VOLUMES OF HYPERBOLIC HAKEN MANIFOLDS, II

Marc Culler and Peter B. Shalen

University of Illinois at Chicago

Abstract. We show that if M is a closed hyperbolic 3-manifold and if π_{1}(M) has a nonabelian free quotient, then the volume of M is greater than 0.92. If, in addition, π_{1}(M) contains no genus 2 surface groups, then the volume of M is greater than 1.02. Using these results we show that if there are infinitely many primitive homology classes in H_{2}(M; ℤ) which are not represented by fibroids, then the volume of M is greater than 0.83.

Introduction

In the paper [CHS] it is shown that if M is a closed hyperbolic 3-manifold of minimal volume then the first Betti number of M is at most 2. A lower bound for the volume of a manifold with first Betti number 2 was given in [CS2]. In this paper we give stronger lower bounds which apply to certain classes of hyperbolic 3-manifolds, including many manifolds with first Betti number 2.

Our first lower bound, given by Theorem 1.3, applies to any closed hyperbolic 3-manifold whose fundamental group has a non-abelian free quotient. Theorem 1.3 asserts that if π_{1}(M) has a non-abelian free quotient then M contains a ball of radius (log 3)/2. Estimates for the density of a sphere-packing in hyperbolic space then imply that the volume of M is greater than 0.92. We also point out, in Theorem 1.4, that the conclusion of Theorem 1.3 can be strengthened under the additional hypothesis that π_{1}(M) has no subgroup isomorphic to the fundamental group of a genus-2 surface. Specifically, if π_{1}(M) has a non-abelian free quotient and contains no genus-2 surface groups, then the volume of M is greater than 1.02. An even stronger estimate applies if the free quotient has rank > 2.

Our second main result applies to many manifolds of Betti number 2. To indicate the statement, we recall some definitions from [CS3]. A compact, connected, orientable topological 3-manifold (with boundary) N is called a book of I-bundles if it has the form N = E ∪ V, where

(i) E is an I-bundle over a non-empty compact 2-manifold-with-boundary B; (ii) each component of V is homeomorphic to D^{2} × S^{1}; (iii) the set A = E ∩V is the inverse image of ∂B under the bundle projection E → B;

and (iv) each component of A is an annulus in ∂V which is homotopically non-trivial in

V.

Typeset by AMS-TEX

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

Both authors are partially supported by the National Science Foundation.

A fibroid in a closed orientable 3-manifold M is a closed incompressible surface such that each component of the manifold obtained by splitting M along S is a book of I-bundles. (This definition is slightly more general than the one used in [CS3]. A “fibroid” in the sense of [CS3] is a fibroid in the above sense which is connected.) A corollary to one of the main results of this paper, Corollary 3.4, asserts that if M is a closed, orientable hyperbolic 3-manifold whose first Betti number β_{1}(M) is at least 2 and if M contains no non-separating, connected fibroid, then the volume of M is greater than 0.83. The hypothesis that M contains no non-separating, connected fibroid is to be regarded as a slightly stronger version of the condition that M not be a fibered 3-manifold.

The theorem from which Corollary 3.4 is deduced, Theorem 3.3, asserts that the same volume estimate holds under more general conditions. An element of an abelian group A will be termed primitive if it cannot be written in the form nx with x ∈ A and n > 1. Theorem 3.3 asserts that if there are infinitely many primitive homology classes in H_{2}(M; ℤ) which are not represented by connected fibroids then the volume of M is greater than 0.83.

Theorem 3.3 and Corollary 3.4 are interesting only when β_{1}(M) = 2. If β_{1}(M) < 2 then H_{2}(M; ℤ) contains at most two primitive elements, and if β_{1}(M) > 2 then the results of [CS1] and [CHS] give larger lower bounds for the volume of M.

