HYPERBOLIC VOLUME AND MOD p HOMOLOGY

Marc Culler and Peter B. Shalen

University of Illinois at Chicago

Abstract. If M is a closed, orientable hyperbolic 3-manifold such that dim_{Z_{p}} H_{1}(M; Z_{p}) ≥ 5 for some prime p, then M contains a hyperbolic ball of radius (log 5)/4. There is also a related result in higher dimensions.

Introduction

In [8, Proposition 5.4] it was shown that if M is an orientable hyperbolic 3-manifold, and if for some prime p the Z_{p}-vector space H_{1}(M, Z_{p}) has dimension at least 4, then M contains a ball of radius (log 3)/4. This implies that the volume of M is greater than 0.11. In this paper we shall prove:

Theorem A. Let M be a closed, orientable hyperbolic 3-manifold. Suppose that for some prime p, the dimension of the Z_{p}-vector space H_{1}(M; Z_{p}) is at least 5. Then M contains a hyperbolic ball of radius (log 5)/4. In particular, the volume of M is greater than 0.35.

By a hyperbolic ball in a hyperbolic n-manifold M we mean an open subset of M which is path-isometric to an open ball in hyperbolic n-space H^{n}. The volume estimate in the theorem can be deduced from the existence of a hyperbolic ball of radius (log 5)/4 by using density estimates for sphere-packings as in [7] (see also [3]).

Theorem 6.1 of this paper asserts that the conclusion of Theorem A remains true under the hypothesis that any three elements of π_{1}(M) generate an infinite-index subgroup of π_{1}(M). The latter hypothesis is actually weaker than that of Theorem A; this is because, according to [8, Proposition 1.1], if k is a positive integer and M is a closed, orientable 3-manifold such that dim_{Z_{p}} H_{1}(M; Z_{p}) ≥ k + 2 for some prime p, then any k elements of π_{1}(M) generate an infinite-index subgroup of π_{1}(M). Thus Theorem A is in fact a special case of Theorem 6.1.

We will also prove a related result in higher dimensions. Recall that the rank of a finitely generated group F is defined to be the minimal cardinality of a generating set for F. A group Γ is said to be k-free, where k is a non-negative integer, if every subgroup of Γ with rank at most k is free. We have:

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 57M50; Secondary 57N10.

Both authors are partially supported by the National Science Foundation.

Theorem B. Let M be a closed hyperbolic manifold of dimension n ≥ 3. Suppose that π_{1}(M) is 3-free. Then M contains a hyperbolic ball of radius

\frac{log 5}{2(n − 1)}.

The first five sections of the paper are devoted to the proof of Theorem B. One regards the hyperbolic n-manifold M as a quotient H^{n}/Γ, where Γ is a discrete, torsion-free group of isometries of H^{n}. For each maximal cyclic subgroup X of Γ and each λ > 0 one considers the set Z_{λ}(X) consisting of all points of H^{n} that are moved a distance less than λ by some non-trivial element of X. It is an elementary observation (Proposition 3.2) that if M contains no hyperbolic ball of radius λ/2 then the non-empty sets of the form Z_{λ}(X) constitute an open covering of H^{n}. The nerve of this covering is a simplicial complex K. The geometric properties of the sets in the covering—which are fairly well-behaved neighborhoods of the axes of the corresponding cyclic subgroups—impose topological restrictions on K: it is connected, and the link of every vertex is connected.

As the sets in the covering are determined by certain maximal cyclic subgroups of Γ, the vertices of K have a natural labeling by maximal cyclic subgroups. As Γ is 3-free, the vertices of any 2-simplex of K generate a free group. However, if λ = (log 5)/(n − 1), the discreteness of Γ can be used to show that this free group is never of rank 3: this depends on Proposition 3.5, which is an elementary geometric argument based on ideas that appeared in [4] and [8]. Thus in the labeling of the vertices of K by cyclic groups, the three cyclic groups labeling the vertices of any given 2-simplex generate a free group of rank 2. Using the topological properties of K and elementary facts about free groups, one can conclude that the group generated by all the labeling cyclic groups—i.e. by all cyclic groups X for which Z_{λ}(X) ≠ ∅—is locally a free group of rank 2. By pushing the group theory a bit further one can then deduce that Γ is itself a free group of rank 2, and this is impossible as Γ is the fundamental group of a closed aspherical manifold.

In Section 1 we prove some elementary properties of the sets Z_{λ}(X), for X any cyclic group of loxodromic isometries of H^{n}. In Section 2 we prove a purely topological result about nerves of coverings of topological spaces. In Section 3 the results of the two preceding sections are combined to establish the relevant topological properties of the complex K. The proof of the geometric result alluded to above, Proposition 3.5, is also given. In Section 4 we establish the relevant facts about free groups and labeled complexes of groups. In Section 5 the results of Sections 3 and 4 are combined to give the proof of Theorem B.

Actually this is all done in a somewhat more refined setting, and gives a result, Theorem 5.1, which is more technical than Theorem B but includes it as a special case. In Section 6 we will combine Theorem 5.1 with the specifically 3-dimensional results of [3] to deduce Theorem 6.1 and hence Theorem A.

The following conventions will be used throughout the paper. The hyperbolic distance in H^{n} will be denoted dist. If S is a subset of H^{n} and r is a positive number, nbhd_{r}(P) will denote the open r-neighborhood of S, i.e. the set of all points whose minimum distance from S is strictly less than r.

If σ is a simplex in the simplicial complex K, the link of σ in K, denoted by link_{K}(σ), consists of all simplices τ such that (i) σ ∩ τ = ∅ and (ii) σ and τ span a simplex of K. The support of a simplex σ in K is the subcomplex of K consisting of σ and all its faces; it will be denoted by |σ|.

