Hyperbolic volume, mod 2 homology, and k-freeness

TL;DR

This paper establishes new volume bounds for hyperbolic 3-manifolds based on their mod 2 homology and k-freeness of their fundamental groups, improving previous results and introducing a novel method for volume estimation.

Contribution

It introduces a new technique for deriving lower volume bounds for hyperbolic 3-manifolds with k-free fundamental groups, enhancing existing bounds and understanding.

Findings

For manifolds with volume ≤ 3.69, the mod 2 homology dimension is at most 7.

For volume ≤ 3.77, the mod 2 homology dimension is at most 10.

If the fundamental group is 4-free, volume exceeds 3.57; if 5-free, exceeds 3.77.

Abstract

We show that if $M$ is any closed, orientable hyperbolic $3$-manifold with ${\rm vol}\ M\le3.69$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le7$. This may be regarded as a qualitative improvement of a result due to Culler and Shalen, because the constant $3.69$ is greater than the ordinal corresponding to $\omega^2$ in the well-ordered set of finite volumes of hyperbolic $3$-manifolds. We also show that if ${\rm vol}\ M\le 3.77$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le10$. These results are applications of a new method for obtaining lower bounds for the volume of a closed, orientable hyperbolic $3$-manifold such that $\pi_1(M)$ is $k$-free for a given $k\ge4$. Among other applications we show that if $\pi_1(M)$ is $4$-free we have ${\rm vol}\ M>3.57$ (improving the lower bound of $3.44$ given by Culler and Shalen), and that if $\pi_1(M)$ is $5$-free we have ${\rm vol}\ M>3.77$.