𝔉; or, equivalently, that f(ξ, η) ≥ log 3 for all (ξ, η) ∈ \overline{𝔊𝔉}.

Every local minimum of the function f on V occurs at a point (ξ, η) such that either ξ or η fixes z. Indeed, if (ξ, η) is a point of V where ξ(z) ≠ z and η(z) ≠ z, then there exist a sequence (z_{i}) of points on the open hyperbolic line segment between z and ξ(z) which converges to ξ(z), and a sequence (w_{i}) of points on the open segment between z and η(z) which converges to η(z). If we choose sequences (α_{i}) and (β_{i}) of hyperbolic isometries converging to the identity, such that α_{i}(z) = z_{i} and β_{i}(z) = w_{i}, then (α_{i}ξ, β_{i}η) converges to (ξ, η) as i → ∞, and f(α_{i}ξ, β_{i}η) < f(ξ, η) for all i. Hence (ξ, η) is not a local minimum of f.

For any point (ξ, η) ∈ 𝔇, the elements ξ and η of PSL_{2}(C) have infinite order in the discrete group ⟨ξ, η⟩, and hence have no fixed points in H^{3}. It follows that f has no local minimum on the open subset 𝔊𝔉 of V.

On the other hand, f is a proper function on V; indeed, for any B ≥ 0, the set f^{−1}([0, B]) consists of all pairs (ξ, η) such that ξ and η each move the point z a distance at most B, and this set is clearly compact. Hence f must take a minimum value on the closed set \overline{𝔊𝔉}. Since it has no local minimum on the open set 𝔊𝔉, it must take its minimum value on 𝔅. Hence we need only show that f(ξ, η) ≥ log 3 for all (ξ, η) ∈ 𝔅.

Let ℭ ⊂ 𝔅 denote the dense G_{δ} given by Theorem 8.2. Since f is continuous and ℭ is dense in 𝔅, we need only prove that f(ξ, η) ≥ log 3 for all (ξ, η) ∈ ℭ. But for any (ξ, η) ∈ ℭ, the Kleinian group ⟨ξ, η⟩ is girded; hence by Proposition 7.4, every positive superharmonic function on the manifold M(Γ) is constant. Thus according to Proposition 3.9, every ⟨ξ, η⟩-invariant conformal density is a constant multiple of the area density. By Proposition 5.2 we therefore have

f(ξ, η) = max ( dist(z, ξ · z), dist(z, η · z)) ≥ log 3.

□

Proof of Theorem 9.1. The theorem follows immediately from Propositions 9.2 and 9.3 and Lemmas 9.4 and 9.5. □

§10. Applications to closed hyperbolic manifolds

10.1. The following result is an immediate consequence of Theorem 9.1 and the definition of a Margulis number given in Section 1.

Proposition. Let M = H^{3}/Γ be a closed, orientable hyperbolic 3-manifold such that every 2-generator subgroup of Γ = π_{1}(M) has non-empty set of discontinuity in S_{∞}. Then log 3 is a Margulis number for M. □

10.2. In this section we shall prove the

Proposition. Let M be a closed, orientable hyperbolic 3-manifold such that the rank of H_{1}(M; Q) is at least 3. Then every 2-generator subgroup of π_{1}(M) is non-co-compact and topologically tame.

10.3. By combining Propositions 10.1 and 10.2 one immediately obtains the following result.

Theorem. Let M be a closed, orientable hyperbolic 3-manifold such that the rank of H_{1}(M; Q) is at least 3. Then log 3 is a Margulis number for M. □

10.4. Corollary. Let M be a closed, orientable hyperbolic 3-manifold such that the rank of H_{1}(M; Q) is at least 3. Then M contains an isometric copy of a ball of radius \frac{1}{2} log 3 in H^{3}. Hence the volume of M is greater than 0.92.

Proof. By Theorem 10.3, the number ϵ = log 3 is a Margulis number for M. Thus M_{(0,ϵ]} is a disjoint union of tubes in M, and in particular M_{(ϵ,∞]} ≠ ∅. It follows from the definition of M_{(ϵ,∞]} that the (ϵ/2)-neighborhood of any point of M_{(ϵ,∞]} is isometric to a ball of radius ϵ/2 in H^{3}. This proves the first assertion.

