Free Kleinian groups and volumes of hyperbolic 3-manifolds

James W. Anderson^{∗}, Richard D. Canary^{†}, Marc Culler^{‡}, and Peter B. Shalen^{§}

January 15, 1996

1 Introduction

The central result of this paper, Theorem 6.1, gives a constraint that must be satisfied by the generators of any free, topologically tame Kleinian group without parabolic elements. The following result is case (a) of Theorem 6.1.

Main Theorem Let k ≥ 2 be an integer and let Φ be a purely loxodromic, topologically tame discrete subgroup of Isom_{+}(H^{3}) which is freely generated by elements ξ_{1},…, ξ_{k}. Let z be any point of H^{3} and set d_{i} = dist(z, ξ_{i} · z) for i = 1,…, k. Then we have

∑_{i=1}^{k} \frac{1}{1 + e^{d_{i}}} ≤ \frac{1}{2}.

In particular there is some i ∈ {1,…, k} such that d_{i} ≥ log(2k − 1).

The last sentence of the Main Theorem, in the case k = 2, is equivalent to the main theorem of [14]. While most of the work in proving this generalization involves the extension from rank 2 to higher ranks, the main conclusion above is strictly stronger than the main theorem of [14] even in the case k = 2 .

∗partially supported by an NSF-NATO postdoctoral fellowship

†partially supported by a Sloan Foundation Fellowship and an NSF grant

‡partially supported by an NSF grant

§partially supported by an NSF grant

§1. Introduction 2

Like the main result of [14], Theorem 6.1 has applications to the study of large classes of hyperbolic 3-manifolds. This is because many subgroups of the fundamental groups of such manifolds can be shown to be free by topological arguments. The constraints on these free subgroups impose quantitative geometric constraints on the shape of a hyperbolic manifold. As in [14] these can be applied to give volume estimates for hyperbolic 3-manifolds satisfying certain topological restrictions. The volume estimates obtained here, unlike those proved in [14], are strong enough to have qualitative consequences, as we shall explain below.

The following result is proved by combining the case k = 3 of the Main Theorem with the techniques of [15].

Corollary 9.2 Let N be a closed orientable hyperbolic 3-manifold. Suppose that the first betti number β_{1}(N) is at least 4, and that π_{1}(N) has no subgroup isomorphic to the fundamental group of a surface of genus 2. Then N contains a hyperbolic ball of radius \frac{1}{2} log 5. Hence the volume of N is greater than 3.08.

There is no reason to expect these estimates to be sharp. For instance, empirical evidence based on Weeks census [39] suggests that the conclusion of the corollary may hold under the hypothesis that β_{1}(N) is at least 2, with no assumption on the surface subgroups of π_{1}(N). However, the significance of our results lies elsewhere. The point is that these results imply that certain topological conditions on the manifold follow from an upper bound on the volume. More specifically, the volumes of hyperbolic 3-manifolds are known to form a well-ordered set of ordinal type ω^{ω}. If one lists the closed hyperbolic manifolds in ascending order of volume, the topological complexity of the manifolds tends to grow as one progresses through the list. We are interested in understanding this phenomenon in an explicit way.

The above result provides explicit information of this type. The volume of a cusped manifold is larger than that of any of its Dehn fillings, and is a limit point of the set of volumes of such fillings. There are 8 distinct volumes less than 3.08 among the volumes of orientable cusped manifolds in the Weeks census. Thus the result implies that each of the manifolds realizing the first 8ω volumes either has betti number at most 3 or has a fundamental group containing an isomorphic copy of a genus-2 surface group. (This conclusion is stated as Corollary 9.3.) It was not possible to deduce qualitative consequences of this sort in [14] because the lower bound of 0.92, obtained there for the volume of a closed hyperbolic 3-manifold of first betti number at least 3, is smaller than the least known volume of any hyperbolic 3-manifold.

Corollary 9.4 is similar to the above corollary but illustrates the applicability of our techniques to the geometric study of infinite-volume hyperbolic 3-manifolds. It asserts that a non-compact, topologically tame, orientable hyperbolic 3-manifold N without cusps always contains a hyperbolic ball of radius \frac{1}{2} log 5 unless π_{1}(N) either is a free group of rank 2 or contains an isomorphic copy of a genus-2 surface group.

Another application of Theorem 6.1 to non-compact finite-volume manifolds is the following result, which uses only the case k = 2 of the Main Theorem, but does not follow from the weaker form of the conclusion which appeared in [14].

Theorem 11.1 Let N = H^{3}/Γ be a non-compact hyperbolic 3-manifold. If N has betti number at least 4, then N has volume at least π.

Theorem 11.1 is deduced via Dehn surgery techniques from Proposition 10.1 and its Corollary 10.3, which are of independent interest. These results imply that if a hyperbolic 3-manifold satisfies certain topological restrictions, for example if its first betti number is at least 3, then there is a good lower bound for the radius of a tube about a short geodesic, from which one can deduce a lower bound for the volume of the manifold in terms of the length of a short geodesic. This lower bound approaches π as the length of the shortest geodesic tends to 0. Corollary 10.3 will be used in [13] as one ingredient in a proof of a new lower bound for the volume of a hyperbolic 3-manifold of betti number 3. This lower bound is greater than the smallest known volume of a hyperbolic 3-manifold, and therefore has the qualitative consequence that any smallest-volume hyperbolic 3-manifold has betti number at most 2.

The proof of the Main Theorem follows the same basic strategy as the proof of the main theorem of [14]. The Main Theorem is deduced from Theorem 6.1(d), which gives the same conclusion under somewhat different hypotheses. In 6.1(d), rather than assuming that the free Kleinian group Φ is topologically tame and has no parabolics, we assume that the manifold H^{3}/Φ admits no non-constant positive superharmonic functions. As in [14], the estimate is proved in this case by using a Banach-Tarski-style decomposition of the area measure based on a Patterson construction. The deduction of the Main Theorem from 6.1(d) is based on Theorem 5.2, which asserts that, in the variety of representations of a free group F_{k}, the boundary of the set CC(F_{k}) of convexcocompact discrete faithful representations contains a dense G_{δ} consisting of representations whose images are “analytically tame” Kleinian groups without parabolics. This was proved in [14] in the case k = 2.

By definition, a rank-k free Kleinian group Γ without parabolics is analytically tame if the convex core of H^{3}/Γ can be exhausted by a sequence of geometrically well behaved compact submanifolds (a more exact definition is given in Section 5). The case k = 2 of Theorem 5.2 was established in [14] by combining a theorem of McMullen’s [29] on the density of maximal cusps on the boundary of CC(F_{k}) with a special argument involving the canonical involution of a 2-generator Kleinian group. The arguments used in the proof of Theorem 5.2 make no use of the involution. This makes possible the generalization to arbitrary k, while also giving a new proof in the case k = 2. The ideas needed for the proof are developed in sections 2 through 5, and will be sketched here.

In Section 3 we prove a general fact, Proposition 3.2, about a sequence (ρ_{n}) of discrete faithful representations of a finitely generated, torsion-free, non-abelian group G which converges to a maximal cusp ω. (For our purposes a maximal cusp is a discrete faithful representation ω of G into PSL_{2}(C) such that ω(G) is geometrically finite and every boundary component of the convex core of H^{3}/ω(G) is a thrice-punctured sphere.) After passing to a subsequence one can assume that the Kleinian groups ρ_{n}(G) converge geometrically to a Kleinian group Γ̂, which necessarily contains ω(G) as a subgroup. Proposition 3.2 then asserts that the convex core of N = H^{3}/ω(G) embeds isometrically in H^{3}/Γ̂. To prove this, we use Proposition 2.7, which combines an algebraic characterization of how conjugates of ω(G) can intersect in the geometric limit (Lemma 2.4), and a description of the intersection of the limit sets of two topologically tame subgroups of a Kleinian group (Theorem 2.5).

In Section 4 we construct a large submanifold D of the convex core of N which is geometrically well-behaved in the sense that ∂D has bounded area and the radius-2 neighborhood of ∂D has bounded volume. We use Proposition 3.2 to show that if ρ is a discrete faithful representation near enough to ω, then H^{3}/ρ(G) contains a nearly isometric copy of D. This copy is itself geometrically well-behaved in the same sense.

In Section 5 we specialize to the case G = F_{k}. We show that if a discrete faithful representation ρ is well-approximated by infinitely many maximal cusps, then its associated quotient manifold contains infinitely many geometrically well-behaved submanifolds. In fact, we show that the resulting submanifolds exhaust the convex core of the quotient manifold and hence that the quotient manifold is analytically tame. We then apply McMullen’s theorem to prove that there is a dense G_{δ} in the boundary of CC(F_{k}) consisting of representations which can be well approximated by maximal cusps.

In the argument given in [14], the involution of a 2-generator Kleinian group is used not only in the deformation argument, but also in the calculation based on the decomposition of the area measure in the case where H^{3}/Φ supports no non-constant superharmonic functions. The absence of an involution in the kgenerator case is compensated for by a new argument based on the elementary inequality established in Lemma 6.2. This leads to the stronger conclusion of the main theorem in the case k = 2.

Section 6 is devoted to the proof of Theorem 6.1.

We have mentioned that the application of Theorem 6.1 to the geometry of hyperbolic manifolds depends on a criterion for subgroups of fundamental groups of such manifolds to be free. The first such criterion in the case of a 2-generator subgroup was proved in [19] and independently in [37]. A partial generalization to k-generator subgroups, applying only when the given manifold is closed, was given in [3]. In Section 7 of this paper we give a criterion that includes the above results as special cases and is adapted to the applications in this paper.

In Section 8 we introduce a generalization of the notion of a Margulis number. We say that a positive number λ is a k-Margulis number for a Kleinian group Γ if the following condition holds: if ξ_{1},…, ξ_{k} are elements of Γ and if there exists a point z ∈ H^{3} which is displaced less than λ by each ξ_{i} then the group ⟨ξ_{1},…, ξ_{k}⟩ can be generated by k − 1 abelian subgroups. In the case k = 2 the group ⟨ξ_{1}, ξ_{2}⟩ would have to be abelian; thus the notion of a 2-Margulis number coincides with that of a Margulis number as defined in [14] and [33]. This notion, and the related notion of a k-strong Margulis number proves useful for organizing the applications of the results of the earlier sections to the study of hyperbolic manifolds. The applications are presented in Sections 9, 10 and 11.

The pictures of limit sets of maximal cusps which appear in Section 3 were created by Yair Minsky, and were based on some earlier pictures by Chris Bishop. We are grateful to them for allowing us to use them here.

We thank Bryan Mosher for supplying a number of corrections to the manuscript.

§2. On the sphere at infinity 6

We close the introduction by mentioning a few notational conventions which are used throughout. We use H ≤ G to denote that H is a subgroup of G, and H < G to denote that H is a proper subgroup of G. The translate of a set X by a group element γ is denoted γ · X. Finally, we use dist(z, w) to denote the hyperbolic distance between points z and w in H^{3}.

2 On the sphere at infinity

In this section we introduce the notion of the geometric limit of a sequence of Kleinian groups. We will consider a convergent sequence of discrete faithful representations into PSL_{2}(C) whose images converge geometrically. In general, the geometric limit of the images contains the image of the limit as a subgroup. The results in this section characterize the intersection of two conjugates of this subgroup, and the intersection of their limit sets.

The group PSL_{2}(C) will be considered to act either by isometries on H^{3} or, via extension to the sphere at infinity, by Möbius transformations on the Riemann sphere \overline{C}. The action of a discrete subgroup Γ of PSL_{2}(C) partitions \overline{C} into two pieces, the domain of discontinuity Ω(Γ) and the limit set Λ(Γ). The domain of discontinuity is the largest open subset of \overline{C} on which Γ acts properly discontinuously. If Λ(Γ) contains two or fewer points, we say Γ is elementary. If Γ has an invariant circle in \overline{C} and preserves an orientation of the circle then we say that Γ is Fuchsian.

By a Kleinian group we will mean a discrete non-elementary subgroup Γ of PSL_{2}(C). We will say that a Kleinian group Γ is purely parabolic if every nontrivial element is parabolic, or purely loxodromic if every non-trivial element is loxodromic.

Given a finitely generated group G, let Hom(G, PSL_{2}(C)) denote the variety of representations of G into PSL_{2}(C). A choice of k elements which generate G determines a bijection from Hom(G, PSL_{2}(C)) to an algebraic subset of (PSL_{2}(C))^{k}. We give Hom(G, PSL_{2}(C)) the topology that makes this bijection a homeomorphism onto the algebraic set with its complex topology. This topology on Hom(G, PSL_{2}(C)) is independent of the choice of generators of G.

For the rest of this section, and throughout section 3, we will assume that G is a finitely generated, non-abelian, torsion-free group.

Let D(G) denote the subspace of Hom(G, PSL_{2}(C)) which consists of those representations which are injective and have discrete image. It is a fundamental result of Jørgensen’s [20] that D(G) is a closed subset of Hom(G, PSL_{2}(C)). The proof of Jørgensen’s result is based on an inequality for discrete subgroups of PSL_{2}(C). A second consequence of this inequality is the following lemma.

Lemma 2.1 (Lemma 3.6 in [21]) Let (ρ_{n}) be a convergent sequence in D(G). If (g_{n}) is a sequence of elements of G such that (ρ_{n}(g_{n})) converges to the identity, then there exists n_{0} such that g_{n} = 1 for n ≥ n_{0}.

2.1

A sequence of discrete subgroups (Γ_{n}) is said to converge geometrically to a Kleinian group Γ̂ if and only if

(1) for every γ ∈ Γ̂, there exist elements γ_{n} ∈ Γ_{n} such that the sequence (γ_{n}) converges to γ, and

(2) whenever (Γ_{n_{j}}) is a subsequence of (Γ_{n}) and γ_{n_{j}} ∈ Γ_{n_{j}} are elements such that the sequence (γ_{n_{j}}) converges to a Möbius transformation γ, we have γ ∈ Γ̂.

We call Γ̂ the geometric limit of (Γ_{n}).

The following basic fact is proved in Jørgensen-Marden [21].