The hypothesis of Theorem 3.3 is related to the norm on H_{2}(M; ℝ) constructed by Thurston in [T]. We regard H_{2}(M; ℤ) as the integer lattice in H_{2}(M; ℝ). In the case that M is hyperbolic, Thurston’s construction gives a norm on H_{2}(M; ℝ) whose unit sphere is a finite-sided polyhedron. He shows that there is a finite collection of top-dimensional faces of this polyhedron, called fibered faces, such that a ray from the origin contains a primitive class in H_{2}(M; ℤ) which is represented by a fiber if and only if it meets the unit sphere in the interior of a fibered face. In the case that β_{1}(M) = 2 the primitive classes (up to sign) in H_{2}(M; ℤ) are identified with the points of P^{1}(ℚ) ⊂ P^{1}(ℝ), and it follows from Thurston’s theorem that the classes that are represented by fibers form the intersection of P^{1}(ℚ) with a finite union of open intervals. L. Mosher has recently proved a partial analogue of this result: he has shown that those primitive classes in H_{2}(M; ℤ) (up to sign) which are represented by fibroids form an open subset of P^{1}(ℚ) ⊂ P^{1}(ℝ) in the subspace topology. If this set has an infinite complement, then there are infinitely many primitive classes in H_{2}(M; ℤ) which are not represented by fibroids. In particular, there are infinitely many primitive classes which are not represented by connected fibroids, so that Theorem 3.3 gives a lower bound of 0.83 for the volume of M.

The proof of Theorem 3.3 is divided into two cases, according to whether π_{1}(M) has a non-abelian free quotient. In the case where it has a non-abelian free quotient, the conclusion follows immediately from Theorem 1.3. In the case where π_{1}(M) has no nonabelian free quotient, it follows from the elementary arguments given in Section 2 that every primitive element of H_{2}(M; ℤ) is represented by some connected incompressible surface. In this case, the conclusion follows from a result, Theorem 3.1, which, in this somewhat more restricted setting, is stronger than Theorem 3.3: it asserts that if there are infinitely many primitive homology classes that can be represented by connected incompressible surfaces that are not fibroids, then the volume of M is greater than 0.83.

The proofs in this paper are conceptual and qualitative. They are based on more technical results that were proved in [CS1], [CS2], [CS3], [ACCS] and [CHS], and we hope that they illustrate the power of the techniques developed in those papers.

Hyperbolic distance in ℍ^{3} will be denoted dist. A hyperbolic ball in a hyperbolic 3manifold is defined to be a connected open subset which, with its intrinsic metric, is isometric to a ball in ℍ^{3}.

We thank Lee Mosher for explaining his work on the set of homology classes represented by fibroids, which we discussed above.

Section 1. Free quotients and hyperbolic volume

The main result of this section is Theorem 1.3, which applies to a manifold whose fundamental group has a non-abelian free quotient. We also establish a result, Theorem 1.4, that gives a stronger estimate under the additional assumption that the fundamental group contains no genus-2 surface groups. The statements of these results are related but the proofs are quite different.

The following two group-theoretical results are needed for the proof of Theorem 1.3. We will use the notation |S| to denote the cardinality of a set S.

Proposition 1.1. Let Γ be a group which admits a homomorphism onto a non-cyclic finite group G, and let S be a collection of elements of Γ. Suppose that every pair of distinct elements of S generate Γ. Then |S| ≤ |G| − 1.

Proof. Since G is non-cyclic we certainly have |G| > 1. Thus the assertion is true if |S| ≤ 1. Now assume that |S| > 1, and let h denote a homomorphism of Γ onto G. For any two distinct elements s and s′ of S, the hypothesis implies that s and s′ generate Γ, and hence that h(s) and h(s′) generate G. Since G is non-cyclic it follows that h(s) ≠ h(s′) and that h(s) ≠ 1. Thus we have shown that h|S is one-to-one and that h(S) ⊂ G − {1}. Thus |S| ≤ |G| − 1. □

Proposition 1.2. Let Γ be a group which has a non-abelian free quotient. Let S be a subset of Γ with |S| = 4. Then there exist a subgroup K of finite index in Γ and an epimorphism ϕ: K → ℤ such that at least two elements of S are contained in the kernel of ϕ.