To deduce the second assertion from the first, we use an observation due to Meyerhoff [Me]. Using a result due to Böröczky [Bör], Meyerhoff shows that if a hyperbolic manifold M contains a hyperbolic ball B(r) of a given radius r then the volume of M is at least 2 Vol(T)/(6α − π), where α = \frac{1}{2} arcsec (2 + sech 2r), and T is a regular tetrahedron in H^{3} whose edges have length 2r. (This is equivalent to the inequality Vol(M) ≥ Vol(B(r))/d(r) given on p. 277 of [Me]: the factor sinh(2r) − 2r in the definition of d(r) on p. 275 of [Me] is equal by [F, p. 209] to \frac{1}{2} Vol(B(r)).) Taking r = \frac{1}{2} log 3, and using Lobachevsky functions as in [F, Sect. IX.4] to compute the volume of a regular hyperbolic tetrahedron with edges of length log 3, one obtains the estimate Vol M > 0.92. □

Proof of Proposition 10.2. Let F be any 2-generator subgroup of Γ = π_{1}(M). Since H_{1}(M; Q) has rank at least 3, there is an epimorphism η : π_{1}(M) → Z such that η(F) = {0}. In particular, F has infinite index in Γ and is therefore non-co-compact. It remains to show that F is topologically tame.

It follows from [He, Lemma 6.5] that η is induced by some map ϕ : M → S^{1} such that ϕ is transverse to 1 ∈ S^{1} and the surface ϕ^{−1}(1) is incompressible; that is, for each component Σ of ϕ^{−1}(1), we have π_{1}(Σ) ≠ {1}, and the inclusion homomorphism π_{1}(Σ) → π_{1}(M) is injective.

Let (M̃_{1}, p_{1}) denote the covering space of M corresponding to the subgroup Γ_{1} = ker η of Γ. Then ϕ is covered by a map ϕ̃ from M̃_{1} to the universal cover R of S^{1}. Let G be a graph whose fundamental group is free of rank 2, and let ψ : G → M be a basepoint-preserving map such that ψ_{#}(π_{1}(G)) = F. Since F ⊂ ker η, the map ψ admits a lift ψ̃ : G → M̃_{1}. Since G is compact, we have ψ̃(G) ⊂ ϕ̃^{−1}([−D, D]) for some positive integer D. The component of ϕ̃^{−1}([−D, D]) containing ψ̃(G) is a compact 3-manifold-with-boundary K_{1} with ∂K_{1} ⊂ p_{1}^{−1}(ϕ^{−1}(1)); hence ∂K_{1} is incompressible in M̃_{1}. We define Γ_{2} ≤ Γ_{1} ≤ Γ to be the image of the inclusion homomorphism π_{1}(K_{1}) → π_{1}(M̃_{1}), so that F ≤ Γ_{2}.

Let (M̃_{2}, p_{2}) denote the covering space of M̃_{1} corresponding to the subgroup Γ_{2}. Then M̃_{2} contains a submanifold K_{2} which is mapped homeomorphically onto K_{1} by p_{2}. Thus if we denote the boundary components of K_{2} by Σ̃^{1},…, Σ̃^{k}, each Σ̃^{j} is mapped homeomorphically by p_{2} onto a component Σ^{j} of ∂K. It is clear that K_{2} is a compact core of M̃_{2} in the sense that the inclusion homomorphism π_{1}(K_{2}) → π_{1}(M̃_{2}) is an isomorphism. It is also clear that the Σ̃^{j} are incompressible.

Thus M̃_{2} is a hyperbolic 3-manifold having a compact core whose boundary is incompressible. It follows from the main theorem of [Bo] that any such hyperbolic manifold is topologically tame. On the other hand, Γ_{2} is contained in Γ_{1} and therefore has infinite index in Γ; hence M̃_{2} has infinite volume. According to [Ca, Proposition 3.2], any covering space, with finitely generated fundamental group, of an infinite-volume topologically tame hyperbolic 3-manifold is itself topologically tame. Since F ≤ Γ_{2}, it follows that F is topologically tame, as required. □

A stronger version of Proposition 10.2 can be established: namely that under the same hypothesis, every 2-generator subgroup of π_{1}(M) is in fact geometrically finite. This can be proved by combining results of [Th. Chapter 9] and [ScS]; see also [Ca2].

References

[A] L. Ahlfors, Finitely generated Kleinian groups, Amer J. Math. 86 (1964), 413–429.

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