Proposition 2.2 (Proposition 3.8 in [21]) Let (ρ_{n}) be a sequence of elements of D(G) converging to ρ. Then (ρ_{n}(G)) has a geometrically convergent subsequence. If Γ̂ is the geometric limit of any such subsequence, then ρ(G) ≤ Γ̂.

The following fact will also be used.

Lemma 2.3 Let (ρ_{n}) be a convergent sequence in D(G) such that (ρ_{n}(G)) converges geometrically to Γ̂. Then Γ̂ is torsion-free.

Proof of 2.3: Suppose that γ ∈ Γ̂ has finite order d. Let (g_{n}) be a sequence of elements of G such that ρ_{n}(g_{n}) converges to γ. Then ρ_{n}(g_{n}^{d}) converges to the identity. Hence by Lemma 2.1 we have g_{n}^{d} = 1 for large n. Since G is torsion-free, we have g_{n} = 1 for large n and therefore γ = 1.

2.3

The following lemma characterizes the intersection of two conjugates of ρ(G) in the geometric limit Γ̂.

Lemma 2.4 Let (ρ_{n}) be a sequence of elements of D(G) converging to ρ. Suppose that the groups ρ_{n}(G) converge geometrically to Γ̂. Then ρ(G)∩γρ(G)γ^{−1} is a (possibly trivial) purely parabolic group for each γ ∈ Γ̂ − ρ(G).

Proof of 2.4: Suppose that ρ(a) is a nontrivial element of ρ(G)∩γρ(G)γ^{−1} for some γ ∈ Γ̂ − ρ(G). We will show that ρ(a) is parabolic.

Assume to the contrary that ρ(a) is loxodromic. We may write ρ(a) = γρ(b)γ^{−1} for some b ∈ G. Choose g_{n} ∈ G such that (ρ_{n}(g_{n})) converges to γ. We then have that (ρ_{n}(g_{n}bg_{n}^{−1})) converges to ρ(a), and hence that (ρ_{n}(a^{−1}g_{n}bg_{n}^{−1})) converges to 1. It follows from Lemma 2.1 that there exists an integer n_{0} such that a = g_{n}bg_{n}^{−1} for all n ≥ n_{0}. Hence g_{n_{0}}^{−1}g_{n} is contained in the centralizer of b for all n ≥ n_{0}. Applying ρ_{n} and passing to the limit, we have that ρ(g_{n_{0}}^{−1})γ commutes with ρ(b).

Since the Kleinian group Γ̂ is torsion-free by Lemma 2.3 and the element ρ(b) ∈ Γ̂ is loxodromic, the centralizer of ρ(b) in Γ̂ is cyclic. Thus there are integers j and k such that (ρ(g_{n_{0}}^{−1})γ)^{j} = ρ(b)^{k}. A second application of Lemma 2.1 shows that for some n_{1} ≥ n_{0} we have (g_{n_{0}}^{−1}g_{n})^{j} = b^{k} for all n ≥ n_{1}. But since G is isomorphic to a torsion-free Kleinian group, each element of G has at most one j^{th} root. Hence b^{k} has a unique j^{th} root c and we have g_{n_{0}}^{−1}g_{n} = c for n ≥ M. Thus g_{n} = g_{n_{0}}c for large n, so the sequence (g_{n}) is eventually constant. Since (ρ_{n}(g_{n})) converges to γ, this implies that γ is an element of ρ(G), which contradicts our hypothesis that γ ∈ Γ̂ − ρ(G).

2.4

Next we consider the intersection of the limit set of ρ(G) with its image under an element of the geometric limit Γ̂.

The following definition will be useful. Let Γ_{1} and Γ_{2} be subgroups of the Kleinian group Γ. We will say that a point p ∈ Λ(Γ_{1}) ∩ Λ(Γ_{2}) is in P(Γ_{1}, Γ_{2}) if and only if Stab_{Γ_{1}}(p) ≅ Z, Stab_{Γ_{2}}(p) ≅ Z, and ⟨Stab_{Γ_{1}}(p), Stab_{Γ_{2}}(p)⟩ ≅ Z⊕Z. In particular, it must be that p is a parabolic fixed point of both Γ_{1} and Γ_{2}.

We will make use of the following result, due to Soma [34] and Anderson [1], which provides the link between the intersection of the limit sets of a pair of subgroups and the limit set of the intersection of the subgroups.

Recall that a Kleinian group Γ is topologically tame if H^{3}/Γ is homeomorphic to the interior of a compact 3-manifold.

Theorem 2.5 Let Γ_{1} and Γ_{2} be nonelementary, topologically tame subgroups of the Kleinian group Γ. Then

Λ(Γ_{1}) ∩ Λ(Γ_{2}) = Λ(Γ_{1} ∩ Γ_{2}) ∪ P(Γ_{1}, Γ_{2}).

The next lemma shows that the term P(Γ_{1}, Γ_{2}) may be ignored in the case where Γ_{1} and Γ_{2} are distinct conjugates of ρ(G) by elements of the geometric limit Γ̂.

Lemma 2.6 Let (ρ_{n}) be a sequence in D(G) converging to ρ. Suppose that the groups ρ_{n}(G) converge geometrically to Γ̂. Then for each γ ∈ Γ̂ − ρ(G), the set P(ρ(G), γρ(G)γ^{−1}) is empty.

Proof of 2.6: The argument runs along much the same line as the proof of Proposition 2.4. Suppose that p ∈ P(ρ(G), γρ(G)γ^{−1}), that Stab_{ρ(G)}(p) ≅ Z is generated by ρ(a) and that Stab_{γρ(G)γ^{−1}}(p) ≅ Z is generated by γρ(b)γ^{−1}. Note that each of the elements a and b generates its own centralizer.

Choose g_{n} ∈ G so that (ρ_{n}(g_{n})) converges to γ. Since ρ(a) commutes with γρ(b)γ^{−1}, we conclude as in the proof of Lemma 2.4 that there exists an integer n_{0} such that a commutes with g_{n}bg_{n}^{−1} for all n ≥ n_{0}. Since a generates its centralizer in G, each of the elements g_{n}bg_{n}^{−1} for n ≥ n_{0} must be a power of a. But these are all conjugate elements, while distinct powers of a are not conjugate. Therefore we must have that g_{n}bg_{n}^{−1} = g_{n_{0}}bg_{n_{0}}^{−1} for all n ≥ n_{0}.

Thus g_{n_{0}}^{−1}g_{n} commutes with b for all n ≥ n_{0}. Since the centralizer of b is cyclic, we may now argue exactly as in the proof of 2.4 that the sequence (g_{n}) must be constant for n ≥ n_{1} ≥ n_{0}, obtaining a contradiction to our hypothesis that γ ∈ Γ̂ − ρ(G).

2.6

As an immediate consequence of Lemma 2.4, Theorem 2.5, and Lemma 2.6, we have the following proposition.

§3. In the convex core 10

Proposition 2.7 Let (ρ_{n}) be a sequence in D(G) converging to ρ. Suppose that the groups ρ_{n}(G) converge geometrically to Γ̂ and that ρ(G) is topologically tame. Then for any γ ∈ Γ̂−ρ(G) the group ρ(G)∩γρ(G)γ^{−1} is purely parabolic and

Λ(ρ(G)) ∩ γ · Λ(ρ(G)) = Λ(ρ(G) ∩ γρ(G)γ^{−1}).

Hence if Λ(ρ(G))∩γ ·Λ(ρ(G)) is non-empty then it must contain only the fixed point of ρ(G) ∩ γρ(G)γ^{−1}.

3 In the convex core

We continue to assume that G is a finitely generated nonabelian torsion-free group. We consider a sequence (ρ_{n}) in D(G) which converges to a representation ω which is a maximal cusp (defined below). If we assume that the groups ρ_{n}(G) converge geometrically to Γ̂ then the hyperbolic manifold N = H^{3}/ω(G) is a covering space of N̂ = H^{3}/Γ̂. The main result of this section says that in this situation the restriction of the covering projection gives an embedding of the convex core of N into N̂.

It may be helpful to recall the most basic situation in which the convex core of a manifold does not embed in a manifold which it covers. Let N̂ be a hyperbolic 3-manifold and let f : S → N̂ be a totally geodesic isometric immersion of a finite area surface S. Let N be the cover of N̂ associated to π_{1}(S), so that f lifts to a totally geodesic embedding f̃: S → N. Since f̃(S) is totally geodesic, Λ(π_{1}(S)) is a circle and f̃(S) is the convex core of N. The convex core of N embeds in N̂ if and only if f is an embedding. Notice that f is not an embedding if and only if there exists an element γ of π_{1}(N̂) such that γ(Λ(π_{1}(S))) intersects Λ(π_{1}(S)) transversely, hence in at least two points.

We will be dealing with the case where the algebraic limit is a maximal cusp and hence each boundary component of the convex core of our algebraic limit is a totally geodesic thrice-punctured sphere (Lemma 3.1). We will see, as in the example above, that if the convex core of the algebraic limit does not embed then there must be an element γ of Γ̂ − ω(G) such that γ(Λ(ω(G))) intersects Λ(ω(G)) in at least two points. An application of Proposition 2.7 will complete the proof.

Given a Kleinian group Γ, define its convex hull CH(Γ) in H^{3} to be the smallest non-empty convex set in H^{3} which is invariant under the action of Γ. Thus CH(Γ) is the intersection of all half-spaces in H^{3} whose closures in the compactification H^{3} ∪ \overline{C} contain Λ(Γ). (Recall that a Kleinian group is, by definition, non-elementary so that its limit set has more than two points.)

The convex core of N = H^{3}/Γ is C(N) = CH(Γ)/Γ. We say that N, or equivalently Γ, is geometrically finite if Γ is finitely generated and C(N) has finite volume.

The injectivity radius inj_{N}(x) of N at the point x is half the length of the shortest homotopically non-trivial closed loop passing through x. Note that injectivity radius increases under lifting to a covering space. That is, if N is a cover of N̂ with covering map π : N → N̂, and x is any point of N, then

inj_{N}(x) ≥ inj_{N}(π(x)).

Given a hyperbolic 3-manifold N, define the ϵ-thick part of N as

N_{thick(ϵ)} = \{x ∈ N| inj_{N}(x) ≥ \frac{ϵ}{2}\}

and the ϵ-thin part of N as

N_{thin(ϵ)} = \{x ∈ N| inj_{N}(x) ≤ \frac{ϵ}{2}\}.

We recall that C(N) ∩ N_{thick(ϵ)} is compact for all ϵ > 0 if and only if N is geometrically finite (see Bowditch [7]). Hence, for a geometrically finite hyperbolic 3-manifold N, the sets C(N)∩N_{thick(1/m)} for m ≥ 1 form an exhaustion of C(N) by compact subsets.

We say that a representation ω in D(G) is a maximal cusp if N = H^{3}/ω(G) is geometrically finite and every component of the boundary ∂ C(N) of its convex core is a thrice-punctured sphere. We will further require that ω(G) not be a Fuchsian group. (This rules out only the case that ω(G) is in the (unique) conjugacy class of finite co-area Fuchsian groups uniformizing the thrice-punctured sphere.) Maximal cusps are discussed at length by Keen, Maskit and Series in [22], where the image groups are termed maximally parabolic.

A proof of the following lemma appears in [22]; since we will be using the lemma heavily, we include a sketch of the proof here.

Lemma 3.1 Let ω ∈ D(G) be a maximal cusp, and let N = H^{3}/ω(G). Then each component of ∂ C(N) is totally geodesic.

Proof of 3.1: Since the universal cover of C(N) is CH(ω(G)), it suffices to show that each component of ∂ CH(ω(G)) is a totally geodesic hyperplane or, equivalently, that each component of Ω(ω(G)) is a disk bounded by a circle on the sphere at infinity.

Figure 1: The domain of discontinuity of a maximal cusp is a union of round disks

Recall, for example from Epstein-Marden [17], that ∂ C(N) is homeomorphic to Ω(ω(G))/ω(G). Moreover, by the Ahlfor’s Finiteness theorem Ω(ω(G))/ω(G) has finite area. Thus each component S of Ω(ω(G))/ω(G) must be a thrice-punctured sphere. Write S = Δ/Γ_{Δ}, where Δ is a component of Ω(ω(G)) and Γ_{Δ} is the subgroup of ω(G) stabilizing Δ.

Since Δ/Γ_{Δ} is a thrice-punctured sphere, the group Γ_{Δ} must be a Fuchsian group and Δ must be a disk bounded by a circle on the sphere at infinity. (For a proof, see Chapter IX.C of Maskit’s book [26]; for a picture see figure 1 below.)

3.1

We are now ready to prove the main result of this section. A map between locally compact spaces will be called an embedding if it is proper and one-to-one.

Proposition 3.2 Let (ρ_{n}) be a sequence of elements of D(G) converging to a maximal cusp ω. Suppose that the groups ρ_{n}(G) converge geometrically to Γ̂.

Let N = H^{3}/ω(G), N̂ = H^{3}/Γ̂, and let π : N → N̂ be the covering map. Then π|_{C(N)} is an embedding.

Figure 2: The limit set of a maximal cusp may intersect its translate in at most one point.

Proof of 3.2: We first note that if (x_{i}) is a sequence in C(N) leaving every compact set, then lim_{i↦∞} inj_{N}(x_{i}) = 0. Hence, lim_{i↦∞} inj_{N}(π(x_{i})) = 0, which implies that (π(x_{i})) leaves every compact subset of N̂. Thus, π|_{C(N)} is a proper mapping.

It remains to show that π is injective. The universal cover of C(N) is CH(ω(G)). Thus it suffices to show that CH(ω(G)) ∩ γ · CH(ω(G)) is empty for each γ ∈ Γ̂ − ω(G). For notational convenience, set X = CH(ω(G)).

Since ω is assumed to be a maximal cusp, each component of ∂ C(N) is totally geodesic. Hence each component of ∂X is a plane H in H^{3} whose boundary at infinity is a circle C which lies in Λ(ω(G)).

If X ∩ γ · X is not empty, there are two possibilities. Either there is a point in ∂X ∩ γ · ∂X, or a component of ∂X lies entirely within γ · X (or vice versa).