Proof. Let h be a homomorphism of Γ onto a free group Γ̄ of rank n ≥ 2. We claim that there exist two distinct elements s and s′ of S such that h(s) and h(s′) generate a proper subgroup of Γ̄. This is clear if the restriction of h to S fails to be one-to-one. If h|_{S} is oneto-one then |h(S)| = 4 and, since Γ̄ admits a homomorphism onto the non-cyclic group G of order 4, it follows from Proposition 1.1 that there exist two distinct elements of h(S) which generate a proper subgroup F of Γ. (Otherwise 1.1 would imply that |S| = |h(S)| ≤ 3.)

We now apply M. Hall’s theorem (see [H], [KS]) that every finitely generated subgroup of a finite-rank free group Γ̄ is a free factor of some finite-index subgroup of Γ̄. Thus there is a subgroup K̄ of some index d < ∞ in Γ̄ such that F ≤ K̄ and F is a free factor of K̄.

We claim that F is a proper free factor of K̄. Suppose to the contrary that F = K̄. Then F also has index d in Γ̄. Since F is a proper subgroup of Γ we must have d > 1. On the other hand the rank of F is (n − 1)d + 1 and, since F is generated by two elements, we must have (n − 1)d + 1 ≤ 2. But this is impossible since n ≥ 2.

Since F is a proper free factor of K̄ there is an epimorphism ϕ: K̄ → ℤ whose kernel contains F. If we set K = h^{−1}(K̄) then K has finite index in Γ and ϕ∘h is an epimorphism from K to ℤ whose kernel contains h^{−1}(F), and hence contains two distinct elements of S. □

Theorem 1.3. Let M be a closed, orientable hyperbolic 3-manifold such that π_{1}(M) has a non-abelian free quotient. Then M contains a hyperbolic ball of radius (log 3)/2. Hence the volume of M exceeds 0.92.

Proof. According to [CHS, Proposition 2.5], if Γ is a co-compact, torsion-free, discrete subgroup of Isom_{+}(ℍ^{3}), and Δ is a positive real number, then either (i) the closed hyperbolic 3-manifold M = ℍ^{3}/Γ contains a hyperbolic ball of radius Δ/2, or (ii) there exist a point z_{0} ∈ ℍ^{3} and pairwise non-commuting (and hence distinct) elements ξ_{0}, ξ_{1}, ξ_{2}, ξ_{3} of Γ such that dist_{h}(z_{0}, ξ_{i} · z_{0}) < Δ for i = 0, 1, 2, 3.

Now suppose that M is a closed hyperbolic 3-manifold such that π_{1}(M) has a nonabelian free quotient, but that M contains no hyperbolic ball of radius (log 3)/2. Let us write M = ℍ^{3}/Γ, where Γ ≤ Isom_{+}(ℍ^{3}) is co-compact, discrete and torsion-free. Then we may fix a point z_{0} ∈ ℍ^{3} such that (ii) holds with Δ = log 3.

Since Γ ≅ π_{1}(M) has a non-abelian free quotient, we may apply Proposition 1.2 to the family (ξ_{i})_{i=1}^{4}. This shows that, after re-indexing the ξ_{i} if necessary, we may assume that there exist a subgroup K of finite index in Γ and an epimorphism ϕ: K → ℤ such that ξ_{1}, ξ_{2} ∈ K and ϕ(ξ_{1}) = ϕ(ξ_{2}) = 0.

We wish to apply Theorem 9.1 of [CS1] to ξ_{1} and ξ_{2}. The latter theorem applies to two non-commuting elements of Isom_{+}(ℍ^{3}) which generate a discrete, torsion-free nonco-compact, purely loxodromic, topologically tame group. We know that ξ_{1} and ξ_{2} do not commute. Since ξ_{1} and ξ_{2} belong to Γ, they generate a discrete, torsion-free, purely loxodromic group F. The group F has infinite index in Γ because it is contained in the kernel of the epimorphism ϕ: K → ℤ; hence F is non-co-compact. Finally, by [CS2, Prop. 2.2] and [CS1, Prop. 10.2], if ϕ is any epimorphism from the fundamental group of a closed hyperbolic 3-manifold to ℤ, then any two generator subgroup of the kernel of ϕ is topologically tame. Thus ξ_{1} and ξ_{2} satisfy all the hypotheses of [CS1, Theorem 9.1]. The latter theorem now asserts that for any point z ∈ ℍ^{3} we have

max{dist(z, ξ_{1} · z), dist(z, ξ_{2} · z)} ≥ log 3.

