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 $\text{dim}{H}_1(M;{\mathbf{F}}_p)<168.602\cdot \text{vol}M$. 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 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 ${\pi }_1(M)$ 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 $167.79$

arXiv comment32 pages. This version incorporates a number of corrections and stylistic improvements and is to be published in Journal of Topology and Analysis