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The ratio of homology rank to hyperbolic volume, I

Rosemary K. Guzman

https://orcid.org/0000-0003-4555-0607

Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA

E-mail Address: rguzma1@illinois.edu

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and

Peter B. Shalen

Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA

E-mail Address: petershalen@gmail.com

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https://doi.org/10.1142/S1793525323500176Cited by:0 (Source: Crossref)

Abstract

We show that for every finite-volume hyperbolic $3$ -manifold $M$ and every prime $p$ we have $dim H_{1} (M; F_{p}) < 168.602 \cdot vol (M)$ . 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 $π_{1} (M)$ is $2$ -semifree, then $rank π_{1} (M) < 1 + λ 0 \cdot vol M$ , where $λ 0$ is a certain constant less than $167.79$ .

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