But for z = z_{0} this contradicts the choice of the elements ξ_{1} and ξ_{2}.

This completes the proof that, under the hypotheses of Theorem 1.3, the manifold M contains a hyperbolic ball of radius (log 3)/2. The lower bound 0.92 for the volume of M now follows from sphere-packing arguments as in CS1, Corollary 10.4. □

The following result gives stronger estimates than Theorem 1.3 under somewhat stronger hypotheses.

Theorem 1.4. Let M be a closed orientable hyperbolic 3-manifold such that π_{1}(M) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus 2.

(i) If π_{1}(M) has a non-abelian free quotient, then the volume of M exceeds 1.02.

(ii) If π_{1}(M) admits a homomorphism onto a free group of rank 3 then the volume

of M exceeds 1.54. (iii) If π_{1}(M) admits a homomorphism onto a free group of rank 4 then the volume

of M exceeds 3.08.

Proof. Assertion (iii) is a special case of [ACCS, Corollary 9.2], which asserts that if the first Betti number of the closed, orientable, hyperbolic 3-manifold M has rank at least 4 and if π_{1}(M) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus 2, then the volume of M is greater than 3.08. To prove (i) we consider a homomorphism h of π_{1}(M) onto a free group F of rank 2, and we choose any index-3 subgroup F_{0} of F. Then Γ_{0} = h^{−1}(F_{0}) has index 3 in π_{1}(M), and hence defines a 3-sheeted covering space M_{0} of M. The group π_{1}(M_{0}) ≅ Γ_{0} admits a homomorphism onto F_{0}, which is a free group of rank 4. Of course π_{1}(M_{0}) contains no genus-2 surface groups since π_{1}(M) contains none. Hence by assertion (iii) we have vol(M_{0}) > 3.08, and hence

vol(M) = \frac{vol(M_{0})}{3} > 1.02.

Assertion (ii) is deduced in the same way from assertion (iii), since an index-2 subgroup of a rank-3 free group has rank 5 ≥ 4. □

Section 2. Free quotients vs. connected surfaces

In this section we establish the following elementary result, which will be used in the proof of Theorem 3.3.

Proposition 2.1. Let M be a closed, orientable 3-manifold such that π_{1}(M) has no nonabelian free quotient. Then every primitive element of H_{2}(M; ℤ) is represented by a connected, oriented, incompressible surface.

Proof. Let c be any element of H_{2}(M; ℤ). According to [T, Lemma 1], c can be represented by some oriented 2-manifold S. (To say that S is oriented means that each of its components is oriented and hence represents an element of H_{2}(M; ℤ); the sum of these elements is equal to c.) After doing a finite number of surgeries on S, which do not change the homology class represented by S, we may assume that each component of S is incompressible. We may also assume that no component of S separates the manifold M, since discarding the separating components of S does not change the homology class that S represents. If c is primitive, then in particular c ≠ 0, so that S ≠ ∅.

Suppose that S_{1} and S_{2} are distinct components of S. We claim that M − (S_{1} ∪ S_{2}) is not connected. To prove this, we consider disjoint tubular neighborhoods N_{1} and N_{2} of S_{i}, and we set X = M −int(N_{1} ∪N_{2}). If M −(S_{1} ∪S_{2}) is connected, so is X. Then by Van Kampen’s theorem, π_{1}(M) is obtained from a free product π_{1}(X) ∗ ⟨t_{1}⟩ ∗ ⟨t_{2}⟩, where ⟨t_{1}⟩ and ⟨t_{2}⟩ are infinite cyclic groups, by adding relations of the form t_{i}α_{i}(x)t_{i}^{−1} = α′_{i}(x) for each x ∈ π_{1}(S_{i}) and for i = 1, 2, where α_{i} and α′_{i} are suitable homomorphisms from π_{1}(S_{i}) to π_{1}(X). It is apparent from this presentation that π_{1}(M) admits a homomorphism onto a free group of rank 2. As this contradicts the hypothesis that π_{1}(M) has no free quotient, it follows that M − (S_{1} ∪ S_{2}) is indeed not connected.

