The ratio of homology rank to hyperbolic volume, I

Abstract

We show that for every finite-volume hyperbolic [Formula: see text]-manifold [Formula: see text] and every prime [Formula: see text] we have [Formula: see text]. There are slightly stronger estimates if [Formula: see text] or if [Formula: see text] is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of [Formula: see text] in place of [Formula: see text]. It also improves on the analogous result with a coefficient of about [Formula: see text], 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 [Formula: see text] is a finite-volume orientable hyperbolic [Formula: see text]-manifold such that [Formula: see text] is [Formula: see text]-semifree, then [Formula: see text], where [Formula: see text] is a certain constant less than 167.79.