We begin with the case that there is a point x in ∂X ∩ γ · ∂X. There then exist a plane H in ∂X and a plane H′ in γ ·∂X with x ∈ H ∩H′. If two planes

Figure 3: The convex hull of the limit set of a maximal cusp cannot intersect its translate

in H^{3} meet, either their intersection is a line or they are equal. If we let C denote the boundary at infinity of H and C′ the boundary at infinity of H′, then either C ∩ C′ contains exactly two points or C = C′. However, C ∩ C′ is contained in Λ(ω(G)) ∩ γ · Λ(ω(G)), so that Λ(ω(G)) ∩ γ · Λ(ω(G)) contains at least two points, which contradicts Proposition 2.7.

The second possibility is that a component H of ∂X lies entirely within γ · X. In this case, the boundary at infinity C of H lies in the boundary at infinity of γ · X, which is exactly γ · Λ(ω(G)). However, C also lies in Λ(ω(G)); hence Λ(ω(G)) ∩ γ · (Λ(ω(G))) contains C. This also contradicts Proposition 2.7. See figures 2 and 3 for an illustration of how Λ(ω(G)) can meet its translate by γ.

3.2

Remark 3.3 The conclusion of Proposition 3.2 holds, by the same argument, whenever N is topologically tame and ∂ C(N) is totally geodesic.

In general, the convex core of the algebraic limit need not embed in the geometric limit. However, one can define the visual hull of a Kleinian group Γ to be the set of all points in H^{3} such that the visual area of every component of Ω(Γ) is at most \frac{1}{2}. The visual core is then defined to be the quotient of the visual hull. Notice that if Γ is a maximal cusp, then its visual core and its convex core coincide.

§4. Near a maximal cusp 15

Anderson and Canary [2] use techniques similar to those developed in this section to prove that the visual core embeds whenever the algebraic limit has connected limit set and no accidental parabolics. They also show that, under the same assumptions, there is a compact core for the algebraic limit which embeds in the geometric limit. It has recently been discovered that the visual core of the algebraic limit need not embed in the algebraic limit even if the algebraic limit has connected limit set.

4 Near a maximal cusp

In this section, we will prove that if a representation in D(G) is near enough to a maximal cusp, then its associated hyperbolic 3-manifold contains a nearly isometric copy of an ϵ-truncated convex core of the maximal cusp. We first define this ϵ-truncated object and describe some of its useful attributes.

We recall that it follows from the Margulis lemma that there exists a constant λ_{0}, such that if ϵ < λ_{0} and N is a hyperbolic 3-manifold, then every component P of N_{thin(ϵ)} is either a solid torus neighborhood of a closed geodesic, or the quotient of a horoball H by a group Θ of parabolic elements fixing H (see for example [4]). In the second case, Θ is isomorphic either to Z or to Z ⊕ Z. Moreover, H is precisely invariant under Θ < Γ, by which we mean that if γ ∈ Γ and γ · H ∩ H ≠ ∅, then γ ∈ Θ and γ · H = H. If Θ ≅ Z, we call P a rank-one cusp, and if Θ ≅ Z ⊕ Z, we call P a rank-two cusp. Recall also that there exists L(ϵ) > 0, such that any two components of N_{thin(ϵ)} are separated by a distance of at least L(ϵ).

The next lemma gives the structure of N_{thin(ϵ)} for sufficiently small ϵ. For a proof see section 6 of Morgan [31].

Lemma 4.1 Let N be a geometrically finite hyperbolic 3-manifold. There exists δ(N) < λ_{0}, such that if ϵ ≤ δ(N) and P is a component of N_{thin(ϵ)}, then

(i) P is non-compact,

(ii) ∂P meets C(N) orthogonally along each component of their intersection.

(iii) E = ∂P ∩ C(N) is a Euclidean surface with geodesic boundary, and diam E ≤ 1;

(iv) if P is a rank-one cusp then E is an annulus, and if P is a rank-two cusp then E is a torus; and (v) C(N) ∩ P is homeomorphic to E × [0, ∞).

In particular, for any ϵ ≤ δ(N) the set N_{thick(ϵ)}∩C(N) is a compact 3-manifold with piecewise smooth boundary.

Given a geometrically finite hyperbolic 3-manifold N we may define its ϵ-truncated convex core D_{ϵ}(N) to be the intersection of its convex core C(N) with the ϵ-thick part N_{thick(ϵ)} of N. The above lemma completely characterizes C(N) − D_{ϵ}(N) when ϵ < δ(N).

Recall that a compact submanifold of N is said to be a compact core for N if the inclusion map is a homotopy equivalence.

Lemma 4.2 Suppose that N is a geometrically finite hyperbolic 3-manifold, and δ(N) > ϵ > 0. Then D_{ϵ}(N) is a compact core for N.

Proof of 4.2: First recall that the inclusion of C(N) into N is a homotopy equivalence. Moreover, each component of C(N) − D_{ϵ}(N) is homeomorphic to E × (0, ∞) for some Euclidean surface E. Thus, the inclusion of D_{ϵ}(N) into C(N) is a homotopy equivalence. The result follows.

4.2

For any subset X of a hyperbolic manifold N we will denote by N_{r}(X) the closed neighborhood of radius r of X. In the case N = H^{3}/ω(G), where ω is a maximal cusp, Proposition 4.4 will provide bounds for both the area of ∂D_{ϵ}(N) and the volume of N_{2}(∂D_{ϵ}(N)). These bounds will depend only on the topological type of N and not on ϵ. We first recall the following special case of Lemma 8.2 in [8] (see also Proposition 8.12.1 of Thurston [36]).

Lemma 4.3 There is a constant κ > 0, such that for any maximal cusp ω ∈ D(G) and any collection S of components of the boundary of the convex core of N = H^{3}/ω(G), the neighborhood N_{3}(S) has volume less than κ area S.

4.3

If N = H^{3}/ω(G), where ω(G) is a maximal cusp, we will denote by σ(N) the number of components of ∂ C(N), and by τ(N) the number of rank-two cusps of N. We set

α(N) = \frac{7}{2}πσ(N) + 2πτ(N)

and

β(N) = 2πκσ(N) + πe^{4}τ(N),

where κ is the constant given by Lemma 4.3.

Lemma 4.4 Let ω be a maximal cusp and let N = H^{3}/ω(G). If δ(N) > ϵ > 0, then

(i) area ∂D_{ϵ}(N) ≤ α(N), and

(ii) vol(N_{2}(∂D_{ϵ}(N))) ≤ β(N).

Proof of 4.4: Notice that ∂D_{ϵ}(N) = S ∪ E where S = ∂ C(N) ∩ N_{thick(ϵ)} and E = C(N) ∩ ∂N_{thick(ϵ)}. Since S ⊂ ∂ C(N) and since each component of ∂ C(N) is a thrice-punctured sphere, we have area S ≤ area ∂ C(N) = 2πσ(N). By Lemma 4.1, each component of E is a Euclidean manifold of diameter at most 1, so each component of E has area at most π. Since each component of ∂ C(N) contains three components of ∂E there are \frac{3}{2}σ(N) annular components of E. Moreover, there are τ(N) toroidal components of E. The first assertion follows.

Let Ŝ denote the union of S with the annular components of E. Since each annular component of E has diameter less than 1,

N_{2}(Ŝ) ⊂ N_{3}(S) ⊂ N_{3}(∂ C(N)).

Thus Lemma 4.3 guarantees that

vol N_{2}(Ŝ) ≤ κ area(∂ C(N)) = 2πκσ(N).

Now, if T is a toroidal component of E, then N_{2}(T) is (the quotient of) a region isometric to T × (−2, 2) with the metric ds^{2} = e^{−2t}ds_{T}^{2} + dt^{2}, which has volume less than 2πe^{4}. The second assertion now follows.

4.4

Our next result, Proposition 4.5, asserts (among other things) that the hyperbolic manifold associated to a representation which is near enough to a maximal cusp contains a biLipschitz copy of the ϵ-truncated convex core of the manifold associated to the maximal cusp.

We first outline the argument. Suppose that (ρ_{n}) is a sequence of representations in D(G) which converges to a maximal cusp ω and that the groups ρ_{n}(G) converge geometrically to Γ̂. Let N = H^{3}/ω(G) and N̂ = H^{3}/Γ̂. If π : N → N̂ is the covering map, Proposition 3.2 implies that π|_{D_{ϵ}(N)} is an embedding. Since (ρ_{n}(G)) converges geometrically to Γ̂, larger and larger chunks of N̂ are nearly isometric to larger and larger chunks of N_{n} = H^{3}/ρ_{n}(G). In particular, for all large enough n there exists a 2-biLipschitz embedding f_{n} : V_{n} → N_{n} where π(D_{ϵ}(N)) ⊂ V_{n} ⊂ N̂. The desired biLipschitz copy of D_{ϵ}(N) is f_{n}(π(D_{ϵ}(N))).

In order to carry out the program outlined above, it will be necessary to make consistent choices of base points in different hyperbolic 3-manifolds. We will use the following convention. If z is a point in H^{3} and Γ is a Kleinian group, we will let z_{Γ} denote the image of z in the hyperbolic manifold H^{3}/Γ. In the case that Γ = ρ(G) for some representation ρ ∈ Hom(G, PSL_{2}(C)) we will write z_{ρ} = z_{ρ(G)}.

If a codimension-0 submanifold X of a hyperbolic manifold N is connected and has piecewise smooth boundary, then it has two natural distance functions. In the extrinsic metric the distance between two points of X is equal to their distance in N, while in the intrinsic metric the distance is the infimum of the lengths of rectifiable paths in X joining the two points. Observe that if X and Y are submanifolds of hyperbolic manifolds and if f: X → Y is a K-biLipschitz map with respect to the extrinsic metrics, then f is also K-biLipschitz with respect to the intrinsic metrics.

Proposition 4.5 Suppose that ω ∈ D(G) is a maximal cusp and set N = H^{3}/ω(G). Let ϵ > 0 be given with the property that ϵ < δ(N) and let z be a point of H^{3} such that z_{ω} lies in the interior of D_{ϵ}(N). Then there is a neighborhood U(ϵ, z, ω) of ω in D(G) such that for each ρ ∈ U(ϵ, z, ω), there exists a map ϕ : D_{ϵ}(N) → N′ = H^{3}/ρ(G), with the following properties:

(1) ϕ maps D_{ϵ}(N) homeomorphically onto a manifold with piecewise smooth boundary, and is 2-biLipschitz with respect to the intrinsic metrics on D_{ϵ}(N) and ϕ(D_{ϵ}(N)),

(2) ϕ(z_{ω}) = z_{ρ},

(3) vol N_{1}(∂(ϕ(D_{ϵ}(N)))) ≤ 8β(N), and (4) ϕ(N_{thin(δ)} ∩ D_{ϵ}(N)) ⊂ N′_{thin(2δ)} for any δ < \frac{λ_{0}}{2} (where λ_{0} is the Margulis

constant).

Proof of 4.5: Let (ρ_{n}) be a sequence in D(G) that converges to ω, and set N_{n} = H^{3}/ρ_{n}(G). It suffices to prove that (ρ_{n}) has a subsequence (ρ_{n_{i}}) such that there exist maps ϕ_{i} : D_{ϵ}(N) → N_{n_{i}} which have properties (1)—(4).

Given any sequence (ρ_{n}) in D(G) converging to ω, Proposition 2.2 guarantees that there exists a subsequence (ρ_{n_{i}}(G)) of (ρ_{n}(G)) which converges geometrically to a group Γ̂ such that ω(G) ⊂ Γ̂. Let N̂ = H^{3}/Γ̂ and let π : N → N̂ be the associated covering map. Proposition 3.2 guarantees that π|_{C(N)} is an embedding and hence that π|_{D_{ϵ}(N)} is an embedding. Let D = π(D_{ϵ}(N)). Since π is a local isometry we find using Lemma 4.4 that

vol N_{2}(∂D) ≤ vol N_{2}(∂D_{ϵ}(N)) ≤ β(N).

Since (ρ_{n_{i}}(G)) converges geometrically to Γ̂, it follows from Corollary 3.2.11 in [10] or Theorem E.1.13 in [4] that there exist smooth submanifolds V_{i} ⊂ N̂, numbers r_{i} and α_{i}, and maps f_{i} : V_{i} → N_{n_{i}} such that

(i) V_{i} contains B(r_{i}, z_{Γ}), the closed radius-r_{i} neighborhood of z_{Γ},

(ii) f_{i}(z_{Γ}) = z_{ρ_{n_{i}}},

(iii) r_{i} converges to ∞, and α_{i} converges to 1,

(iv) f_{i} maps V_{i} diffeomorphically onto f(V_{i}) and is α_{i}-biLipschitz with respect to the extrinsic metrics on V_{i} and f(V_{i}).

Choose d so that D ⊂ B(d, z_{Γ}). Set μ = max{1, λ_{0}/2}. We may assume that the subsequence (ρ_{n_{i}}) has been chosen so that α_{i} < 2 and r_{i} > d + 2μ for all i. This condition on r_{i} implies that N_{2μ}(D) is contained in the interior of V_{i}.

We claim that N_{μ}(f_{i}(D)) ⊂ f_{i}(V_{i}). To prove the claim, we consider the frontier X of N_{2μ}(D) in N̂ and the frontier Y_{i} of f_{i}(N_{2μ}(D)) in N_{n_{i}}. Since f_{i} is a homeomorphism onto its image it follows from invariance of domain that f_{i}(X) = Y_{i}. Since f_{i}^{−1} is extrinsically α_{i}-Lipschitz with α_{i} < 2, and since every point of X has distance 2μ from D, every point of f_{i}(X) = Y_{i} must be a distance greater than μ from f_{i}(D). Thus Y_{i} is disjoint from N_{μ}(f_{i}(D)). Since f_{i}(N_{2μ}(D)) contains f_{i}(D) and is disjoint from the frontier Y_{i} of N_{μ}(f_{i}(D)) we have f_{i}(N_{2μ}(D)) ⊃ N_{μ}(f_{i}(D)), and the claim follows.