We next claim that any two components S_{1} and S_{2} of S represent the same element of H_{2}(M, ℤ) up to sign. We may assume that S_{1} and S_{2} are distinct. We define N_{1}, N_{2} and X as above. Let C be any component of X. Then ∂C is a union of components of the 2-manifold ∂N_{1} ∪ ∂N_{2}, which has four components. Now ∂C cannot be connected, since no component of S separates M; and we cannot have ∂C = ∂N_{1}, for this would imply that C ∪ N_{1} was an open closed subset of M disjoint from N_{2}, in contradiction to the connectedness of M. Hence ∂C ⊄ ∂N_{1}. Similarly ∂C ⊄ ∂N_{2}. Hence ∂C must contain at least one component of each ∂N_{i}. But we have shown that X is disconnected, and if C′ is another component of X then C′ must also contain at least one component of each ∂N_{i}. It follows that ∂C consists precisely of one component of ∂N_{1} and one component of ∂N_{2}. This immediately implies that S_{1} and S_{2} are homologous up to sign, as asserted.

Thus all the components of S represent the same element of H_{2}(M; ℤ) up to sign. More precisely, there is an element c_{0} of H_{2}(M; ℤ) such that each S_{i} represents either c_{0} or −c_{0}. It follows that c is an integer multiple of c_{0}. If we assume that c is primitive, we therefore have c = ±c_{0}. Thus any component of S represents the element c ∈ H_{2}(M; ℤ), after changing its orientation if necessary. Since the components of S are incompressible, this completes the proof of Proposition 2.1. □

Section 3. Non-fibroidal homology classes and hyperbolic volume

The bulk of this section is devoted to proving Theorem 3.1. We will then deduce Theorem 3.3 and Corollary 3.4 by combining Theorem 3.1 with the results of the preceding sections.

Theorem 3.1. Let M be a closed hyperbolic 3-manifold. Suppose that there is an infinite set N of primitive elements of H_{2}(M; ℤ) such that every element of N is represented by some connected, incompressible surface which is not a fibroid. Then the volume of M exceeds 0.83.

The proof of Theorem 3.1 depends on the following lemma, which is a slight variant of [ACCS, Proposition 10.1]. We refer the reader to [ACCS] for the definition of the maximal embedded tube about a closed geodesic, and of the radius of this tube.

Lemma 3.2. Let M be an orientable hyperbolic 3-manifold. Let C be a non-trivial closed geodesic in M, and let l denote its length. Let γ denote an element of π_{1}(M) whose conjugacy class is represented by C. Suppose that γ lies in the kernel of some homomorphism of π_{1}(M) onto a free abelian group of rank 2. 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)}.

Proof. The conclusion of this lemma is the same as that of Proposition 10.1 of [ACCS]. In the latter result however, in place of the hypothesis that γ lies in the kernel of some homomorphism of π_{1}(M) onto a free abelian group of rank 2, one has the assumption that log 3 is a “strong Margulis number” in the sense of [ACCS]. In that proof, one writes M = ℍ^{3}/Γ, where Γ is a discrete, torsion-free, co-compact subgroup of Isom_{+}(ℍ^{3}). If one identifies π_{1}(M) with Γ then γ becomes identified with a generator of a maximal cyclic subgroup of Γ whose axis in ℍ^{3} projects to C. The assumption that log 3 is a strong Margulis number is then used to guarantee that for any element η of Γ which does not commute with γ, and any point z on the axis of γ, the inequality

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

holds if we set α = dist(z, η · z). Hence we need only show that the latter inequality holds under the assumption that γ lies in the kernel of some homomorphism of π_{1}(M) onto a free abelian group of rank 2.

