Volume and Homology for Hyperbolic 3-Orbifolds

TL;DR

This paper establishes explicit upper bounds on the first homology dimension of hyperbolic 3-orbifolds based on their volume, with stronger results under additional geometric and topological assumptions.

Contribution

It provides new volume-to-homology bounds for hyperbolic 3-orbifolds, especially when the singular set is a link or when irreducibility conditions are met.

Findings

Upper bounds on volume imply bounds on homology dimension.

Stronger bounds are obtained when the singular set is a link.

Irreducibility assumptions lead to improved results.

Abstract

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that $\pi_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 23, 43 and 79 for $\dim H_1({\mathfrak M};{\mathbb F}_2 )$. Stronger results hold if we assume that the singular set $\Sigma$ is a link; specifically, under this assumption, strict upper bounds of 0.305, 0.4575, 0.61, 0.7625 and 0.915 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 7, 13, 14, 28 and 29 for ${\rm dim}\ H_1({\mathfrak M};{\mathbb F}_2 )$. Irreducibility assumptions on the underlying manifold $|{\mathfrak M}|$ of ${\mathfrak M}$, and of the underlying manifolds of certain coverings of ${\mathfrak M}$, also give stronger results. The upper bounds on $\dim H_1({\mathfrak M};{\mathbb F}_2 )$ for an orbifold ${\mathfrak M}$ whose volume is subject to a suitable upper bound are deduced from upper bounds on ${\rm dim}\ H_1(|{\mathfrak M}|;{\mathbb F}_2 )$ for an orbifold ${\mathfrak M}$ whose volume is subject to a suitable upper bound.