If S is a subset of a group Γ, we denote by ⟨S⟩ the subgroup of Γ generated by S. (If S = {x_{1},…, x_{r}}, we may also write ⟨x_{1},…, x_{r}⟩ for ⟨S⟩.)

Let z_{1},…, z_{r} be elements of a group Γ. We shall say that z_{1},…, z_{r} are independent if they freely generate a (free, rank-r) subgroup of Γ. (Here we regard {z_{1},…, z_{r}} as an indexed r-tuple; in particular, if two of the z_{i} coincide, then z_{1},…, z_{r} are not independent.)

We are very grateful to Sa’ar Hersonsky for helping us with our 3-dimensional hyperbolic trigonometry.

Section 1. Loxodromic isometries and displacement cylinders

1.1. Recall that an isometry x of H^{n} is loxodromic if there is an x-invariant line A(x) in H^{n}, and x acts on A(x) as a translation through some distance length x > 0. The line A(x) is unique, and is called the axis of x.

If x is an isometry of H^{n} we define a continuous non-negative-valued function D_{x} on H^{n} by D_{x}(P) = dist(P, x · P). Note that D_{x} = D_{x^{−1}}.

1.2. Suppose that x is loxodromic with length l. Let P be any point of H^{n} − A(x), and let Q denote the point of A(x) closest to P. Set P′ = x · P and Q′ = x · Q. We have ∠P QQ′ = ∠P′Q′Q = π/2, dist(Q, Q′) = l, and dist(P, Q) = dist(P′, Q′) = r, where r = r_{x}(P) denotes the perpendicular distance from P to A(x). Let θ = θ_{x}(P) denote the dihedral angle between the planes P QQ′ and P′Q′Q. Setting D = D_{x}(P) = dist(P, P′), one sees by elementary hyperbolic geometry that

cosh dist(P, P′) = cosh l + (sinh^{2} r)(cosh l − cos θ),

i.e.

(1.2.1)  cosh D_{x}(P) = cosh l + (sinh^{2} r_{x}(P))(cosh l − cos θ_{x}(P)).

This formula is clearly valid for P ∈ A(x) if we assign an arbitrary value to θ_{x}(P). In particular we recover the familiar fact that dist(P, x · P) ≥ l for every point P ∈ H^{n}, with equality if and only if P ∈ A(x).

1.3. Note that θ_{x} is constant on every ray which is perpendicular to A(x) and has its endpoint in A. If ρ is such a ray, the function r_{x}|ρ maps ρ homeomorphically onto [0, ∞) and thus defines a coordinate r on ρ. It follows from (1.2.1) that D_{x}|ρ is a strictly monotonically increasing function of r and goes to infinity with r.

(In the case where n = 3 and x preserves orientation, θ_{x} is constant on all of H^{3}. This fact will not be used in the present paper.)

1.4. Now for any loxodromic isometry x of H^{n} we define a non-negative-valued function E_{x} on H^{n} by setting E_{x}(P) = min_{d≥1} D_{x^{d}}(P) for every P ∈ H^{n}.

Proposition. Let x be a loxodromic isometry of H^{n}. Then E_{x} is continuous, and min_{P ∈H^{n}} E_{x}(P) = length x. Furthermore, for every C > 0 there exists R > 0 such that E_{x}(P) > C for every point P such that r_{x}(P) > R.

Proof. Set l = length x. Then length x^{d} = dl for every integer d > 0. Hence for each d we have min_{P ∈H^{n}} D_{x^{d}}(P) = dl, and so

min_{P ∈H^{n}} E_{x}(P) = min_{d>0} dl = l.

To show that E_{x} is continuous on H^{n} it suffices to show that it is continuous on the set H_{α} = D_{x}^{−1}([0, α)) for each α > 0. For any integer d > \frac{α}{l} we have D_{x^{d}}(P) ≥ dl > α for every P ∈ H^{n}. It follows that for any p ∈ H_{α} we have E_{x}(P) = min_{1≤d≤[ \frac{α}{l}]} D_{x^{d}}(P); since each of the functions D_{x^{d}}(P) is continuous on H^{n}, it follows that E_{x} is continuous on H_{α}.

Now let C be any positive constant, and let R be a constant such that (sinh^{2} R)(cosh l− 1) > cosh C. Since x^{d} has translation length dl, it follows from (1.2.1) that D_{x^{d}}(P) > C for every positive integer d > 0 and for every P with r_{x}(P) ≥ R. Hence E_{x^{d}}(P) > C whenever r_{x}(P) ≥ R. □

1.5. Proposition. Let x be a loxodromic isometry of H^{n}. Let ρ be any ray in H^{n} which has its endpoint in A(x) and is perpendicular to A(x); let us identify ρ isometrically with [0, ∞). Then f = E_{x}|ρ is monotonically increasing, and f(P) tends to infinity with P.

Proof. According to 1.3, the function D_{x^{d}}|ρ is strictly monotonically increasing for every d > 0. Hence f = E_{x}|ρ = min_{d≥1} D_{x^{d}}|ρ is also strictly monotonically increasing. The final assertion of Proposition 1.4 implies that f(P) tends to ∞ with P. □

For any loxodromic x and any λ > 0 we set Z_{λ}(x) = E_{x}^{−1}[0, λ).