In particular, N_{1}(f_{i}(∂D)) ⊂ f_{i}(V_{i}). Again using that f_{i}^{−1} is extrinsically 2-Lipschitz we conclude that N_{1}(f_{i}(∂D)) ⊂ f_{i}(N_{2}(∂D)). On the other hand, since f_{i} is extrinsically 2-Lipschitz, it is intrinsically 2-Lipschitz and can therefore increase volume by at most a factor of 8. Therefore

vol N_{1}(∂f_{i}(D)) ≤ 8 vol N_{2}(∂D) ≤ 8β(N).  (1)

We now define ϕ_{i}: D_{ϵ}(N) → N_{n_{i}} to be f_{i} ∘ π. We will complete the proof of the proposition by showing that ϕ_{i} has properties (1)—(4).

Since π is a local isometry and π|_{D_{ϵ}(N)} is an embedding, the map π|_{D_{ϵ}(N)} is an isometry between D_{ϵ}(N) and π(D_{ϵ}(N)) with respect to their intrinsic metrics. Since f_{i}: D → f_{i}(D) is an extrinsically (and hence intrinsically) 2biLipschitz homeomorphism, it follows that ϕ_{i} has property (1).

We have ϕ_{i}(z_{ω}) = f_{i}(π(z_{ω})) = f_{i}(z_{Γ}) = z_{ρ_{n_{i}}}. This is property (2). Property (3) follows from equation (1) since ϕ_{i}(D_{ϵ}(N)) = f_{i}(D).

It remains to check property (4). Suppose that x ∈ N_{thin(δ)}∩D_{ϵ}(N) and δ < \frac{λ_{0}}{2} . Then there exists a homotopically non-trivial loop C (in N) based at x and having length at most δ. Notice that ϕ_{i}(C) has length at most 2δ. Hence ϕ_{i}(x) must lie in the 2δ-thin part of N_{n_{i}} unless ϕ_{i}(C) is homotopically trivial. But since ϕ_{i}(C) has length at most 2δ, it is contained in the closed δ-neighborhood of ϕ_{i}(x) in N_{n_{i}}. Thus if ϕ_{i}(C) were homotopically trivial in N, it would lift to a loop in a ball of radius δ in H^{3} whose center projects to ϕ_{i}(x). Hence ϕ_{i}(C) would be null-homotopic in B(δ, ϕ_{i}(x)) ⊂ N_{δ}(ϕ_{i}(D)) ⊂ N_{μ}(ϕ_{i}(D)) ⊂ f_{i}(V_{i}). This would imply that f_{i}^{−1}(ϕ_{i}(C)) = π(C) is homotopically trivial in N̂, in contradiction to the fact that π is a covering map. Thus, ϕ_{i} has property (4), and the proof is complete.

4.5

Remark 4.6 Lemmas 4.3 and 4.4 have analogues for general geometrically finite hyperbolic 3-manifolds, but the constants would also depend on the minimal length of a compressible curve in ∂ C(N). Proposition 4.5 remains true, by similar arguments, whenever N is geometrically finite and every component of ∂ C(N) is totally geodesic. One may use arguments similar to those in section 3 of [11] to prove that, if ρ is sufficiently near to a maximal cusp ω, then ϕ(D_{ϵ}(N_{ω})) is a compact core for N_{ρ}.

§5. All over the boundary of Schottky space 21

5 All over the boundary of Schottky space

We now restrict to the case where G is the free group F_{k} on k generators, where k ≥ 2. We set D_{k} = D(F_{k}). Recall that D_{k} is a closed subset of Hom(F_{k}, PSL_{2}(C)). Let CC_{k} denote the subset of D_{k} consisting of representations which are convex-cocompact, i.e. are geometrically finite and have purely loxodromic image. Moreover, let B_{k} = \overline{CC_{k}} − CC_{k} ⊂ D_{k}. It is known (see Marden [24]) that CC_{k} is an open subset of Hom(F_{k}, PSL_{2}(C)). (The quotient of CC_{k} under the action of PSL_{2}(C) is often called Schottky space.)

Let M_{k} denote the set of maximal cusps in D_{k}. It is a theorem of Maskit’s [28] that M_{k} ⊂ B_{k}. McMullen has further proved that M_{k} is a dense subset of B_{k}. This result, though not written down, is in the spirit of McMullen’s earlier result [29] that maximal cusps are dense in the boundary of any Bers slice of quasi-Fuchsian space.

The main result of this section, Theorem 5.2, asserts that there is a dense G_{δ}-set of purely loxodromic, analytically tame representations in B_{k}. This theorem generalizes and provides an alternate proof of Theorem 8.2 in [14].

The proof of Theorem 5.2 makes use of Proposition 4.5 and McMullen’s theorem. We use Proposition 4.5 to show that if ρ ∈ B_{k} is well-approximated by an infinite sequence of maximal cusps, then its convex core can be exhausted by nearly isometric copies of the truncated convex cores of the maximal cusps; this implies that ρ is analytically tame. McMullen’s theorem guarantees that the G_{δ}-set of points in B_{k} which are well-approximated by an infinite sequence of maximal cusps is dense.

We now recall the definition of an analytically tame hyperbolic 3-manifold.

Definition: A hyperbolic 3-manifold N with finitely generated fundamental group is analytically tame if C(N) may be exhausted by a sequence of compact submanifolds {M_{i}} with piecewise smooth boundary such that

(1) M_{i} ⊂ M^{∘}_{j} if i < j, where M^{∘}_{j} denotes the interior of M_{j} considered as a subset of C(N),

(2) ∪M^{∘}_{i} = C(N),

(3) there exists a number K > 0 such that the boundary ∂M_{i} of M_{i} has area at most K for all i, and (4) there exists a number L > 0 such that N_{1}(∂M_{i}) has volume at most L

for every i.

While the definition of analytic tameness is geometric in nature, it does have important analytic consequences. In particular, for an analytically tame group Γ one can control the behavior of positive Γ-invariant superharmonic functions on H^{3}. Specifically, we will make extensive use of the following result, which is contained in Corollary 9.2 of [8].

Proposition 5.1 If N = H^{3}/Γ is analytically tame and Λ(Γ) = \overline{C} then all positive superharmonic functions on N are constant.

5.1

We are now in a position to state Theorem 5.2.

Theorem 5.2 For all k ≥ 2, there exists a dense G_{δ}-set C_{k} in B_{k}, which consists entirely of analytically tame Kleinian groups whose limit set is the entire sphere at infinity.

The proof of Theorem 5.2 involves the following three lemmas. The first lemma is contained in Chuckrow [12].

Lemma 5.3 The set U_{k} of purely loxodromic representations in B_{k} is a dense G_{δ}-set in B_{k}. Moreover, if ρ ∈ U_{k}, then Λ(ρ(F_{k})) = \overline{C}.

5.3

The second lemma is an adaptation of Bonahon’s bounded diameter lemma [5].

Lemma 5.4 For every δ > 0, there is a number c_{k}(δ) with the following property. Let ϵ > 0 be given, let ω be any maximal cusp in D_{k}, and set N = H^{3}/ω(F_{k}). If δ(N) > ϵ, then any two points in ∂D_{ϵ}(N) may be joined by a path β in ∂D_{ϵ}(N) such that β ∩ N_{thick(δ)} has length at most c_{k}(δ).

Proof of 5.4: Let δ > 0 be given. In order to define c_{k}(δ) we consider a hyperbolic 2-manifold P which is homeomorphic to a thrice-punctured sphere; there is only one such hyperbolic 2-manifold up to isometry. Since P_{thick(δ)} is a compact subset of the metric space P, it has a finite diameter d(δ). It is clear that any two points in P may be joined by a path β such that β ∩ P_{thick(δ)} has length at most d(δ). We set c_{k}(δ) = (2k − 2)d(δ) + 3k − 3.

Now let ω be any maximal cusp in D_{k}, and set N = H^{3}/ω(F_{k}). We consider an arbitrary component S of ∂ C(N). Then S is a totally geodesic thrice-punctured sphere. Hence S, with its intrinsic metric, is isometric to P. Furthermore, the inclusion homomorphism π_{1}(S) → π_{1}(N) is injective, and hence S_{thin(δ)} ⊂ N_{thin(δ)}. It follows that any two points in S may be joined by a path β in S such that β ∩ N_{thick(δ)} has length at most d(δ).

Now, since D_{ϵ}(N) is a compact core for N and π_{1}(N) is a free group of rank k, we see that D_{ϵ}(N) is a handlebody of genus k. In particular, there are exactly 2k − 2 components of ∂ C(N) and exactly 3k − 3 annular components of ∂D_{ϵ}(N) − ∂ C(N). Also recall, from Lemma 4.1 that each component of ∂D_{ϵ}(N) − ∂ C(N) has diameter at most 1. Thus, since ∂D_{ϵ}(N) is connected, we see that any two points in ∂D_{ϵ}(N) may be joined by a path β such that β ∩ N_{thick(δ)} has length at most (2k − 2)d(δ) + 3k − 3 = c_{k}(δ).

5.4

In the following lemma, and in the rest of the section, we arbitrarily fix a base point z^{0} ∈ H^{3}, and we let X_{k} denote the set of all representations ρ ∈ B_{k} such that z^{0} lies in the interior of CH(ρ(F_{k})) relative to H^{3}.

Lemma 5.5 The set X_{k} is an open dense subset of B_{k}.

Proof of 5.5: Given a representation ρ ∈ X_{k}, we have that z^{0} lies in the interior of CH(ρ(F_{k})). Hence z^{0} lies in the interior of some ideal tetrahedron T with vertices in Λ(ρ(F_{k})). Since the fixed points of elements of ρ(F_{k}) are dense in Λ(ρ(F_{k})), we may assume that the vertices of T are attracting fixed points of elements ρ(g_{1}),…, ρ(g_{4}) of ρ(F_{k}). It follows that for any ρ′ ∈ B_{k} sufficiently close to ρ, the attracting fixed points of ρ′(g_{1}),…, ρ′(g_{4}) span a tetrahedron having z^{0} as an interior point. Hence z^{0} lies in the interior of CH(ρ′(F_{k})). This shows that X_{k} is an open subset of B_{k}.

If ρ ∈ U_{k}, then ρ ∈ X_{k} since Λ(ρ(F_{k})) = \overline{C}. Hence, Lemma 5.3 implies that X_{k} is dense.

5.5

Proof of 5.2: By the result of McMullen’s discussed at the beginning of this section, M_{k} is a dense subset of B_{k}. In view of Lemma 5.5 it follows that M_{k} ∩ X_{k} is also dense in B_{k}. For each ω ∈ M_{k} ∩ X_{k} and each ϵ > 0 we will define a neighborhood V (ϵ, ω) of ω in X_{k} ⊂ B_{k}.

Set N_{ω} = H^{3}/ω(F_{k}). Since ω ∈ X_{k}, we have z^{0} ∈ CH(ω(F_{k}); in the basepoint convention given in Section 4 we have z_{ω}^{0} ∈ C(N_{ω}). Hence either z_{ω}^{0} ∈ D_{ϵ}(N_{ω}), or z_{ω}^{0} lies in the interior of the ϵ-thin part of N_{ω}. Let η(ϵ, ω) = min{ϵ, δ(N_{ω})}. If z_{ω}^{0} ∈ D_{ϵ}(N_{ω}) then we set V (ϵ, ω) = U(η(ϵ, ω), z^{0}, ω) ∩ X_{k}, where U(η(ϵ, ω), z^{0}, ω) is the open set given by Proposition 4.5. If z_{ω}^{0} lies in the ϵ-thin part of N_{ω}, we take V (ϵ, ω) to be a neighborhood of ω in X_{k} such that for every ρ ∈ V the point z_{ρ}^{0} lies in the interior of the ϵ-thin part of H^{3}/ρ(F_{k}). (Such a neighborhood exists because there is an element g ∈ F_{k} such that dist(z^{0}, ω(g) · z^{0}) < ϵ. For any ρ sufficiently close to ω we have dist(z^{0}, ρ(g) · z^{0}) < ϵ.)

We now set W_{k}(ϵ) = ∪_{ω∈M_{k}∩X_{k}}V (ϵ, ω). Since M_{k} ∩ X_{k} is dense in B_{k}, the set W_{k}(ϵ) is an open dense subset of X_{k} for every ϵ > 0. Since U_{k} is a dense G_{δ}-set in B_{k} it follows that

C_{k} = U_{k} ∩ ⋂_{m∈Z_{+}} W_{k}(\frac{1}{m})

is a dense G_{δ}-set in X_{k}. In order to complete the proof, we need only to show that each element of C_{k} is analytically tame and has the entire sphere as its limit set.

Let ρ : F_{k} → PSL_{2}(C) be a representation in C_{k}. Set N = H^{3}/ρ(F_{k}). Lemma 5.3 guarantees that C(N) = N.

By the definition of C_{k}, for every m ∈ Z_{+} there exists a maximal cusp ω_{m} : F_{k} → PSL_{2}(C) such that ρ ∈ V (\frac{1}{m}, ω_{m}). Let N_{m} = H^{3}/ω_{m}(F_{k}) and let m_{0} be a positive integer such that \frac{1}{2m_{0}} is less than the injectivity radius of N at z_{ρ}^{0}. In what follows we consider an arbitrary integer m ≥ m_{0}.

By the definition of m_{0} the point z_{ρ}^{0} lies in the \frac{1}{m}-thick part of N. By the definition of the sets V (\frac{1}{m}, ω_{m}), it follows that ρ ∈ U(η(\frac{1}{m}, δ(N_{m})), z^{0}, ω_{m}). Let D_{m} = D_{η( \frac{1}{m} ,δ(N_{m}))}(N_{m}). Proposition 4.5 guarantees that there is a map ϕ_{m} : D_{m} → N, such that (1) ϕ_{m} maps D_{m} homeomorphically onto a manifold with piecewise smooth

boundary, and is 2-biLipschitz with respect to the intrinsic metrics on D_{m} and ϕ_{m}(D_{m}),

(2) ϕ_{m}(z_{ω_{m}}^{0}) = z_{ρ}^{0},

(3) vol N_{1}(∂(ϕ_{m}(D_{m}))) ≤ 8β(N_{m}) = 32πκ(k − 1), and

(4) ϕ_{m}((N_{m})_{thin(δ)} ∩ D_{m}) ⊂ N_{thin(2δ)} for any δ < \frac{λ_{0}}{2} .