More generally, we shall show that under this assumption, if z is any point of ℍ^{3} and η is any element of Γ not commuting with γ, and if we set α = dist(z, η·z) and β = dist(z, η·z), then

\frac{1}{1 + e^{α}} + \frac{1}{1 + e^{β}} ≤ \frac{1}{2}.

According to the Main Theorem of [ACCS], this will be the case if γ and η freely generate a topologically tame subgroup of the purely loxodromic, discrete, torsion-free group Γ ≤ Isom_{+}(ℍ^{3}). Now suppose that γ lies in the kernel of a homomorphism h of Γ onto a rank-2 free abelian group L. Clearly L/h(η) admits a homomorphism h′ onto ℤ. Thus h′ ∘ h is a homomorphism of Γ onto ℤ whose kernel contains γ and η. It then follows from [CS2, Prop. 2.2] and [CS1, Prop. 10.2]that γ and η generate a topologically tame group, which concludes the proof. □

Proof of Theorem 3.1. Since H_{2}(M; ℤ) has infinitely many primitive elements, the first Betti number β_{1}(M) is at least 2. If β_{1}(M) ≥ 3, then the volume of M exceeds 0.92 according to [CS1, Corollary 10.4]. Thus we may assume that β_{1}(M) = 2. Hence the quotient of H_{1}(M, ℤ) by its torsion subgroup is a free abelian group L of rank 2. We let h: π_{1}(M) → L denote the natural homomorphism.

We set λ = 1.0616, and we distinguish two cases. First consider the case in which M contains a non-trivial closed geodesic C of some length l < λ such that the conjugacy class represented by C is contained in the kernel of h. Then by Lemma 3.2, the radius R of the maximal embedded tube T about C satisfies

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

As is pointed out in [ACCS] at the beginning of Section 10, the volume of T is πl sinh^{2} R. Since sinh^{2} R = \frac{1}{2}(cosh 2R−1), the volume of T is equal to V (l), where V (x) is the function defined 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}.

On the other hand, according to [ACCS, Prop. 10.4], the function V (x) decreases monotonically for x > 0. Since l < λ it follows that V (l) > V (λ) = 0.8364 …. Hence in this case 0.83 is a lower bound for the volume of T, and therefore also for the volume of M.

Now consider the case in which no non-trivial closed geodesic of length < λ represents a conjugacy class contained in the kernel of h. Since M is closed, there are only a finite number n ≥ 0 of conjugacy classes in π_{1}(M) that are represented by closed geodesics of length < l. Let γ_{1},…, γ_{n} be elements belonging to these n conjugacy classes. Then γ̄_{i} = h(γ_{i}) is a non-trivial element of L for i = 1,…, n. Since L is a free abelian group of rank 2, there exists, for each i ∈ {1,…, n}, a homomorphism ϕ_{i} of L onto ℤ such that ϕ_{i}(γ̄_{i}) = 0. Because γ̄_{i} ≠ 0, the epimorphism ϕ_{i} is unique up to sign.

The epimorphism ϕ_{i} ∘ h: π_{1}(M) → ℤ corresponds to a primitive element of H^{1}(M; ℤ), whose Poincaré dual in H_{2}(M; ℤ) we shall denote by c_{i}. Since the set N ⊂ H_{2}(M; ℤ) given by the hypothesis of the theorem is infinite, there is an element c of N which is distinct from ±c_{i} for i = 1,…, n. Since c ∈ N it follows from the hypothesis that there is a connected incompressible surface S ⊂ M which represents the homology class c and is not a fibroid.

We now apply Theorem A of [CS3], which asserts that if S is a connected non-fibroid incompressible surface in a closed, orientable hyperbolic 3-manifold M, and if λ is any positive number, then either (i) M contains a non-trivial closed geodesic of length < λ which is homotopic in M to a closed curve in M − S, or (ii) M contains a hyperbolic ball of radius λ/2. (We repeat that in [CS3], “fibroids” are defined to be connected. However, since the hypothesis of Theorem A of [CS3] refers only to a connected surface S, it remains true under the more general definition of a fibroid.) In the present situation, with λ chosen as above, we claim that alternative (i) of the conclusion of Theorem A of [CS3] cannot hold.