1.6. Proposition. Let x be a loxodromic isometry of H^{n}. For any λ ≤ length x we have Z_{λ}(x) = ∅ . For any λ > length x the set Z_{λ}(x) is an open contractible neighborhood of A(x) and is contained in nbhd_{R} A(x) for some constant R > 0. Furthermore, the frontier of Z_{λ}(x) in H^{n} is the set Q_{λ}(x) = E_{x}^{−1}({λ}), and Q_{λ}(x) is homeomorphic to S^{n−2} × R.

Proof. Set l = length x. Since min_{P ∈H^{n}} E_{x}(P) = l by Proposition 1.4, we have Z_{λ}(x) = ∅ for any λ ≤ l. On the other hand, since D_{x} is identically equal to l on A(x), we have A(x) ⊂ Z_{λ}(x) for any λ > l; in view of the continuity of E_{x} it follows that Z_{λ}(x) is an open neighborhood of A(x). To show that Z_{λ}(x) is contractible in this case, we consider any ray ρ ⊂ H^{n} which has its endpoint in A(x) and is perpendicular to A(x). It follows from Proposition 1.5 that Z_{λ}(x) ∩ ρ is a half-open line segment with the same endpoint as ρ; since this holds for every such ray ρ, the contractibility of Z_{λ}(x) is clear.

It follows immediately from Proposition 1.4 that Z_{λ}(x) ⊂ nbhd_{R} A(x) for some constant R > 0.

The continuity of E_{x} implies that the frontier of Z_{λ}(x) in H^{n} is contained in the set Q_{λ}(x) = E_{x}^{−1}({λ}). To prove the reverse inclusion, we consider any point P ∈ Q_{λ}(x), and we let ρ_{P} denote the unique ray in H^{n} which has its endpoint in A, is perpendicular to A(x) and contains P. According to 1.5, the function f = E_{x}|ρ_{P} is strictly monotonically increasing. Since f(P) = λ, the monotonicity of f implies that P lies in the frontier relative to ρ_{P} of the set f^{−1}[0, λ) = Z_{λ}(x) ∩ ρ_{P}. In particular, P lies in the frontier of Z_{λ}(x) in H^{n}.

It remains to show that Q_{λ}(x) is homeomorphic to S^{n−2} × R. For this purpose we consider the set Q^{∗}(x) ⊂ H^{n} consisting of all points whose perpendicular distance from A(x) is 1. If P is any point of Q_{λ}(x), and ρ_{P} is defined as above, then ρ_{P} ∩ Q^{∗}(x) consists of a single point which we denote h(P). This defines a continuous map h : Q_{λ}(x) → Q^{∗}(x). It follows immediately from Proposition 1.5 that h is a bijection. On the other hand, since we have shown that Z_{λ}(x) ⊂ nbhd_{R} A(x) for some constant R > 0, it is clear that every compact subset of Q^{∗}(x) has bounded pre-image under h. But Q_{λ}(x) is closed in H^{n} since it is the frontier of Z_{λ}(x). Thus Q_{λ}(x) is locally compact and h is a proper map. It follows that h is a homeomorphism. Since Q^{∗}(x) is clearly homeomorphic to S^{n−2} × R, this completes the proof. □

We remark that in the case that n = 3 and x preserves orientation we have Z_{λ}(x) = nbhd_{R} A(x) for some R > 0. This fact will not be used in the present paper.

1.7. For any loxodromic isometry x of H^{n}, it follows from 1.1 that D_{x^{d}} = D_{x^{−d}} for every d > 0. This implies that E_{x} = E_{x^{−1}}, and hence that Z_{λ}(x) = Z_{λ}(x^{−1}) for every λ > 0. Hence if X is any infinite cyclic subgroup of Γ with a loxodromic generator x we may unambiguously write Z_{λ}(X) = Z_{λ}(x) for any positive number λ. This notation will be used extensively in the next two sections.

Section 2. Nerves and connectedness

2.1. By an open covering of a topological space H we shall mean an indexed family (U_{i})_{i∈I} of non-empty open sets in H such that ⋃_{i∈I} U_{i} = H. Note that we may have U_{i} = U_{j} for distinct indices i and j. The nerve of the covering (U_{i})_{i∈I} is an abstract simplicial complex with an indexed vertex set (v_{i})_{i∈I}, where v_{i} = v_{j} if and only if i = j. A collection {v_{i_{0}},…, v_{i_{k}}} of vertices, where i_{0},…, i_{k} are distinct indices in I, spans a k-simplex if and only if U_{i_{0}} ∩ ⋯ ∩ U_{i_{k}} ≠ ∅.

If H is connected, any open covering of H has connected nerve. (This depends on our requirement that the sets in an open covering be non-empty.)

2.2. Proposition. Let (U_{i})_{i∈I} be a covering of a topological space H. Suppose that (i) for every i ∈ I the set U_{i} is connected and has connected frontier, and (ii) for any two distinct indices i, j ∈ I we have U_{i} ⊄ U_{j}. Then the link of every vertex in the nerve of (U_{i})_{i∈I} is connected.

Proof. Let K denote the nerve of (U_{i})_{i∈I}. Suppose that we are given a vertex of K, say v_{s} for some s ∈ I. Set C = link_{K} v_{s}. We are required to show that C is a connected simplicial complex. Let us write the set of vertices of C as an indexed set (v_{i})_{i∈J}, where J is a subset of I. We have j ∈ J if and only if v_{s} and v_{j} span a 1-simplex of K; by the definition of the nerve, this is equivalent to saying that j ≠ s and U_{j} ∩ U_{s} ≠ ∅.

We denote by Q the frontier of U_{s} in H. By hypothesis (i), Q is a connected space. Consider the indexed family (U_{j} ∩ Q)_{j∈J} of open sets in Q. We claim that this family is an open covering of Q.