(Here κ is the constant given by Lemma 4.3. Notice that since π_{1}(N_{m}) is a free group we have τ(N_{m}) = 0 and hence β(N_{m}) = 2πκσ(N_{m}) = 4πκ(k − 1).)

We set δ_{0} = λ_{0}/3 and M_{m} = ϕ_{m}(D_{m}).

Then by (4) we have

M_{m} ∩ ϕ_{m}(D_{m} ∩ (N_{m})_{thin(δ_{0})}) ⊂ N_{thin(2δ_{0})}.

Hence, by Lemma 5.4, any two points in ∂M_{m} may be joined by a path β in ∂M_{m} such that β ∩ N_{thin(2δ_{0})} has length at most 2c_{k}(δ_{0}).

Let r > 0, and let X(r) denote the set of points x ∈ N for which there exists a path β beginning in B(r, z_{ρ}^{0}) and ending at x such that β ∩ N_{thick(2δ_{0})} has length at most 2c_{k}(δ_{0}). Since ρ(F_{k}) is purely loxodromic, each component of N_{thin(2δ_{0})} is compact. Moreover, the components of N_{thin(2δ_{0})} are separated by a distance of at least L(2δ_{0}), so there exist only a finite number of components of N_{thin(2δ_{0})} contained in X(r). Therefore X(r) is compact. Set ζ(r) = min_{x∈X(r)} inj_{N}(x), and set m_{1} = max(m_{0}, 2/ζ(r)).

If m > m_{1} then ∂M_{m} ∩ B(r, z_{ρ}^{0}) = ∅, since any point in ∂M_{m} may be joined by a path of length at most 2c_{k}(δ_{0}) to a point of injectivity radius ≤ \frac{2}{m} (namely any point in ϕ_{m}(∂D_{m} − ∂ C(N_{m}))). Since z_{ρ}^{0} ∈ M_{m} by (2), and since ∂M_{m} ∩ B(r, z_{ρ}^{0}) = ∅, we see that B(r, z_{ρ}^{0}) ⊂ M_{m} for every m > m_{1}.

We therefore have ∪_{m>m_{1}} M^{∘}_{m} = N = C(N). We may pass to a subsequence

M_{m_{j}} such that M_{m_{j}} ⊂ M^{∘}_{m_{j+1}} for all j and ∪_{j∈Z_{+}} M^{∘}_{m_{j}} = N = C(N).

By (3) we have

vol N_{1}(∂M_{m}) ≤ 32πκ(k − 1)

for all m > m_{1}. Moreover, by (1) and Lemma 4.4 we have

area ∂M_{m} ≤ 4α(N_{m}) = 14πσ(N_{m}) = 28π(k − 1).

Thus N is analytically tame.

5.2

§6. Free groups and displacements 26

6 Free groups and displacements

This section is devoted to the proof of the following theorem, which includes the Main Theorem stated in the introduction.

Theorem 6.1 Let k ≥ 2 be an integer and let Φ be a Kleinian group which is freely generated by elements ξ_{1},…, ξ_{k}. Suppose that either

(a) Φ is purely loxodromic and topologically tame, or

(b) Φ is geometrically finite, or

(c) Φ is analytically tame and Λ(Φ) = \overline{C}, or

(d) the hyperbolic 3-manifold H^{3}/Φ admits no non-constant positive superharmonic functions.

Let z be any point of H^{3} and set d_{i} = dist(z, ξ_{i} · z) for i = 1,…, k. Then we have

∑_{i=1}^{k} \frac{1}{1 + e^{d_{i}}} ≤ \frac{1}{2}.

In particular there is some i ∈ {1,…, k} such that d_{i} ≥ log(2k − 1).

Note that if we had d_{i} < log(2k − 1) for i = 1,…, k it would follow that

∑_{i=1}^{k} \frac{1}{1 + e^{d_{i}}} > k · \frac{1}{2k} = \frac{1}{2}.

Thus the last sentence of Theorem 6.1 does indeed follow from the preceding sentence.

Conditions (a)–(d) of Theorem 6.1 are by no means mutually exclusive. In particular, according to Proposition 5.1, condition (c) implies condition (d).

The following elementary inequality will be needed for the proof of Theorem 6.1.

Lemma 6.2 Let x and y be non-negative real numbers with x + y ≤ 1. Set p = \frac{1}{2}(x + y). Then we have

(\frac{1 − x}{x})(\frac{1 − y}{y} ) ≥ (\frac{1 − p}{p})^{2}.

Proof of 6.2: We can write x = p + α and y = p − α for some α ∈ R. We find by direct calculation that p^{2}(1 − x)(1 − y) − (1 − p)^{2}xy = (1 − 2p)α^{2}. But p ≤ 1/2 since x + y ≤ 1. Hence p^{2}(1 − x)(1 − y) ≥ (1 − p)^{2}xy, and the assertion follows.

6.2

A basic step in the proof of Theorem 6.1 is the observation that if the inequality in the the conclusion holds on the boundary B_{k} of CC_{k} then it also holds on CC_{k}. This observation is contained in the following lemma.

Lemma 6.3 For k > 1 let F_{k} denote the free group on the k generators x_{1} … x_{k} and let ρ: F_{k} → PSL_{2}(C) be any representation in CC_{k}. Given any point z in H^{3} there exists a representation ρ_{z} ∈ B_{k} such that

dist(z, ρ(x_{i}) · z) = dist(z, ρ_{z}(x_{i}) · z)

for i = 1,…, k.

Proof of 6.3: We consider the set

R_{z} = {σ ∈ Hom(F_{k}, PSL_{2}(C))| dist(z, σ(x_{i}) · z) = dist(z, ρ(x_{i}) · z)}.

It suffices to show that R_{z} is connected and contains a point of Hom(F_{k}, PSL_{2}(C)) − CC_{k}. The connectedness follows immediately from the easy observation that for any positive number d the set {ξ ∈ Isom_{+}(H^{3})| dist(z, ξ · z) = d} is path-connected. Since R_{z} contains a representation with an invariant line passing through z, R_{z} − CC_{k} is clearly nonempty.

6.3

Proof of Theorem 6.1: We first prove that condition (d) implies the conclusion of the theorem.

We use the terminology of [14]. We set Ψ = {ξ_{1}, ξ_{1}^{−1},…, ξ_{k}, ξ_{k}^{−1}} ⊂ Φ. According to Lemma 5.3 of [14], there exists a number D ∈ [0, 2] such that, for any z_{0} ∈ H^{3}, there exists a Φ-invariant D-conformal 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 condition (d) of the theorem holds, it follows from Proposition 3.9 of [14] that any Φ-invariant conformal density for H^{3} is a constant multiple of the area density 𝔄. In view of condition (i) above we must in fact have 𝔐 = 𝔄. In particular D = 2.

For i = 1,…, k we set ν_{i} = ν_{ξ_{i}} and ν′_{i} = ν_{ξ_{i}^{−1}}. We denote by α_{i} and β_{i} the total masses of the measures ν_{i} and ν′_{i} respectively. After possibly interchanging the roles of ξ_{i} and ξ_{i}^{−1} we may assume that α_{i} ≤ β_{i}. (Interchanging the roles of ξ_{i} and ξ_{i}^{−1} does not affect the truth of the conclusion of the lemma, since d_{i} = dist(z_{0}, ξ_{i} · z_{0}) = dist(z_{0}, ξ_{i}^{−1} · z_{0}).)

By conditions (i) and (ii) above we have ∑_{i=1}^{k}(α_{i} + β_{i}) = 1. In particular for each i we have 0 ≤ β_{i} ≤ 1, and since α_{i} ≤ β_{i} we have 0 ≤ α_{i} ≤ 1/2. Since 𝔐 = 𝔄, condition (ii) also implies that ν_{i} ≤ A_{z_{0}}, where A_{z_{0}} denotes the area measure on S_{∞} centered at z_{0}. By applying condition (iii) above to ψ = ξ_{i}^{−1} we get that ∫_{S_{∞}} λ_{ξ_{i}^{−1},z_{0}}^{2}dν_{i} = 1 − β_{i}. And by definition we have ν_{i}(S_{∞}) = α_{i}. Thus the hypotheses of Lemma 5.5 of [14] hold with ν = ν_{i}, a = α_{i} and b = 1 − β_{i}. Hence by Lemma 5.5 of [14] we have

d_{i} = dist(z_{0}, ξ_{i}^{−1} · z_{0}) ≥ \frac{1}{2} log \frac{b(1 − a)}{a(1 − b)}.

(This is a corrected version of the conclusion of Lemma 5.5 of [14]. In the published version of [14] the inequality appeared with the roles of a and b reversed.)

Thus

d_{i} ≥ \frac{1}{2} log \frac{(1 − β_{i})(1 − α_{i})}{α_{i}β_{i}}.

Since α_{i} + β_{i} ≤ ∑_{i=1}^{k}(α_{i} + β_{i}) = 1, it follows from Lemma 6.2 that

\frac{(1 − α_{i})(1 − β_{i})}{α_{i}β_{i}} ≥ (\frac{1 − p_{i}}{p_{i}})^{2},

where p_{i} = (α_{i} + β_{i})/2. Hence

d_{i} ≥ log \frac{1 − p_{i}}{p_{i}} .

It follows that

\frac{1}{1 + e^{d_{i}}} ≤ p_{i}.

Hence

∑_{i=1}^{k} \frac{1}{1 + e^{d_{i}}} ≤ ∑_{i=1}^{k} p_{i} = \frac{1}{2} ∑_{i=1}^{k}(α_{i} + β_{i}) = \frac{1}{2}.

This completes the proof that condition (d) implies the conclusion of the theorem. In view of Proposition 5.1, it follows that condition (c) also implies the conclusion of the theorem.

Next we assume that condition (a) holds and deduce the conclusion of the theorem.

Since Φ is purely loxodromic and free of finite rank, either Λ(Φ) = \overline{C} or Φ is a Schottky group (see Maskit [27]). In the case that Λ(Φ) = \overline{C}, we use Theorem 8.1 in [8], which states that a topologically tame hyperbolic 3manifold is analytically tame.

Thus in this case condition (c) holds, and hence the conclusion of the theorem is true.

Now suppose that Φ is a Schottky group. Let F_{k} denote the abstract free group generated by {x_{1},…, x_{k}}. Let ρ_{0} : F_{k} → Φ denote the unique isomorphism that takes x_{i} to ξ_{i} for i = 1,…, k. We may regard ρ_{0} as a representation of F_{k} in P SL_{2}(C). Since Φ is a Schottky group we have ρ_{0} ∈ CC_{k}.

We define a continuous, non-negative real-valued function f_{z} on the representation space Hom(F_{k}, PSL_{2}(C)) by setting

f_{z}(ρ) = ∑_{i=1}^{k} \frac{1}{1 + exp dist(z, ρ(x_{i}) · z)} .

§7. Topology and free subgroups 30

We must show that for any point z in H^{3} and any representation ρ in CC_{k} we have f_{z}(ρ) ≤ \frac{1}{2}. Let z and ρ be given. By Lemma 6.3 there exists a representation ρ_{z} ∈ B_{k} such that the point z is displaced the same distance by ρ_{0}(x_{i}) as by ρ_{z}(x_{i}) for i = 1,…, k. Thus f_{z}(ρ_{z}) = f_{z}(ρ_{0}), so it suffices to show that f_{z}(ρ) ≤ \frac{1}{2} for any representation ρ ∈ B_{k}. To see this we consider the dense G_{δ}-set C_{k} ⊂ B_{k} given by Theorem 5.2. Recall that every representation in C_{k} maps F_{k} isomorphically onto an analytically tame Kleinian group whose limit set is the entire sphere at infinity. Thus for any ρ ∈ C_{k} the group ρ(F_{k}) satisfies condition (c) of the present theorem, and we therefore have f_{z}(ρ) ≤ \frac{1}{2}. Since C_{k} is dense in B_{k} and f_{z} is continuous, we have f_{z}(ρ) ≤ \frac{1}{2} for every ρ ∈ B_{k}.

Finally, we prove the conclusion of the theorem under the assumption that condition (b) holds.

We continue to denote by F_{k} the abstract free group on k generators. We fix an isomorphism ρ : F_{k} → Φ, which we regard as a discrete, faithful representation of F_{k} in Isom_{+}(H^{3}). In view of the geometric finiteness of Φ = ρ(F_{k}), a theorem of Maskit’s [28] guarantees that there exists a sequence of discrete faithful representations {ρ_{j} : F → Isom_{+}(H^{3})} such that (for all j) ρ_{j}(F) is geometrically finite and purely loxodromic, and ρ_{j} converges (as a sequence of representations) to ρ. Given z ∈ H^{3}, we set

m_{j} = ∑_{i=1}^{k} \frac{1}{1 + e^{dist(ρ_{j}(x_{i})·z,z)}} and m = ∑_{i=1}^{k} \frac{1}{1 + e^{d_{i}}} .

Since each ρ_{j} satisfies (a), we have m_{j} ≤ \frac{1}{2}. But clearly \{m_{j}\} converges to m, so m ≤ \frac{1}{2}. This is the conclusion of the theorem.

Theorem 6.1

7 Topology and free subgroups

The results of the last section can be used in studying the geometry of a hyperbolic manifold N. One writes N in the form H^{3}/Γ where Γ is a torsionfree Kleinian group, and applies the estimate given by Theorem 6.1 to suitable free subgroups Φ of Γ to deduce geometric information about N. Of course, such applications require that one be able to produce free subgroups of Γ. In this section we concentrate on the problem of giving sufficient conditions for a subgroup of Γ ≅ π_{1}(N) to be free.

This problem can be attacked with purely topological techniques. Results which use 3-manifold theory to deduce, from topological hypotheses, that certain subgroups of π_{1}(N) are free have appeared in [37], in Section VI.4 of [19] and in the appendix to [3]. The first two sources consider the case of a 2-generator subgroup, while the results of the third apply to higher rank subgroups, but require that N be closed. In this section we give a systematic treatment of this topic. We prove a result that includes the results of [3] as special cases and is suitable for the applications in this paper.

We shall follow a couple of conventions that are widely used in lowdimensional topology. Unlabeled homomorphisms between fundamental groups are understood to be induced by inclusion maps. Base points will be suppressed whenever it is clear from the context how to choose consistent base points.

