Volume and Homology for Hyperbolic $3$-Orbifolds, I

TL;DR

This paper establishes bounds on the first homology dimension over Z2 for certain hyperbolic 3-orbifolds based on their volume and topological properties, aiding understanding of their geometric and algebraic structure.

Contribution

It provides new volume-dependent bounds on the first homology dimension for hyperbolic 3-orbifolds with specific singular set and fundamental group conditions.

Findings

For volume ≤ 1.72, dim H_1 ≤ 15.

For volume ≤ 1.22, dim H_1 ≤ 11.

For volume ≤ 0.61, dim H_1 ≤ 7.

Abstract

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $\pi_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is irreducible, and $|{\mathfrak M}|$ is irreducible for every two-sheeted (orbifold) covering $\widetilde{\mathfrak M}$ of ${\mathfrak M}$, and if ${\rm vol} {\mathfrak M}\le1.72$, then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 15$. Furthermore, if ${\rm vol} {\mathfrak M}\le1.22$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 11$, and if ${\rm vol} {\mathfrak M}\le0.61$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 7$. The proof is an application of results that will be used in the sequel to this paper to obtain qualitatively similar results without the assumption of irreducibility of $|{\mathfrak M}|$ and $|\widetilde{\mathfrak M}|$.