First we must show that U_{j} ∩ Q ≠ ∅ for any j ∈ J. Since for j ∈ I we have j ≠ s, hypothesis (ii) implies that U_{j} ⊄ U_{s}. Since U_{j} is connected by hypothesis (i), and since U_{j} ∩ U_{s} ≠ ∅ when j ∈ J, it follows that U_{j} meets the frontier Q of U_{s} as required. Next we must show that Q = ⋃_{j∈J}(Q ∩ U_{j}), i.e. that Q ⊂ ⋃_{j∈J}(U_{j}). Given any point q ∈ Q, we have q ∈ U_{j} for some j ∈ I; since q ∉ U_{s} we have s ≠ j. But since q ∈ \overline{U}_{s} we must have U_{j} ∩ U_{s} ≠ ∅. This shows that j ∈ J, and completes the proof that (U_{j} ∩ Q)_{j∈J} is an open covering of Q.

Let E denote the nerve of the covering (U_{j} ∩ Q)_{j∈J}. Let w_{j} denote the vertex of E corresponding to the index j ∈ J. If w_{j_{0}},…, w_{j_{k}} span a k-simplex of E then (Q ∩ U_{j_{0}}) ∩ ⋯ ∩ (Q ∩ U_{j_{k}}) ≠ ∅; hence in particular, (U_{j_{0}} ∩ ⋯ ∩ U_{j_{k}}) ∩ U_{s} ≠ ∅, so that v_{j_{0}},…, v_{j_{k}}, v_{s} span a (k + 1)-simplex of K. This means that v_{j_{0}},…, v_{j_{k}} span a k-simplex of C. This shows that E is simplicially isomorphic to a subcomplex of C containing all the vertices of C.

Since Q is connected, the nerve E of the covering (U_{j} ∩ Q)_{j∈J} is connected. Thus there is a connected subcomplex of C containing all the vertices of C. It follows that C is itself connected. □

Section 3. Discrete groups and coverings of hyperbolic space

3.1. In this section, M will denote a closed hyperbolic manifold of some dimension n ≥ 2. We may regard M as the quotient of H^{n} by a co-compact, discrete, torsion-free group Γ of isometries. We recall some elementary properties of Γ. Since Γ is co-compact, each non-trivial element x of Γ is loxodromic. The centralizer C(x) of x is cyclic and consists of all elements having the same axis as x. In particular C(x) is the unique maximal cyclic subgroup containing x. For two non-trivial elements x and y of Γ we have C(x) = C(y) if and only if x and y commute, or equivalently if and only if A(x) = A(y). Thus there is a natural one-one correspondence between maximal cyclic subgroups of Γ and axes of elements of Γ.

3.2. Proposition. Suppose that λ is a positive number such that M contains no hyperbolic ball of radius \frac{λ}{2}. Then we have

H^{n} = ⋃_{X} Z_{λ}(X),

where X ranges over all maximal cyclic subgroups of Γ.

Proof. Let P be any point of H^{n}. The hypothesis that M contains no hyperbolic ball of radius \frac{λ}{2} implies that dist(P, x_{0} · P) < λ for some x_{0} ∈ Γ − 1. (Indeed, if dist(P, x · P) ≥ λ for every x ∈ Γ − {1} then by the triangle inequality B = nbhd_{λ/2}(P) is disjoint from x·B = nbhd_{λ/2}(x·P) for every x ∈ Γ−{1}; hence the covering projection maps B injectively into M, and its image is a hyperbolic ball of radius \frac{λ}{2}.) Now by 3.1, X_{1} = C(x_{0}) is a maximal cyclic subgroup of Γ, and x_{0} is a positive power of some generator x_{1} of X_{1}. By the definitions we have E_{x_{1}}(P) ≤ D_{x_{0}}(P) = dist(P, x_{0} ·P) < λ, so that P ∈ Z_{x_{1}}(P) = Z_{X_{1}}(P). Since P ∈ H^{n} was arbitrary, the conclusion of the lemma follows. □

3.3. Proposition. Suppose that X and X′ are maximal cyclic subgroups of Γ, and suppose that for some λ > 0 we have ∅ ≠ Z_{λ}(X) ⊂ Z_{λ}(X′). Then X = X′.

Proof. Let x and x′ be generators of X and X′ respectively, and set A = A_{x}, A′ = A_{x′}. By Proposition 1.6 we have Z_{λ}(X′) ⊂ nbhd_{R} A′ for some R > 0. Hence Z_{λ}(X) ⊂ nbhd_{R} A′. Now let \overline{H}^{n} denote the union of H^{n} with the sphere at infinity S_{∞}^{n−1}. We give \overline{H}^{n} the natural topology, in which it is homeomorphic to a closed n-ball. Let \overline{Z} denote the closure of Z_{λ}(X) in \overline{H}^{n}. Since Z_{λ}(X) ⊂ nbhd_{R} A′, we have \overline{Z} ∩ S_{∞}^{n−1} ⊂ A′ ∩ S_{∞}^{n−1} = \{P, Q\}, where P and Q are the fixed points of x′ in S_{∞}^{n−1}. Thus {P, Q} is invariant under x. Hence x^{2} fixes P and Q. Since x^{2} is loxodromic with axis A it follows that P and Q are the endpoints of A and hence that A = A′. By 3.1 this implies X = X′. □

3.4. Suppose that λ is a positive number such that M contains no hyperbolic ball of radius λ/2. Let X = X_{λ}(M) denote the set of all maximal cyclic subgroups X of Γ such that Z_{λ}(X) ≠ ∅. Proposition 3.2 implies that the indexed family (Z_{λ}(X))_{X∈X} is an open covering of H^{n} (see 2.1). We will denote the nerve of this covering by K_{λ}(M).