Recall that an orientable piecewise linear 3-manifold N is said to be irreducible if every PL 2-sphere in N bounds a PL ball. We shall say that N is simple if N is irreducible and if for every rank-2 free abelian subgroup A of π_{1}(N), there is a closed PL subspace E of N, piecewise linearly homeomorphic to T^{2} × [0, ∞), such that A is contained in a conjugate of im(π_{1}(E) → π_{1}(N)). (The subgroup im(π_{1}(E) → π_{1}(N)) of π_{1}(N) is itself well-defined up to conjugacy.)

We shall say that an orientable PL 3-manifold N without boundary has cusp-like ends if it is PL homeomorphic to the interior of a compact manifoldwith-boundary M such that (i) every component of ∂M is a torus and (ii) for every component B of ∂M, the inclusion homomorphism π_{1}(B) → π_{1}(M) is injective. In particular, note that if N is closed then it has cusp-like ends.

Recall that the rank of a group Γ is the minimal cardinality of a generating set for Γ.

A group Γ is termed freely indecomposable if it is non-trivial and is not a free product of two non-trivial subgroups.

For any non-negative integer g we denote by S_{g} the closed orientable surface of genus g.

Theorem 7.1 Let N be a simple orientable PL 3-manifold without boundary. Suppose that k = rank π_{1}(N) < ∞, that π_{1}(N) is freely indecomposable, and that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. Then either π_{1}(N) is a free abelian group, or N has cusp-like ends.

Proof of 7.1: According to [32], there is a compact PL manifold-with-boundary M ⊂ N such that π_{1}(M) → π_{1}(N) is an isomorphism. Among all compact PL manifolds-with-boundary with this property we may suppose M to have been chosen so as to minimize the number of components r = r_{M} of ∂M. We may assume if we like that r > 0, for if r = 0 then N = M is closed, and in particular it has cusp-like ends. Let B_{1},…, B_{r} denote the components of ∂M. Let g_{i} denote the genus of B_{i} for i = 1,…, r. Since π_{1}(M) ≅ π_{1}(N) has rank k, the first betti number of M is at most k.

It follows from Poincaré-Lefschetz duality and the exact homology sequence of (M, ∂M) that the total genus ∑ g_{i} of ∂M is at most the first betti number of M. Thus ∑ g_{i} ≤ k.

We must have g_{i} > 0 for i = 1,…, r. Indeed, if B_{i} is a 2-sphere for some i, then since N is irreducible, B_{i} bounds a PL ball K ⊂ N. We have either K ⊃ M or K ∩ M = B_{i}. If K ⊃ M then since π_{1}(M) → π_{1}(N) is an isomorphism we have π_{1}(N) = 1, in contradiction to the free indecomposability of π_{1}(N). If K ∩ M = B_{i} then M′ = M ∪ K has fewer boundary components than M and π_{1}(M′) → π_{1}(N) is an isomorphism; this contradicts our choice of M.

Let us consider the case in which π_{1}(B_{i}) → π_{1}(M) has a non-trivial kernel for some i ≤ r. According to the Loop Theorem [35], M contains a properly embedded disk D such that ∂D is homotopically non-trivial in ∂M. If D separates M, both components of M − D have boundary components of positive genus and are therefore non-simply connected. This contradicts the free indecomposability of π_{1}(M). Hence D does not separate M, and π_{1}(M) is a free product of an infinite cyclic group with a group isomorphic to π_{1}(M − D). The latter group must be trivial in view of the free indecomposability of π_{1}(M) ≅ π_{1}(N). Thus π_{1}(N) ≅ π_{1}(M) is infinite cyclic in this case, and in particular free abelian.

From this point on we assume that π_{1}(B_{i}) → π_{1}(M) is injective for i = 1,…, r. Since by the hypothesis of the theorem, π_{1}(M) ≅ π_{1}(N) has no subgroup isomorphic to π_{1}(S_{g_{i}}) for 2 ≤ g_{i} ≤ k − 1, each g_{i} is either ≤ 1 or ≥ k. We have seen that the g_{i} are all strictly positive and that their sum is at most k. Hence we must have either (i) r = 1 and g_{1} = k, or (ii) g_{i} = 1 for i = 1,…, r.

Suppose that (i) holds. Then ∂M is a connected surface of genus k. Hence the Euler characteristic χ(∂M) is equal to 2−2k. We have χ(M) = \frac{1}{2}χ(∂M) = 1−k. Now as a compact PL 3-manifold with non-empty boundary, M admits a simplicial collapse to a 2-complex L. In particular M is homotopy-equivalent to L, and hence to the CW-complex L′ obtained from L by identifying a maximal tree in the 1-skeleton of L to a point. If c_{i} denotes the number of i-cells in L′, we have c_{0} = 1 and 1 − c_{1} + c_{2} = χ(L′) = χ(M) = 1 − k, so that c_{1} − c_{2} = k. But π_{1}(L′) ≅ π_{1}(M) ≅ π_{1}(N) has a presentation with c_{1} generators and c_{2} relations, and k = c_{1} − c_{2} is by definition the deficiency of the presentation. On the other hand, k is by hypothesis the rank of π_{1}(N). It is a theorem due to Magnus [23] that if a group Γ has rank k and admits a presentation of deficiency k, then Γ is free of rank k. Since π_{1}(N) is freely indecomposable, we must have k = 1, and π_{1}(N) must be infinite cyclic in this case as well.

Now suppose that (ii) holds. Then B_{1},…, B_{r} are tori. Since the inclusion homomorphisms π_{1}(B_{i}) → π_{1}(M) are injective, the groups A_{i} = im(π_{1}(B_{i}) → π_{1}(M)) are free abelian groups of rank 2. Since N is simple, there are closed PL subspaces E_{1},…, E_{r} of N, each piecewise linearly homeomorphic to T^{2} × [0, ∞), such that A_{i} is contained in a conjugate of im(π_{1}(E_{i}) → π_{1}(N)) for i = 1,…, r. It follows from Proposition 5.4 of [38] that B_{i} is isotopic to ∂E_{i} for i = 1,…, r. Hence we may suppose the E_{i} to have been chosen so that ∂E_{i} = B_{i}. For each i ≤ r we have either M ⊂ E_{i} or M ∩ E_{i} = B_{i}.

If M ∩ E_{i} = B_{i} for i = 1,…, r, we have N = M ∪ E_{1} … ∪ E_{r}. It follows that in this case N is PL homeomorphic to the interior of M, and hence that N has cusp-like ends.

There remains the case in which M ⊂ E_{i} for some i. In the sequence of inclusion homomorphisms

π_{1}(B_{i}) → π_{1}(M) → π_{1}(E_{i}) → π_{1}(N)

the composition of the first two arrows (from the left) is the isomorphism π_{1}(B_{i}) → π_{1}(E_{i}), and the composition of the last two arrows is the isomorphism π_{1}(M) → π_{1}(N). It follows that the entire sequence consists of isomorphisms, and hence that π_{1}(N) is a rank-2 free abelian group in this case.

7.1

We shall say that a group is semifree if it is a free product of abelian groups.

Corollary 7.2 Let N be an orientable hyperbolic 3-manifold of infinite volume. Suppose that k = rank π_{1}(N) < ∞, and that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. Then π_{1}(N) is semifree.

Proof of 7.2: Let us write N = H^{3}/Γ, where Γ ≅ π_{1}(N) is a discrete torsion-free subgroup of Isom_{+}(H^{3}). Since Γ is finitely generated it can be written as a free product Γ_{1} ∗ … ∗ Γ_{n} of freely indecomposable subgroups. Since N has infinite volume, so does the manifold N_{i} = H^{3}/Γ_{i} for i = 1,…, n. The rank k_{i} of Γ_{i} ≅ π_{1}(N_{i}) is at most k. Hence the hypothesis of the corollary implies that Γ_{i} has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. Applying Theorem 7.1 with N_{i} in place of N, we conclude that for each i ≤ n, either Γ_{i} is free abelian or N_{i} has cusp-like ends. But the latter alternative is impossible because a hyperbolic manifold with cusp-like ends has finite volume (see [4], D.3.18). Thus all the Γ_{i} are free abelian and hence Γ is semifree.

7.2

Recall that a group Γ is termed k-free, where k is a cardinal number, if every subgroup of Γ whose rank is at most k is free. We shall say that Γ is k-semifree if every subgroup of Γ whose rank is at most k is semifree.

Corollary 7.3 Let N be an orientable hyperbolic 3-manifold and let k be a non-negative integer. Suppose that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. In addition suppose that either

(i) N has infinite volume, or

(ii) every subgroup of π_{1}(N) whose rank is at most k is of infinite index in π_{1}(N).

Then π_{1}(N) is k-semifree.

Proof of 7.3: Let Δ be any subgroup of π_{1}(N) whose rank is at most k. Let Ñ denote the covering space of N associated to Δ. If either (i) or (ii) holds, Ñ has infinite volume. Hence by Corollary 7.2, Δ is semifree.

7.3

§8. Strong Margulis numbers and k-Margulis numbers 35

Hypothesis (ii) of 7.3 clearly holds if the first betti number of N is at least k + 1. According to Proposition 1.1 of [33], it also holds if H_{1}(N, Z/p) has rank at least k + 2 for some prime p. Thus we have:

Corollary 7.4 Let N be an orientable hyperbolic 3-manifold and let k be a non-negative integer. Suppose that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. In addition suppose that either

(i) the first betti number of N is at least k + 1, or

(ii) H_{1}(N, Z/p) has rank at least k + 2 for some prime p.

Then π_{1}(N) is k-semifree.

Remark 7.5 If the orientable hyperbolic 3-manifold N has no cusps, then every abelian subgroup of π_{1}(N) is infinite cyclic; thus π_{1}(N) is k-semifree for a given k if and only if it is k-free. Thus if N has no cusps we may replace “semifree” by “free” in the conclusions of Corollaries 7.3 and 7.4.

8 Strong Margulis numbers and k-Margulis numbers

In order to unify the different applications of the results of the last two sections it is useful to introduce a little formalism. Let Γ be a discrete torsion-free subgroup of Isom_{+}(H^{3}). Recall from [33] and [14] that a positive number λ is termed a Margulis number for the group Γ, or for the orientable hyperbolic 3-manifold N = H^{3}/Γ, if whenever ξ and η are non-commuting elements of Γ, and z ∈ H^{3}, we have max{dist(ξ · z, z), dist(η · z, z)} ≥ λ. We shall say that λ is a strong Margulis number for Γ, or for N, if whenever ξ and η are non-commuting elements of Γ, we have

\frac{1}{1 + e^{dist(ξ·z,z)}} + \frac{1}{1 + e^{dist(η·z,z)}} ≤ \frac{2}{1 + e^{λ}} .

Notice that if λ is a strong Margulis number for Γ, then λ is also a Margulis number for Γ.

More generally, let k ≥ 2 be an integer, and let λ be a positive real number. We shall say that λ is a k-Margulis number for the discrete torsion-free group Γ ≤ Isom_{+}(H^{3}), or for N = H^{3}/Γ, if for any k elements ξ_{1},…, ξ_{k} ∈ Γ, and for any z ∈ H^{3}, we have that either (i) max_{i=1}^{k} dist(ξ_{i} · z, z) ≥ λ, or

(ii) the group ⟨ξ_{1},…, ξ_{k}⟩ is generated by at most k − 1 abelian subgroups.

We say that λ is a strong k-Margulis number for Γ, or for N, if for any k elements ξ_{1},…, ξ_{k} ∈ Γ, and for any z ∈ H^{3}, we have that either

(i)

∑ \frac{1}{1 + e^{dist(ξ_{i}·z,z)}} ≤ \frac{k}{1 + e^{λ}} ,

or

(ii) the group ⟨ξ_{1},…, ξ_{k}⟩ is generated by at most k − 1 abelian subgroups.

Note that λ is a (strong) 2-Margulis number for Γ if and only if it is a (strong) Margulis number for Γ. Note also that if λ is a strong k-Margulis number for Γ, then λ is also a k-Margulis number for Γ.

In this section we will use Theorem 6.1 and the corollaries of Theorem 7.1 to prove that under various conditions a hyperbolic 3-manifold has log(2k − 1) as a strong k-Margulis number. In the following three sections these results will be used to obtain lower bounds for the volume of various classes of hyperbolic 3-manifolds.

Our first result is an easy consequence of Theorem 6.1(a). We shall say that a Kleinian group Γ is k-tame, where k is a positive integer, if every subgroup of Γ having rank at most k is topologically tame.

Proposition 8.1 Let k ≥ 2 be an integer and let Γ be a discrete subgroup of Isom_{+}(H^{3}). Suppose that Γ is k-free, k-tame and purely loxodromic. Then log(2k − 1) is a strong k-Margulis number for Γ.

Proof of 8.1: If ξ_{1},…, ξ_{k} ∈ Γ are elements of Γ, the group ⟨ξ_{1},…, ξ_{k}⟩ is topologically tame, purely loxodromic and free of some rank ≤ k. If its rank is k then it is freely generated by ξ_{1},…, ξ_{k}; hence for any z ∈ H^{3}, we have

∑ \frac{1}{1 + e^{dist(ξ_{i}·z,z)}} ≤ \frac{1}{2} = \frac{k}{1 + e^{log(2k−1)}}

by Theorem 6.1(a). If ⟨ξ_{1},…, ξ_{k}⟩ has rank ≤ k − 1, then in particular it is generated by at most k − 1 abelian subgroups.

8.1 Corollary 8.2 Let k ≥ 2 be an integer and let N be a non-compact, topologically tame orientable hyperbolic 3-manifold without cusps. Suppose that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. Then log(2k − 1) is a strong k-Margulis number for N.

Proof of 8.2: Let us write N = H^{3}/Γ, where Γ is a discrete, noncocompact, purely loxodromic subgroup of Isom_{+}(H^{3}). According to Proposition 3.2 in [8], every finitely generated subgroup of Γ is topologically tame. In particular Γ is k-tame. On the other hand, since N has infinite volume and π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1, Corollary 7.3 and Remark 7.5 guarantee that Γ ≅ π_{1}(N) is k-free. The desired conclusion therefore follows from Proposition 8.1.

8.2

It is worth pointing out that the following corollary can be deduced from Proposition 8.1, although a more general result, Corollary 8.7, will be proved below by a slightly different argument.