Indeed, suppose that C is a non-trivial closed geodesic of length < λ with the properties stated in (i). Since C has length < λ, the conjugacy class represented by C contains γ_{i} for some i ∈ {1,…, n}. Since C is homotopic to a closed curve in M − S it follows that the image of γ_{i} in H_{1}(M; ℤ) has homological intersection number 0 with c. Thus if ψ: π_{1}(M) → ℤ is the homomorphism corresponding to the Poincaré dual of c, we have ψ(γ_{i}) = 0. Now since L is the quotient of H_{1}(M) by its torsion subgroup, ψ factors as ϕ ∘ h, where ϕ is some homomorphism from L to ℤ. Since c is primitive, ψ is surjective, and hence so is ϕ. But we have ϕ(γ̄_{i}) = ψ(γ_{i}) = 0. In view of the uniqueness that we observed above for ϕ_{i}, it follows that ϕ = ±ϕ_{i}, so that ψ = ±ϕ_{i} ∘ h and hence c = ±c_{i}. This contradicts our choice of c.

Hence (ii) must hold. The lower bound 0.83 for the volume of M now follows from sphere-packing arguments as in CS1, Corollary 10.4. □

It is now easy to deduce the theorem and corollary that were explained in the introduction.

Theorem 3.3. Let M be a closed hyperbolic 3-manifold. Suppose that there is an infinite set N of primitive elements of H_{2}(M; ℤ) such that no element of N is represented by a connected fibroid. Then the volume of M exceeds 0.83.

Proof. If π_{1}(M) has a non-abelian free quotient, then by Theorem 1.3 the volume of M exceeds 0.92. Now suppose that π_{1}(M) has no non-abelian free quotient. If N is the set given by the hypothesis of Theorem 3.3, it now follows from Proposition 2.1 that every element of N is represented by a connected incompressible surface, which by hypothesis cannot be a fibroid. Thus N has the properties stated in the hypothesis of Theorem 3.1. The latter result therefore implies that the volume of M exceeds 0.83. □

Note that if M is a 3-manifold whose first Betti number is at least 2, then H_{2}(M; ℤ) has infinitely many primitive elements. Hence Theorem 3.3 implies:

Corollary 3.4. Let M be a closed hyperbolic 3-manifold. Suppose that the first betti number of M is at least 2, and that M contains no non-separating, connected fibroid. Then the volume of M exceeds 0.83. □

References

[ACCS] J. W. Anderson, R. Canary, M. Culler, P. B. Shalen, Free kleinian groups and volumes of hyperbolic 3-manifolds,, J. Diff. Geom. (to appear.).

[CHS] M. Culler, S. Hersonsky and P. B. Shalen, The first Betti number of the smallest closed hyperbolic 3-manifold,, preprint.

[CS1] M. Culler and P. B. Shalen, Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds, J. Amer. Math. Soc. 5 (1992), 231-288.

[CS2] M. Culler and P. B. Shalen, The volume of a hyperbolic 3-manifold with Betti number 2,, Proc. Amer. Math. Soc. 120 (1994), 1281-1288.

[CS3] M. Culler and P. B. Shalen, Volumes of hyperbolic Haken manifolds, I, Invent. math 118 (1994), 285-329.

[H] M. Hall, Jr., Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421-432.

[KS] A. Karass and D. Solitar, On finitely generated subgroups of a free group, Proc. Amer. Math. Soc. 22 (1969), 209-213.

[T] W. P. Thurston, A norm for the volume of 3-manifolds, Memoirs of the Amer. Math. Soc. No. 339, Volume 59, ????, pp. 99-130.

Department of Mathematics (M/C 249), University of Illinois at Chicago, 851 S. Morgan St. Chicago, IL 60607-7045