Proposition. Let λ be a positive number such that M contains no hyperbolic ball of radius λ/2. Then K = K_{λ}(M) is a connected complex with more than one vertex, and the link of every vertex of K is connected.

Proof. Since H^{n} is connected, the nerve of any open covering of H^{n} is connected. Set X = X_{λ}(M) and K = K_{λ}(M). If K had only one vertex, we would have H^{n} = Z_{λ}(X) for some X ∈ X. This is impossible since by Proposition 1.6 we have Z_{λ}(X) ⊂ nbhd_{R}(A) where A is the axis of a generator of X and R is some positive number. To show that the link of every vertex of K is connected we apply Proposition 2.2. According to Proposition 1.6, for each X ∈ X the set Z_{λ}(X) is contractible and hence connected, and its frontier is homeomorphic to S^{n−2} × R and is therefore connected since n ≥ 3. Thus hypothesis (i) of Proposition 2.2 holds. That hypothesis (ii) holds is precisely the content of Proposition 3.3. □

3.5. As we explained in the introduction, the following result is the basic geometric fact underlying the proofs of Theorems A and B. The proof is a slight variant of the proof of [8, Proposition 5.2 and Corollary 5.3]; see also [4].

Proposition. Let x_{1},…, x_{r} be independent elements of Γ. Set

λ = \frac{log(2r − 1)}{n − 1}.

Then Z_{λ}(x_{1}) ∩ ⋯ ∩ Z_{λ}(x_{r}) = ∅.

Proof. Suppose that P is a point of Z_{λ}(x_{1}) ∩ ⋯ ∩ Z_{λ}(x_{r}). For each i ∈ {1,…, r} we have E_{x_{i}}(P) < λ, and hence D_{y_{i}}(P) < λ for some positive power y_{i} = x_{i}^{d_{i}} of x_{i}. Clearly y_{1},…, y_{r} are independent. We fix a number λ′ < λ such that dist(P, y_{i} ·P) = D_{y_{i}}(P) < λ′ for i = 1,…, r. It then follows by induction on m ≥ 1, using the triangle inequality and the fact that the y_{i} are isometries, that if γ ∈ Γ is given by a word of length m in y_{1},…, y_{r} then dist(P, γ · P) < mλ′.

For each m ≥ 1, let S_{m} denote the set of all elements of Γ that are expressible as reduced words of length m in y_{1},…, y_{r}. Since y_{1},…, y_{r} are independent, S_{m} has cardinality exactly (2r)(2r − 1)^{m−1}. Let b be an open ball about P such that γ · b ∩ b = ∅ for every γ ∈ Γ − {1}. Let ρ denote the radius of b, and v its volume. Then the balls γ · b for γ ∈ S_{m} are pairwise disjoint and are contained in nbhd_{mλ′+ρ}(P). Hence

(2r)(2r − 1)^{m−1}v ≤ vol nbhd_{mλ′+ρ}(P) < C exp (n − 1)(mλ′ + ρ),

where C is a constant depending only on the dimension n. Hence

(2r − 1)^{m} < C′ exp (n − 1)mλ′,

where C′ is a constant depending on n and ρ but independent of m. If in the last inequality we take logarithms of both sides, divide by m and take limits as m → ∞, we obtain log(2r − 1) ≤ (n − 1)λ′, which is impossible since

λ′ < λ = \frac{log(2r − 1)}{n − 1}.

□

Section 4. Structure of 3-free groups

4.1. Let W be a subgroup of a group Γ, and let k be positive integer. We shall say that Γ is k-free over W if every subgroup of Γ which contains W and has rank ≤ k is free (of some rank ≤ k). Note that a group is k-free if and only if it is k-free over the trivial subgroup; and that a k-free group is k-free over every subgroup.

A group Γ will be said to have local rank ≤ k, where k is a positive integer, if every finitely generated subgroup of Γ is contained in a subgroup of rank ≤ k. The local rank is the smallest integer k with this property, and is defined to be ∞ if no such integer exists. Note that for a finitely generated group, the local rank is equal to the rank.

The following result plays the role of an induction step in the proofs of the two main results of this section, Propositions 4.3 and 4.4.

4.2. Lemma. Let x, y and z be elements of a group Γ. Suppose that x and y do not commute, and that x, y and z are not independent. Let A be a subgroup of Γ which contains x and y and has local rank 2. Suppose that Γ is 3-free over some finitely generated subgroup J of A. Then the group ⟨A ∪ {z}⟩ is also of local rank 2.

Proof. Since ⟨A ∪ {z}⟩ contains the non-commuting elements x and y, it must have local rank > 1. We must show that it has local rank ≤ 2.

Set B = ⟨A ∪ {z}⟩. Let B_{0} be any finitely generated subgroup of B. Then there is a finitely generated subgroup A_{1} of A such that B_{0} is contained in the subgroup B_{1} = ⟨{z} ∪ A_{1}⟩. After possibly replacing A_{1} by a larger finitely generated subgroup we may assume that x, y ∈ A_{1} and that J ≤ A_{1}. Since A has local rank ≤ 2, we may assume after further enlarging A_{1} that A_{1} has rank ≤ 2. Hence B_{1} has rank at most 3. Since J ≤ B_{1}, and since Γ is 3-free over J, it follows that B_{1} is free of some rank at most 3. We claim that B_{1} cannot have rank 3. This will imply that B_{1} ≥ B_{0} has rank ≤ 2, and will complete the proof that B has local rank ≤ 2.

