The ratio of homology rank to hyperbolic volume, I

TL;DR

This paper establishes a new upper bound relating the homology rank and hyperbolic volume of finite-volume hyperbolic 3-manifolds, improving previous inequalities with tighter coefficients.

Contribution

It introduces a sharper inequality connecting homology rank to volume and extends results to cases with prime p, refining bounds on fundamental group rank.

Findings

Homology rank is less than 168.602 times the volume for all prime p.

Improved bounds for non-compact manifolds and p=2 cases.

Fundamental group rank is less than 1 plus a constant times the volume.

Abstract

We show that for every finite-volume hyperbolic $3$-manifold $M$ and every prime $p$ we have $\text{dim}\ H_1(M;\mathbf{F}_p)< 168.602\cdot\text{vol}\ M$. There are slightly stronger estimates if $p = 2$ or if $M$ is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of $334.08$ in place of $168.602$. It also improves on the analogous result with a coefficient of about $260$, which could have been obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to B\"or\"oczky. 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 $\pi_1(M)$ is $2$-semifree, then $\text{rank}\ \pi_1(M)<1+\lambda_{0}\cdot\text{vol}\ M$, where $\lambda_{0}$ is a certain constant less than $167.79$