Volume and homology of one-cusped hyperbolic 3–manifolds

Abstract

Let M be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that π 1 (M ) has no subgroup isomorphic to a genus-2 surface group, and that either (a) dim Zp H 1 (M ; Z p ) ≥ 5 for some prime p, or (b) dim Z2 H 1 (M ; Z 2 ) ≥ 4, and the subspace of H 2 (M ; Z 2 ) spanned by the image of the cup product H 1 (M ; Z 2 ) × H 1 (M ; Z 2 ) → H 2 (M ; Z 2 ) has dimension at most 1, then vol M > 5.06. If we assume that dim Z2 H 1 (M ; Z 2 ) ≥ 7, and that the compact core N of M does not contain a genus-2 closed incompressible surface, then vol M > 5.06.

4-free groups and hyperbolic geometry

“…The lower bound for vol N provided by Lemma 7.6 is in particular a lower bound for vol M, but this lower bound decreases as D decreases, and by itself it turns out to be insufficient to give the conclusion of Theorem 1.6 except for rather large values of D. To compensate for this we obtain a lower bound for vol(M − N) which increases as D decreases, and use vol N + vol(M − N) as a lower bound for vol M. To obtain a lower bound for vol(M − N), we exploit the fact that there are non-commuting elements of π 1 (M, P ) represented by loops of length log 7 and D, and again use the hypothesis that π 1 (M) is 4-free. These pieces of information are used, via results proved in [11] and adapted to the context of this paper in Section 8, to show that there is a point Y ∈ M whose distance from P is ρ, where the quantity ρ is explicitly defined as a monotonically decreasing function of D. The results from [11] also guarantee that we may take Y to lie in the µ-thick part of M if µ is any Margulis number for M. (One could, for example, take µ to be log 3, which by [3, Corollary 4.2] is a Margulis number for any closed orientable hyperbolic 3-manifold with 2-free fundamental group.) When ρ is sufficiently large it is easy to use the existence of the point Y to give a nontrivial lower bound for vol(M − N).…”

“…We mentioned above that log 3 is a Margulis number for M provided that π 1 (M) is 2-free. In Section 10 we use the methods of [11] to give a stronger result when π 1 (M) is k-free for a given k > 2 and the diameter ∆ of M is known. Corollary 10.3 asserts that a certain quantity defined as a function of k and ∆, which is monotonically increasing in k and monotonically decreasing in ∆, is a Margulis number.…”

Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

“…The following lemma is a strict improvement on Proposition 10.1 of [7]. The improvement is made possible by the results of [6].…”

“…We now invoke Proposition 3.5 of [7], which asserts that if M is a closed, aspherical 3manifold, if r = dim Z 2 H 1 (M ; Z 2 ), and if t denotes the dimension of the image of the cup product map H 1 (M ; Z 2 ) ⊗ H 1 (M ; Z 2 ) → H 2 (M ; Z 2 ), then for any integer m ≥ 0 and any regular covering M of M with covering group (Z 2 ) m , we have dim Z 2 H 1 ( M ; Z 2 ) ≥ (m + 1)r − m(m + 1)/2 − t. Taking M and M as above, the hypotheses of of [7,Proposition 3.5] hold with m = 1, and by the hypothesis of the present theorem we have r ≥ 4 and t ≤ 1. Hence dim Z 2 H 1 ( M ; Z 2 ) ≥ 6.…”

Volume and topology of bounded and closed hyperbolic $3$-manifolds

“…Otherwise there is a subgroup G of π 1 M which has rank at most 4 and is not free. The homological hypotheses and Proposition 3.5 of [9] ensure that there is a two-fold cover M → M , with dim Z2 H 1 ( M ; Z 2 ) ≥ 8, such that G < π 1 M . Theorem 1.1 of [7] implies that M contains an incompressible surface of genus 2 or 3.…”