Assume that B_{1} is free of rank 3. Since A_{1} has rank ≤ 2 it is generated by two elements u and v. Then u, v and z generate the rank-3 free group B_{1}. Hence by [6, p. 59], B_{1} is freely generated by these three elements. Thus we may regard B_{1} as a free product A_{1} ∗ ⟨z⟩. Now T = ⟨x, y⟩ ≤ B_{1} is free by the Nielsen-Schreier theorem, and has rank ≤ 2; since x and y do not commute, T must be free of rank exactly 2, and must therefore be free on x and y. But since B_{1} has been identified with a free product A_{1} ∗ ⟨z⟩, the subgroup ⟨x, y, z⟩ = ⟨T ∪{z}⟩ is identified with a free product T ∗⟨z⟩, and is therefore freely generated by x, y and z. This is a contradiction since x, y and z are not independent. □

4.3. Let Γ be a group. By a Γ-labeled complex we shall mean 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 Γ-labeled complex then for any subcomplex L of K we shall denote by Θ(L) the subgroup of Γ generated by all the groups X_{v}, where v ranges over the vertices of L.

Proposition. Let (K, (X_{v})_{v}) be a Γ-labeled complex. 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 Γ is 3-free over Θ(|e|). Suppose also that there is no 2-simplex σ of K such that Θ(|σ|) is free of rank 3. Then Θ(K) has local rank 2.

Proof. We shall say that a subcomplex L of K is good if (i) L is connected and contains more than one vertex, (ii) Θ(L) has local rank 2, and (iii) Γ is 3-free over some finitely generated subgroup of Θ(L).

If e is any 1-simplex of K, then Θ(|e|) is by definition generated by two elements, and the hypothesis of the lemma implies that Θ(|e|) is non-abelian; in particular Θ(|e|) has local rank 2. The hypothesis also implies that Γ is 3-free over Θ(|e|). Thus |e| is a good subcomplex of K. It now follows from Zorn’s Lemma that there exists a maximal good subcomplex L_{0} of K. We shall complete the proof by showing that L_{0} = K.

4.3.1. Claim. The complex L_{0} is a full subcomplex of K. (This means that any simplex whose vertices lie in L_{0} is itself a simplex of L_{0}.)

To prove this claim, suppose that σ is a simplex whose vertices lie in L_{0}. Set L = L_{0}∪|σ|. It is clear that L satisfies condition (i) of the definition of a good complex, since L_{0} does, and that Θ(L) = Θ(L_{0}). Hence L is good; by the maximality of L_{0} we have L_{0} = L, so that σ ∈ L_{0}. This proves Claim 4.3.1.

4.3.2. Claim. If e is any 1-simplex of L_{0}, then link_{K}(e) ⊂ L_{0}.

To prove this claim, let u and v denote the vertices of e, and let w be any vertex in the link of e. Let x_{u}, x_{v} and x_{w} be generators of X_{u}, X_{v} and X_{w} respectively. The vertices u, v and w span a 2-simplex σ. We set L = L_{0} ∪ |σ|. We shall show that L is good; by the maximality of L_{0} this means that L = L_{0}, so that w ∈ L_{0}. Since L_{0} is full in K, the claim will then follow.

Condition (i) of the definition of a good complex is clear. Condition (iii) is also clear since Θ(L) ≥ Θ(L_{0}). To verify condition (ii), note that Θ(L) = ⟨Θ(L_{0}) ∪ {x_{w}}⟩. We shall apply Lemma 4.2, with A = Θ(L_{0}) and with x = x_{u}, y = x_{v}, z = x_{w}, to show that Θ(L) has local rank 2.

By the hypothesis of the proposition, the group Θ(|e|) = ⟨x_{u}, x_{v}⟩ is non-abelian; that is, x_{u} and x_{v} do not commute. Since L_{0} is good, Γ is 3-free over some finitely generated subgroup J of Θ(L_{0}). Finally, since u, v and w span a 2-simplex σ, the hypothesis of the proposition implies that Θ(σ) = ⟨x_{u}, x_{v}, x_{w}⟩ is not free of rank 3, and so x_{u}, x_{v} and x_{w} are not independent. It now follows from Lemma 4.2 that Θ(L) = ⟨Θ(L_{0}) ∪ {x_{w}}⟩ has local rank at most 2, and the proof of Claim 4.3.2 is complete.

We now proceed to the proof that L_{0} = K, which will complete the proof of the proposition. Note that since K is connected by the hypothesis of the proposition, and since L_{0} is full and non-empty(!), we need only prove that for any vertex v_{0} of L_{0} we have link_{K}(v_{0}) ⊂ L_{0}. Set C = link_{K}(v_{0}) and D = C ∩ L_{0}; we must show that D = C. Since L_{0} is connected and contains more than one vertex, we must have D ≠ ∅. Note also that D is a full subcomplex of C since L_{0} is a full subcomplex of K. But C is also connected by the hypothesis of the proposition. Hence in order to prove that D = C we need only prove that for any vertex v ∈ D we have link_{C}(v) ⊂ D. If e denotes the 1-simplex joining v_{0} to v, we have link_{C}(v) = link_{K}(e) ⊂ L_{0} by Claim 4.3.2, and hence link_{C}(v) ⊂ D as required. □

4.4. Proposition. Let Θ be a normal subgroup of a finitely generated group Γ. Suppose that Θ has local rank 2, and that Γ is 3-free over some finitely generated subgroup of Θ. Suppose also that Θ contains an element x_{0} with the property that for every element γ ∈ Γ which is not a power of x_{0}, the element γx_{0}γ^{−1} does not commute with x_{0}. Then Γ is a free group of rank 2.

