The ratio of homology rank to hyperbolic volume, II

TL;DR

This paper establishes new linear upper bounds relating the homology rank and hyperbolic volume of finite-volume orientable hyperbolic 3-manifolds, using innovative methods including the Four Color Theorem, under certain topological restrictions.

Contribution

It introduces novel linear bounds for homology dimensions of hyperbolic 3-manifolds that depend on volume and topological restrictions, improving previous bounds and employing the Four Color Theorem.

Findings

Bound for dim H_1(M;F_p) < 157.763 * volume(M) under specific conditions.

Previous bounds had coefficients around 168 without restrictions.

New bounds are achieved with topological restrictions on surface subgroups.

Abstract

Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$ manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) $\pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2, 3$ or $4$, or (b) $p = 2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2, 3$ or $4$, then $\text{dim}\, H_1(M;F_p) < 157.763 \cdot \text{vol}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that $\pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g = 2, \dots,8$. These results should be compared with those of our previous paper $The\ ratio\ of\ homology\ rank\ to\ hyperbolic\ volume,\ I$, in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of $\pi_1(M)$ in terms of $\text{vol}\,M$, assuming that either $\pi_1(M)$ is $9$-free, or $M$ is closed and $\pi_1(M)$ is $5$-free.