Corollary 8.3 Let k ≥ 2 be an integer and let N be a closed orientable hyperbolic 3-manifold. Suppose that the first betti number of N is at least k + 1 and that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. Then log(2k − 1) is a strong k-Margulis number for N.

Proof of 8.3: Let us write N = H^{3}/Γ, where Γ is a discrete, cocompact, purely loxodromic subgroup of Isom_{+}(H^{3}). By Corollary 7.4 and Remark 7.5, Γ ≅ π_{1}(N) is k-free. On the other hand, since N has betti number at least k + 1, any subgroup Γ′ of Γ having rank at most k is contained in the kernel of a surjective homomorphism β : Γ → Z. According to Proposition 8.4 of [9], it follows that Γ′ is topologically tame. Thus Γ is k-tame and the desired conclusion follows from Proposition 8.1.

8.3

The following result gives information not contained in Proposition 8.1 because the group Γ is allowed to have parabolic elements.

Proposition 8.4 Let k ≥ 2 be an integer and let Γ be a discrete subgroup of Isom_{+}(H^{3}). Suppose that Γ is k-semifree. Suppose in addition that for every subgroup Γ′ of Γ having rank at most k, either

(i) Γ′ is geometrically finite, or

(ii) N′ = H^{3}/Γ′ admits no non-constant positive superharmonic functions.

Then log(2k − 1) is a strong k-Margulis number for Γ.

Proof of 8.4: If ξ_{1},…, ξ_{k} ∈ Γ are elements of Γ, the group ⟨ξ_{1},…, ξ_{k}⟩ is semifree. Thus we may write it as a free product A_{1} ∗ … ∗ A_{r}, where r is an integer ≤ k and A_{1},…, A_{r} are free abelian groups. The sum of the ranks of the A_{i} is at most k. If r < k, or if some A_{i} has rank > 1, then ⟨ξ_{1},…, ξ_{k}⟩ is generated by at most k − 1 abelian subgroups. Now suppose that r = k and that the A_{i} are all cyclic. Then ⟨ξ_{1},…, ξ_{k}⟩ is free of rank k and is therefore freely generated by ξ_{1},…, ξ_{k}. If condition (i) of the hypothesis of the proposition holds, it follows from Theorem 6.1(b) that for any z ∈ H^{3} we have

∑ \frac{1}{1 + e^{dist(ξ_{i}·z,z)}} ≤ \frac{1}{2} = \frac{k}{1 + e^{log(2k−1)}} .

If condition (ii) holds, the same conclusion follows from Theorem 6.1(d). 8.4

If a torsion-free Kleinian group Γ is geometrically finite and has infinite covolume, then a theorem of Thurston’s (see Proposition 7.1 in Morgan [31]) guarantees that every finitely generated subgroup of Γ is geometrically finite. This yields the following corollary to Proposition 8.4.

Corollary 8.5 Let k ≥ 2 be an integer and let Γ be a discrete subgroup of Isom_{+}(H^{3}) which is geometrically finite and k-semifree and has infinite covolume. Then log(2k − 1) is a strong k-Margulis number for Γ.

8.5

This result can also be combined with the results from Section 7 as in the following corollary.

Corollary 8.6 Let k ≥ 2 be an integer and let N be a geometrically finite orientable hyperbolic 3-manifold of infinite volume. Suppose that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. Then log(2k − 1) is a strong k-Margulis number for N.

Proof of 8.6: We write N = H^{3}/Γ, where Γ is a geometrically finite Kleinian group. Since N has infinite volume and π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1, Corollary 7.3 guarantees that Γ ≅ π_{1}(N) is k-semifree. The assertion now follows from Corollary 8.5.

8.6

The next corollary generalizes Corollary 8.3.

Corollary 8.7 Let k ≥ 2 be an integer and let N be an orientable hyperbolic 3-manifold of finite volume. Suppose that the first betti number of N is at least k + 1 and that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. Then log(2k − 1) is a strong k-Margulis number for N.

Proof of 8.7: We write N = H^{3}/Γ, where Γ is a Kleinian group of finite covolume. It follows from Corollary 7.4 that Γ ≅ π_{1}(N) is k-semifree. To complete the proof it suffices to show that for every subgroup Γ′ of Γ whose rank is at most k, one of the hypotheses (i) or (ii) of Proposition 8.4 holds.

Since N has betti number at least k+1, the subgroup Γ′ is contained in the kernel of a surjective homomorphism β : π_{1}(N) → Z. Therefore, by Corollary E in [9], Γ′ is either geometrically finite, or N has a finite cover N̂ which fibers over the circle and Γ′ is topologically tame and contains the fiber subgroup Γ′′ of N̂. In the latter case we have Λ(Γ′) = \overline{C}, since Λ(Γ′′) = \overline{C}. Corollary 9.2 in [8] then guarantees that N′ = H^{3}/Γ′ admits no non-constant positive superharmonic functions.

8.7

Finally, by specializing some of the results stated above to the case k = 2 we obtain some sufficient conditions for log 3 to be a Margulis number for a hyperbolic 3-manifold.

§9. Geometric estimates for closed manifolds 40

Corollary 8.8 Let N = H^{3}/Γ be an orientable hyperbolic 3-manifold, such that either

(i) N is geometrically finite and has infinite volume,

(ii) N is topologically tame, purely loxodromic, and has infinite volume, or

(iii) N has finite volume and its first betti number is at least 3.

Then log 3 is a strong Margulis number for Γ.

Proof of 8.8: As we observed at the beginning of this section, a Margulis number is the same thing as a 2-Margulis number. Under the hypothesis (i), (ii) or (iii), the assertion follows respectively from Corollary 8.6, Corollary 8.2, or Corollary 8.7. The general version of each of these corollaries included the assumption that π_{1}(N) has no subgroup isomorphic to any of the groups π_{1}(S_{g}) for 2 ≤ g ≤ k − 1. For k = 2 this condition is vacuously true.

8.8

Remark 8.9 Given corollary 8.8 it seems reasonable to conjecture that log 3 is a strong Margulis number for any infinite volume hyperbolic 3-manifold. We notice that our conjecture would follow from the conjecture that every free 2-generator Kleinian group is a limit of Schottky groups. There appear to exist closed hyperbolic 3-manifolds for which log 3 is not even a Margulis number: computations by Hodgson and Weeks give strong evidence that the Weeks manifold [40] does not contain a ball of radius (log 3)/2.

9 Geometric estimates for closed manifolds

In this section we will prove the results promised in the introduction concerning balls of radius \frac{1}{2} log 5 and volume estimates for closed manifolds of betti number at least 4. This will be done by combining the results of the last section with the following result, which illustrates the use of the notion of a k-Margulis number for k > 2.

Theorem 9.1 Let N be an orientable hyperbolic 3-manifold without cusps. Suppose that π_{1}(N) is 3-free. Let λ be a 3-Margulis number for N. Then either N contains a hyperbolic ball of radius λ/2, or π_{1}(N) is a free group of rank 2.

Before giving the proof of Theorem 9.1 we shall point out how to use it to prove the corollaries stated in the introduction.

Corollary 9.2 Let N be a closed orientable hyperbolic 3-manifold. Suppose that the first betti number β_{1}(N) is at least 4, and that π_{1}(N) has no subgroup isomorphic to π_{1}(S_{2}). Then N contains a hyperbolic ball of radius \frac{1}{2} log 5. Hence the volume of N is greater than 3.08.

Proof of 9.2: According to Corollary 7.4 and Remark 7.5, the group π_{1}(N) is 3-free. According to Corollary 8.3, log 5 is a strong 3-Margulis number, and a fortiori a 3-Margulis number, for N. It therefore follows from Theorem 9.1 that either N contains a hyperbolic ball of radius \frac{1}{2} log 5 or π_{1}(N) is a free group of rank 2. The latter alternative is impossible, because Γ, as the fundamental group of a closed hyperbolic 3-manifold, must have cohomological dimension 3, whereas a free group has cohomological dimension 1. Thus N must contains a hyperbolic ball of radius \frac{1}{2} log 5.

The lower bound on the volume now follows by applying Böröczky’s density estimate for hyperbolic sphere-packings as in [14].

9.2

Let W denote the set of all finite volumes of orientable hyperbolic 3manifolds. Then W is a set of positive real numbers, and by restricting the usual ordering of the real numbers we can regard W as an ordered set. It is a theorem of Thurston’s, based on work due to Jorgensen and Gromov, that W is a well-ordered set having ordinal type ω^{ω} and that there are at most a finite number of isometry types of hyperbolic 3-manifolds with a given volume. (See [4], E.1) Thus there is a unique order-preserving bijection between W and the set of ordinal numbers less than ω^{ω}. Let us denote by v_{c} the element of W corresponding to the ordinal number c.

Corollary 9.3 Let c be any ordinal number less than 8ω and let N be any orientable hyperbolic 3-manifold with vol N = v_{c}. Then either the first betti number of N is at most 3, or π_{1}(N) contains an isomorphic copy of π_{1}(S_{2}).

Proof of 9.3: Assume that N has first betti number at least 4 and contains no isomorphic copy of π_{1}(S_{2}). Then by Corollary 9.2 we have v_{c} = vol N > 3.08. On the other hand, Weeks census (see [18] and [39]) lists 8 distinct volumes less than 3.08 for orientable manifolds with one cusp. The volume of such a cusped manifold is the limit, from below, of the volumes of its Dehn fillings (see Theorem E.7.2 in [4]). The result follows.

9.3

Corollary 9.4 Let N be a non-compact, topologically tame, orientable hyperbolic 3-manifold without cusps. Suppose (i) that π_{1}(N) is not a free group of rank 2, and (ii) that π_{1}(N) has no subgroup isomorphic to π_{1}(S_{2}). Then N contains a hyperbolic ball of radius \frac{1}{2} log 5.

Proof of 9.4: According to Corollary 8.2 and Remark 7.5, the group π_{1}(N) is 3-free. According to Corollary 8.3, log 5 is a strong 3-Margulis number, and a fortiori a 3-Margulis number, for N. It therefore follows from Theorem 9.1 (and hypothesis (ii)) that N contains a hyperbolic ball of radius \frac{1}{2} log 5.

9.4

The rest of this section is devoted to the proof of Theorem 9.1. The essential ideas of the proof appear in the proof of Theorem B in [15]. We begin by reviewing and extending a few notions from [15].

As in [15], we shall say that elements z_{1},…, z_{r} of a group Γ are independent if they freely generate a (free, rank-r) subgroup of Γ. Recall that the rank of a finitely generated group G to be the minimal cardinality of a generating set for G.

As in [15], a Γ-labeled complex, where Γ is a group, is defined to be an ordered pair (K, (X_{v})_{v}), where K is a simplicial complex and (X_{v})_{v} is a family of cyclic subgroups of Γ indexed by the vertices of K. If (K, (X_{v})_{v}) is a Γlabeled complex then for any subcomplex L of K we denote by Θ(L) the subgroup of Γ generated by all the groups X_{v}, where v ranges over the vertices of L.

In this paper we shall use one notion which appeared only implicitly in [15]. Let Γ be a group and let (K, (X_{v})_{v}) be a Γ-labeled complex. By a natural action of Γ on (K, (X_{v})_{v}) we shall mean a simplicial action of Γ on K such that for each vertex v of K we have X_{γ·v} = γX_{v}γ^{−1}. The following result could have been stated and used in [15].

Proposition 9.5 Let Γ be a finitely generated 3-free group in which every non-trivial element has a cyclic centralizer. Let (K, (X_{v})_{v}) be a Γ-labeled complex which admits a natural Γ-action. Suppose that X_{v} is a maximal cyclic subgroup of Γ for every vertex v of K. Suppose that K is connected and has more than one vertex, and that the link of every vertex of K is connected. Suppose that for every 1-simplex e of K the group Θ(|e|) is non-abelian, and that there is no 2-simplex σ of K such that Θ(|σ|) is free of rank 3. Then Θ(K) is a free group of rank 2.

Proof of 9.5: The hypotheses of the above proposition include those of Proposition 4.3 of [15]. According to the latter result, Θ(K) has local rank 2: according to the definitions given in [15], this means that every finitely generated subgroup of Θ(K) is contained in a subgroup of rank ≤ 2, but that not every finitely generated subgroup of Θ(K) is contained in a subgroup of rank ≤ 1. On the other hand, the existence of a natural action of Γ on (K, (X_{v})_{v}) clearly implies that Θ(K) is a normal subgroup of Γ.

Now choose any vertex v_{0} of K and let x_{0} denote a generator of X_{0} = X_{v_{0}}. Since x_{0} has a cyclic centralizer and X_{0} is a maximal cyclic subgroup of Γ, the element x_{0} generates its own centralizer in Γ. Now it is a special case of Proposition 4.4 of [15] that if Θ is a normal subgroup of a finitely generated 3-free group Γ, if Γ is 3-free over some finitely generated subgroup of Θ, and if Θ has local rank 2 and contains an element x_{0} which generates its own centralizer in Γ, then Γ is a free group of rank 2.

This completes the proof.

9.5

Proof of Theorem 9.1: As in [15], for any infinite cyclic group X of isometries of H^{3}, generated by a loxodromic isometry, and for any λ > 0, we denote by Z_{λ}(X) denote the set of points z ∈ H^{3} such that dist(z, ξ · z) < λ for some non-trivial element ξ of X.

Suppose that N satisfies the hypotheses of Theorem 9.1 but contains no ball of radius λ/2. We shall prove the theorem by showing that π_{1}(N) is a free group of rank 2. Let us write N = H^{3}/Γ, where Γ is a purely loxodromic Kleinian group. Then according to the discussion in subsection 3.4 of [15], the indexed family (Z_{λ}(X))_{X∈X}, where X = X_{λ}(N) denotes the set of all maximal cyclic subgroups X of Γ such that Z_{λ}(X) ≠ ∅, is an open covering of H^{3}, and the nerve K = K_{λ}(N) of this covering is a simplicial complex. By definition the vertices of K are in natural one-one correspondence with the maximal cyclic subgroups in the set X. If we denote by X_{v} ∈ X the maximal cyclic subgroup corresponding to a vertex v, then (K, (X_{v})_{v}) is a Γ-labeled complex.