Proof. Let γ_{1},…, γ_{r} be a finite generating set for Γ, and set Θ_{k} = ⟨Θ ∪ {γ_{1},…, γ_{k}}⟩ for k = 0,…, r. (In particular Θ_{0} = Θ.)

4.4.1. Claim. The group Θ_{k} has local rank 2 for k = 0,…, r.

By hypothesis, this claim holds for k = 0. We proceed by induction on k. Suppose that 0 < k ≤ r and that Θ_{k−1} has local rank 2. Since Θ is normal in Γ and contains x_{0}, the elements x_{0} and γ_{k}x_{0}γ_{k}^{−1} belong to Θ and hence to Θ_{i−1}. We now wish to apply Lemma 4.2, taking x = x_{0}, y = γ_{k}x_{0}γ_{k}^{−1}, z = γ_{k} and A = Θ_{k−1}. By hypothesis Γ is 3-free over some finitely generated subgroup J of Θ. It is obvious that x, y and z are not independent. Thus if x and y do not commute, Lemma 4.2 guarantees that Θ_{k} = ⟨Θ_{k−1} ∪ {γ_{k}}⟩ has local rank ≤ 2, and the induction is complete in this case.

There remains the case in which x = x_{0} and y = γ_{k}x_{0}γ_{k}^{−1} commute. In this case, the property of x_{0} given in the hypothesis of the theorem implies that γ_{k} is a power of x_{0}. But in this case we have γ_{k} ∈ Θ ≤ Θ_{k−1}, so that Θ_{k} = Θ_{k−1}, and the induction step is trivial. The proof of Claim 4.4.1 is therefore complete.

It is clear from the definition of the Θ_{k} that Θ_{r} = Γ. Applying Claim 4.4.1 with k = r we conclude that Γ has local rank at most 2. Since Γ is finitely generated this means rank Γ ≤ 2 < 3. But by hypothesis Γ is 3-free over some finitely generated subgroup of Θ. Hence Γ is a free group. Since its local rank is 2, it is in fact free of rank 2. □

Section 5. The proof of Theorem B

5.1. The goal of this section is to prove the following theorem.

Theorem. Let M be a closed hyperbolic manifold of dimension n ≥ 3. Let us write M = H^{n}/Γ, where Γ is a co-compact, discrete, torsion-free group of isometries of H^{n}. Set

λ = \frac{log 5}{n − 1},

and suppose that the following condition holds: (⋆) If x and y are non-commuting elements of Γ such that Z_{λ}(x) ∩ Z_{λ}(y) ≠ ∅, then Γ is

3-free over ⟨x, y⟩. Then M contains a hyperbolic ball of radius λ/2.

5.2. As we observed in 4.1, a group which is k-free is k-free over any subgroup. Hence condition (⋆) of Theorem 5.1 always holds if Γ ≅ π_{1}(M) is 3-free. Thus Theorem B of the Introduction is a special case of Theorem 5.1. In Section 6 we will show how to deduce Theorem A of the Introduction from Theorem 5.1.

Proof of Theorem 5.1. Suppose that M satisfies the hypotheses of Theorem 5.1 but contains no ball of radius λ/2, where

λ = \frac{log 5}{n − 1}.

Then in the notation of 3.4 we have a covering (Z_{λ}(X))_{X∈X} of H^{n} with index set X = X_{λ}(M) and nerve K = K_{λ}(M). 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 in the sense of 4.3.

We shall show that (K, (X_{v})_{v}) satisfies the hypotheses of Proposition 4.3. By Proposition 3.4, 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}; hence by 3.1 the elements x_{v} and x_{w} do not commute, and the group Θ(|e|) = ⟨x_{v}, x_{w}⟩ is non-abelian. On the other hand, by the definition of the nerve K we have Z_{λ}(X_{v}) ∩ Z_{λ}(X_{w}) ≠ ∅, and so the hypothesis of the Theorem implies that Γ is 3-free over ⟨x_{v}, x_{w}⟩. Finally, let σ be any 2-simplex of K, and let u, v and w denote its vertices. Let x_{u}, x_{v} and x_{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}) ≠ ∅. Hence by Proposition 3.5, the elements x_{u}, x_{v} and x_{w} are not independent. By [6, p. 59] this means that Θ(|σ|) = ⟨x_{u}, x_{v}, x_{w}⟩ is not a free group of rank 3.

Thus Proposition 4.3 applies and we conclude that Θ(K) has local rank 2. We claim that Θ = Θ(K) in fact satisfies all the hypotheses of Proposition 4.4. To show that Θ is normal, observe that by definition Θ is generated by all the maximal subgroups in X. If a maximal cyclic subgroup X belongs to X, i.e. if Z_{λ}(X) ≠ ∅, then for any γ ∈ Γ we have Z_{λ}(γXγ^{−1}) = γ · Z_{λ}(X) ≠ ∅. Thus Θ is a normal subgroup of Γ.

We saw above that for any edge |e| of K the group Γ is 3-free over the 2-generator subgroup Θ(|e|) of Θ(K). The only hypothesis of Proposition 4.4 left to check is the existence of the element x_{0}. We take x_{0} to be a generator of any group X_{0} ∈ X. If γ is an element of Γ such that x = x_{0} and y = γ_{k}x_{0}γ_{k}^{−1} commute, then by 3.1, the elements x_{0} and γ_{k}x_{0}γ_{k}^{−1} generate the same maximal cyclic subgroup of Γ, so γ_{k}x_{0}γ_{k}^{−1} = x_{0}^{±1}. Hence γ_{k}^{2} commutes with x_{0}. Thus γ_{k} and x_{0} belong to C(γ_{k}^{2}), which by 3.1 is a maximal cyclic subgroup containing x_{0} and is therefore generated by x_{0}. Hence γ_{k} is a power of x_{0}.

