The ratio of homology rank to hyperbolic volume, I

Abstract

We show that for every finite-volume hyperbolic 3-manifold M and every prime p we have dim H 1 (M ; F p ) < 168.05 • vol M . This improves on a result proved by Agol, Leininger and Margalit giving the same inequality with the coefficient 168.05 replaced by 334.08, and on the analogous result with a coefficient of about 260 which would be obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if M is a finite-volume orientable hyperbolic 3-manifold such that π 1 (M ) is 2-semifree, then rank π 1 (M ) < 1 + λ 0 • vol M , where λ 0 is a certain constant less than 167.79.

The ratio of homology rank to hyperbolic volume, II

“…Theorems 7.8 and 7.9 are partial improvements on Theorem 5.4 of [16], while Theorems 7.6 and 7.7 are partial improvements on Proposition 5.2 of [16]. Theorem 5.4 and Proposition 5.2 of [16] have weaker hypotheses than their counterparts in this paper, in that they do not require restrictions on incompressible surfaces in the manifold M or surface subgroups of π 1 (M).…”

“…Theorems 7.8 and 7.9 are partial improvements on Theorem 5.4 of [16], while Theorems 7.6 and 7.7 are partial improvements on Proposition 5.2 of [16]. Theorem 5.4 and Proposition 5.2 of [16] have weaker hypotheses than their counterparts in this paper, in that they do not require restrictions on incompressible surfaces in the manifold M or surface subgroups of π 1 (M). On the other hand, the results in [16] have weaker conclusions, in that the coefficients in the inequalities relating volume to the dimension of H 1 (M; F p ) or the rank of π 1 (M) are in the range of 168 rather than 158.…”

“…Theorem 5.4 and Proposition 5.2 of [16] have weaker hypotheses than their counterparts in this paper, in that they do not require restrictions on incompressible surfaces in the manifold M or surface subgroups of π 1 (M). On the other hand, the results in [16] have weaker conclusions, in that the coefficients in the inequalities relating volume to the dimension of H 1 (M; F p ) or the rank of π 1 (M) are in the range of 168 rather than 158. The reason for the improvement by a factor of about 15/16 over the constants in [16] will emerge in the sketch given below of the proofs of Theorems 7.6 and 7.7, and will be seen to involve the Four Color Theorem.…”

“…On the other hand, the results in [16] have weaker conclusions, in that the coefficients in the inequalities relating volume to the dimension of H 1 (M; F p ) or the rank of π 1 (M) are in the range of 168 rather than 158. The reason for the improvement by a factor of about 15/16 over the constants in [16] will emerge in the sketch given below of the proofs of Theorems 7.6 and 7.7, and will be seen to involve the Four Color Theorem.…”

“…In [16] it was shown that the inequality (1.0.1) holds with a significantly smaller value of λ than the one given in [4]. In this paper we introduce a surprising new idea that gives further improvements on the value of λ under mild topological hypotheses on the manifold M. In a future paper we will show that when M is closed and the prime p is equal to 2, a similar improvement on the result of [16] can be obtained without any additional topological hypothesis, but the arguments needed to prove this are much more involved than those of the present paper.…”