We shall show that the group Γ and the Γ-labeled complex (K, (X_{v})_{v}) satisfy the hypotheses of Proposition 9.5. By the hypothesis of the theorem, Γ is 3-free. Since Γ is a purely loxodromic Kleinian group, it has the property that each of its non-trivial elements has a cyclic centralizer.

In order to construct a natural action of Γ on (K, (X_{v})_{v}), we first define an action of Γ on the set of vertices of K by X_{γ·v} = γX_{v}γ^{−1}. If v_{0},…, v_{m} are the vertices of an m-simplex of K we have

⋂_{0≤i≤m} Z_{λ}(γXγ^{−1}) = ⋂_{0≤i≤m} γ · Z_{λ}(X) = γ · ⋂_{0≤i≤m} Z_{λ}(X) ≠ ∅,

so that γ · v_{0},…, γ · v_{m} are the vertices of an m-simplex of K. Thus the action of Γ on the vertex set extends to a simplicial action on K. It is immediate from the definitions that this is a natural action on (K, (X_{v})_{v}).

By Proposition 3.4 of [15], K is a connected simplicial complex with more than one vertex, and the link of every vertex of K is connected. Now let e be any 1-simplex of K, and let v and w denote its vertices. Let x_{v} and x_{w} be generators of X_{v} and X_{w}. We have v ≠ w and hence X_{v} ≠ X_{w}; that is, the elements x_{v} and x_{w} generate distinct maximal cyclic subgroups of Γ. Since the abelian subgroups of Γ are cyclic, it follows that Θ(|e|) = ⟨x_{v}, x_{w}⟩ is non-abelian.

Finally, we claim that if σ is a 2-simplex of K, the group Θ(|σ|) cannot be free of rank 3. To prove this, let u, v and w denote the vertices of σ, and let ξ_{u}, ξ_{v} and ξ_{w} be generators of X_{u}, X_{v} and X_{w}. By the definition of the nerve K we have Z_{λ}(X_{u}) ∩ Z_{λ}(X_{v}) ∩ Z_{λ}(X_{w}) ≠ ∅. Let z be any point of Z_{λ}(X_{u}) ∩ Z_{λ}(X_{v}) ∩ Z_{λ}(X_{w}). By definition there are non-trivial elements of X_{u}, X_{v} and X_{w}, say η_{u} = ξ_{u}^{n_{u}}, η_{v} = ξ_{v}^{n_{v}} and η_{w} = ξ_{w}^{n_{w}}, such that dist(z, η_{u} · z), dist(z, η_{v} · z) and dist(z, η_{w} · z) are less than λ. Since λ is a 3-Margulis number for Γ, it follows that ⟨η_{u}, η_{v}, η_{w}⟩ is generated by at most two abelian subgroups. Now if Θ(|σ|) were free of rank 3 then ξ_{u}, ξ_{v} and ξ_{w} would be independent, and so η_{u}, η_{v} and η_{w} would also be independent. This would mean that ⟨η_{u}, η_{v}, η_{w}⟩ would be free of rank 3, and thus could not be generated by two abelian subgroups. This proves the claim.

§10. Volumes and short geodesics 45

Thus Γ and (K, (X_{v})_{v}) satisfy all the hypotheses of Proposition 9.5. Hence Γ is a free group of rank 2, as required.

Theorem 9.1

Remark 9.6 It is possible to drop the hypothesis that N has no cusps in Theorem 9.1. Because π_{1}(N) is 3-free, N could only have rank 1 cusps. The construction of the Γ-labeled complex in the proof of 9.1 can still be carried out, although the arguments in [15] must be extended to account for the fact that some of the sets Z_{λ}(X) will be horoballs instead of cylinders.

10 Volumes and short geodesics

Let C be a non-trivial closed geodesic in a closed hyperbolic 3-manifold N. Let us write N = H^{3}/Γ, where Γ is a cocompact, torsion-free, discrete group of isometries of H^{3}. Then C is the image in N of the axis A_{γ} of some non-trivial (and hence loxodromic) element γ ∈ Γ which is uniquely determined up to conjugacy. Let us set

R = \frac{1}{2} min_{δ} dist(A_{γ}, δ · A_{γ}),

where δ ranges over all elements of Γ which do not commute with γ. If we denote by Z the set of all points in H^{3} whose distance from A_{γ} is less than R, it follows from the definition of R that Z ∩ δ · Z = ∅ for every δ ∈ Γ not commuting with γ; hence the quotient Z/⟨γ⟩ embeds in N. The resulting isometric copy of Z/⟨γ⟩ in N is called the maximal embedded tube about the geodesic C, and the number R is called the radius of the tube. If the geodesic C has length l then the volume of the maximal embedded tube about C is given by the formula

πl sinh^{2} R,  (2)

which is therefore a lower bound for the volume of N.

In this section we prove the following result.

Proposition 10.1 Let N be an orientable hyperbolic 3-manifold having log 3 as a strong Margulis number. Let C be a closed geodesic in N, and let l denote its length. If R denotes the radius of the maximal embedded tube about C, we have

cosh 2R ≥ \frac{e^{2l} + 2e^{l} + 5}{(cosh \frac{l}{2})(e^{l} − 1)(e^{l} + 3)}.

Combining this with Corollary 8.8 we immediately obtain:

Corollary 10.2 Let N be an orientable hyperbolic 3-manifold of finite volume whose first betti number is at least 3. Let C be a closed geodesic in N, and let l denote its length. If R denotes the radius of the maximal embedded tube about C, we have

cosh 2R ≥ \frac{e^{2l} + 2e^{l} + 5}{(cosh \frac{l}{2})(e^{l} − 1)(e^{l} + 3)}.

The above results will also be used to give volume estimates for hyperbolic 3-manifolds containing short geodesics (see 10.3, 10.5 and 10.6 below).

Proof of 10.1: By the definition of R there is an element δ of Γ, not commuting with γ, such that the distance from A_{γ} to δ · A_{γ} = A_{δγδ^{−1}} is 2R. Let B denote the common perpendicular to the lines A_{γ} and δ · A_{γ}, and let z and w denote the points of intersection of B with A_{γ} and δ · A_{γ} respectively. Then dist(z, w) = 2R. Let us write w = δ · u where u is a point of A_{γ}. Since γ acts on A_{γ} as a translation of length l, there is an integer m such that dist(u, γ^{m} · z) ≤ l/2. Hence dist(w, δγ^{m} · z) ≤ l/2. The triangle with vertices z, w and δγ^{m} · z has a right angle at w. Writing α = dist(z, δγ^{m} · z) for the hypotenuse of this right triangle and applying the Hyperbolic Pythagorean Theorem, we obtain

cosh α = cosh 2R cosh dist(w, δγ^{m} · z) ≤ cosh \frac{l}{2} cosh 2R.  (3)

Since γ and δ do not commute, the elements γ and δγ^{m} of Γ also do not commute. Applying the definition of a strong Margulis number with ξ = γ and η = δγ^{m}, and using that α = dist(z, δγ^{m} · z) and that l = dist(z, γ · z), we obtain

\frac{1}{1 + e^{α}} + \frac{1}{1 + e^{l}} ≤ \frac{1}{2},

which we rewrite in the form

e^{α} ≥ \frac{e^{l} + 3}{e^{l} − 1}.  (4)

On the other hand, using (2) we find that

e^{α} = cosh α + sinh α
= cosh α + \sqrt{cosh^{2} α − 1}
≤ cosh 2R cosh \frac{l}{2} + \sqrt{cosh^{2} 2R cosh^{2} \frac{l}{2} − 1}.

Combining this with (3) we get

cosh 2R cosh \frac{l}{2} + \sqrt{cosh^{2} 2R cosh^{2} \frac{l}{2} − 1} ≥ \frac{e^{l} + 3}{e^{l} − 1}.  (5)

The equation

x cosh \frac{l}{2} + \sqrt{x^{2} cosh^{2} \frac{l}{2} − 1} = \frac{e^{l} + 3}{e^{l} − 1}

has the solution

x_{0} = \frac{e^{2l} + 2e^{l} + 5}{(cosh \frac{l}{2})(e^{l} − 1)(e^{l} + 3)}.

Since the function x cosh \frac{l}{2} +\sqrt{x^{2} cosh^{2} \frac{l}{2} − 1} is monotone increasing for x ≥ 1, it follows from (4) that

cosh 2R ≥ x_{0}.

This is the conclusion of Proposition 10.1.

10.1

Let us define a function V (x) for x > 0 by

V (x) = \frac{πx}{e^{x} − 1}(\frac{e^{2x} + 2e^{x} + 5}{2(cosh \frac{x}{2})(e^{x} + 3)}) − \frac{πx}{2}.

Note that

lim_{x→0} V (x) = π.

Since sinh^{2} R = \frac{1}{2}(cosh 2R − 1), Proposition 10.1 and the above formula (1) for the volume of a maximal tube now imply:

Lemma 10.3 Let N be an orientable hyperbolic 3-manifold having log 3 as a strong Margulis number. Let C be a closed geodesic in N, and let l denote its length. Then the maximal embedded tube about C has volume at least V (l).

The following result will permit us to put the information given by the above lemma in a more useful form.

Proposition 10.4 The function V (x) is monotonically decreasing for x > 0.

Proof of 10.4: For x ≥ 0 we set

f(x) = \frac{V (2x)}{π}.

We have

(cosh^{2} x)(e^{4x} + 2e^{2x} − 3)^{2}f′(x) =
(cosh x − x sinh x)(e^{4x} + 2e^{2x} − 3)(e^{4x} + 2e^{2x} + 5)
−(cosh^{2} x)(e^{4x} + 2e^{2x} − 3)^{2}
−32x(cosh x)(e^{4x} + e^{2x}).

Hence

(cosh x)(e^{4x} + 2e^{2x} − 3)^{2}f′(x) ≤ (e^{4x} + 2e^{2x} − 3)(e^{4x} + 2e^{2x} + 5)
−(e^{4x} + 2e^{2x} − 3)^{2} − 32x(e^{4x} + e^{2x})
= 8(e^{4x}(1 − 4x) + 2e^{2x}(1 − 2x) − 3).

But the function e^{4x}(1 − 4x) + 2e^{2x}(1 − 2x) − 3 is negative-valued for x > 0, because it vanishes at 0 and its derivative −xe^{4x} − 4xe^{2x} is negative for x > 0. Thus f′(x) < 0 for x > 0.

10.4

Combining Lemma 10.3 with Proposition 10.4, we immediately obtain the following result.

§11. A volume bound for non-compact manifolds 49

Corollary 10.5 Let N be an orientable hyperbolic 3-manifold having log 3 as a strong Margulis number. Let λ be a positive number, and suppose that N contains a closed geodesic of length at most λ. Then the maximal embedded tube about C has volume at least V (λ). In particular the volume of N is at least V (λ).

Corollary 10.6 Let N be an orientable hyperbolic 3-manifold which has first betti number at least 3. Let λ be a positive number, and suppose that N contains a closed geodesic of length at most λ. Then the volume of N is at least V (λ).

Proof of 10.6: We may assume that N has finite volume, as otherwise the assertion is trivial. It then follows from Corollary 8.8 that log 3 is a Margulis number for N. The assertion now follows from Corollary 10.5.

10.6

We observed above that lim_{x→0} V (x) = π. Thus Corollary 10.6 implies that if an orientable hyperbolic 3-manifold N has betti number at least 3 and contains a very short geodesic, the volume of N cannot be much less than π. Explicitly, we can say for example that if N contains a geodesic of length at most 0.1, then the volume of N is at least V (0.1) = 2.906 . . .. We already get non-trivial information from 10.3 and 10.4 if N contains a closed geodesic of length at most 1: in this case the results imply that N has volume at least V (1) = 0.956 . . .. This is greater than the smallest known volume 0.943 … of a closed orientable hyperbolic 3-manifold, which is in turn greater than the lower bound 0.92 established in [14] for the volume an arbitrary closed orientable hyperbolic 3-manifold of betti number at least 3.

In [13], Corollary 10.6 will be used as one ingredient in a proof that any orientable hyperbolic 3-manifold with betti number at least 3 has a volume exceeding that of the smallest known example, and hence that any smallestvolume orientable hyperbolic 3-manifold has betti number at most 2.

11 A volume bound for non-compact manifolds

Theorem 11.1 Let N = H^{3}/Γ be a non-compact hyperbolic 3-manifold. If N has betti number at least 4, then N has volume at least π.

Proof of 11.1: We may assume that N has finite volume. In this case N is homeomorphic to the interior of a compact 3-manifold M with non-empty boundary ∂M which consists of a finite collection of tori. Let T_{1} be a torus in ∂M and let M_{n} be the result of the (1, n) Dehn filling of M along T_{1}, in terms of some fixed system of coordinates on T_{1} . Notice that M_{n} has betti number at least 3, since N had betti number at least 4.

Thurston’s Hyperbolic Dehn Surgery Theorem (see [36]) guarantees that the interior of M_{n} admits a hyperbolic structure for all large enough n (see also Theorem E.5.1 in [4].) Let N_{n} = H^{3}/Γ_{n} be a hyperbolic manifold homeomorphic to the interior of M_{n}. Then we have vol N_{n} < vol N for all n and vol N_{n} converges to vol N (see Theorem E.7.2 in [4].) Moreover, we may assume that Γ_{n} converges geometrically to Γ (see Theorem E.6.2 in [4].)

Let γ_{n} denote an element of Γ_{n} representing the shortest closed geodesic in N_{n}. Then, since Γ_{n} converges geometrically to Γ, N has k cusps and N_{n} has k − 1 cusps (for every n), we see that l_{n} = l(γ_{n}) converges to 0 (see Theorem E.2.4 in [4]). By Corollary 10.3 we have

vol N_{n} ≥ V (l_{n}) = \frac{πl}{e^{l_{n}} − 1}(\frac{e^{2l_{n}} + 2e^{l_{n}} + 5}{2(cosh \frac{l_{n}}{2} )(e^{l_{n}} + 3)}) − \frac{πl_{n}}{2} .

Recall that V (l_{n}) converges to π, since l_{n} converges to 0. We therefore have vol N ≥ π.

11.1

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