It now follows from Lemma 4.4 that Γ is a free group of rank 2. However, this is impossible, because Γ, as the fundamental group of a closed hyperbolic n-manifold, must have cohomological dimension n ≥ 3, whereas a free group has cohomological dimension 1. This contradiction completes the proof of Theorem 5.1. □

Section 6. The proof of Theorem A

6.1. We shall prove the following result.

Theorem. Let M be a closed orientable hyperbolic 3-manifold. Suppose that every subgroup of π_{1}(M) whose rank is at most 3 is of infinite index in π_{1}(M). Then M contains a hyperbolic ball of radius (log 5)/4.

6.2. Now recall the statement of [8, Proposition 1.1]. Let M be a closed 3-manifold, let p be a prime number, and let k be a positive integer. Suppose that either M is orientable or p = 2. Suppose that the Z_{p}-vector space H_{1}(M; Z_{p}) has dimension at least k + 2. Then every subgroup of π_{1}(M) having rank ≤ k is of infinite index.

In particular, if M is a closed, orientable, hyperbolic 3-manifold, and if H_{1}(M; Z_{p}) has dimension at least 5 for some prime p, then every subgroup of π_{1}(M) having rank ≤ 3 is of infinite index. Combining this with Theorem 6.1 we obtain Theorem A of the Introduction.

6.3. It remains to give the Proof of Theorem 6.1. We can write M = H^{3}/Γ, where Γ is a co-compact, discrete, torsionfree group of orientation-preserving isometries of H^{3}. Recall that since Γ is co-compact, every non-trivial element of Γ is loxodromic. In particular Γ contains no parabolic elements. Set

λ = \frac{log 5}{2}.

We wish to apply Theorem 5.1 to conclude that M contains a hyperbolic ball of radius λ/2. It suffices to show that condition (⋆) of 5.1 holds.

Suppose that x and y are non-commuting elements of Γ such that Z_{λ}(x) ∩ Z_{λ}(y) ≠ ∅. We must show that Γ is 3-free over ⟨x, y⟩. Let Θ be a subgroup of rank ≤ 3 containing x and y. We are required to prove that Θ is free.

The hypothesis of the theorem guarantees that Θ has infinite index in Γ ≅ π_{1}(M). Thus Θ is not co-compact.

Let us choose a point P_{0} ∈ Z_{λ}(x) ∩ Z_{λ}(y). By the definition of Z_{λ}(x) and Z_{λ}(y) there exist positive integers a and b such that D_{x^{a}}(P_{0}) < λ and D_{y^{b}}(P_{0}) < λ; that is,

max(dist(P_{0}, x^{a}(P_{0})), dist(P_{0}, y^{b}(P_{0}))) < λ.

We observe that x^{a} and y^{b} do not commute. Indeed, it follows from 3.1 that C(x^{a}) = C(x) and that C(y^{b}) = C(y). Hence if x^{a} and y^{b} were to commute then x and y would also commute, which they do not.

Before showing that Θ is free we will show that it is freely decomposable, i.e. that it is a free product of two non- trivial subgroups. Assume to the contrary that Θ is freely indecomposable. According to [1], if Θ is any freely indecomposable, discrete, torsion-free group of orientation- preserving isometries of H^{3}, then Θ is topologically tame, that is, the quotient hyperbolic 3-manifold H^{3}/Θ is homeomorphic to the interior of a compact 3-manifold with boundary. Furthermore, according to [2, Proposition 3.2], if Θ is any non-co-compact, topologically tame, discrete, torsion-free group of orientation-preserving isometries of H^{3}, then any finitely generated subgroup of Θ is topologically tame.

In particular, ⟨x^{a}, y^{b}⟩ is topologically tame. Of course, since Θ is non-co-compact, ⟨x^{a}, y^{b}⟩ is also non-co-compact. We now recall the statement of the main theorem of [3]. Let ξ and η be non-commuting orientation-preserving isometries of H^{3}. Suppose that ⟨ξ, η⟩ is discrete, torsion-free, topologically tame and non-co-compact, and contains no parabolic elements. Then for any point P ∈ H^{3} we have

max(dist(P, ξ(P)), dist(P, η(P))) ≥ log 3.

As we have checked the hypotheses of this statement for ξ = x^{a} and η = y^{b} we now have

max(dist(P_{0}, x^{a}(P_{0})), dist(P_{0}, y^{b}(P_{0}))) ≥ log 3.

Since

λ = \frac{log 5}{2} < log 3,

we have a contradiction. This proves that Θ is freely decomposable.

Thus we may write Θ = Θ_{1} ∗ Θ_{2}, where the Θ_{i} are non-trivial. By Grushko’s theorem [9] we have rank Θ_{1} + rank Θ_{2} = rank Θ ≤ 3, and hence each Θ_{i} has rank at most 2. But each Θ_{i} has infinite index in Γ = π_{1}(M), since Θ does; and it follows from [5, Theorem VI.4.1] that any infinite-index subgroup of rank ≤ 2 in the fundamental group of the closed, orientable hyperbolic 3-manifold M is free. It follows that Θ_{1} and Θ_{2} are free, and hence that Θ is free also. This completes the proof. □

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Department of Mathematics (M/C 249), University of Illinois, 851 South Morgan Street Chicago, IL 60607-7045