Homology bounds for hyperbolic 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.