Homology bounds for hyperbolic orbifolds

TL;DR

This paper establishes linear bounds on the Betti numbers and torsion in the homology of hyperbolic orbifolds, using a new simplicial model of the thick part, extending previous work and focusing on arithmetic non-compact cases.

Contribution

It introduces a new simplicial model for the thick part of hyperbolic orbifolds and provides linear bounds on homology, advancing understanding of their topological invariants.

Findings

Linear bounds on Betti numbers of hyperbolic orbifolds

Linear bounds on torsion part of homology

Strongest results for arithmetic non-compact orbifolds

Abstract

We will provide bounds on both the Betti numbers and the torsion part of the homology of hyperbolic orbifolds. These bounds are linear in the volume and are a direct consequence of an efficient simplicial model of the thick part, which we will construct as well. The homology statements complement previous work of Bader, Gelander and Sauer (torsion homology of manifolds), Samet (Betti numbers of orbifolds) and a classical theorem of Gromov (Betti numbers of manifolds). For arithmetic, non-compact hyperbolic orbifolds -- i.e. in the case of arithmetic, non-uniform lattices in $\operatorname{Isom}(\mathbb{H}^n)$ -- the strongest results will be obtained.