Finite Index Rigidity of Relatively Hyperbolic Groups

Abstract: We prove that, given a torsion-free relatively hyperbolic group G with non-relatively-hyperbolic peripherals, isomorphic finite index subgroups of G have the same index. This applies for instance to fundamental groups of finite-volume negatively curved manifolds, to limit groups, and to free-by-cyclic groups. More generally, we show that if two finite index subgroups of a relatively hyperbolic group are isomorphic via a map that respects their peripheral structures, then their indices in the ambient group are equal. The proof relies on demonstrating that the number of simplices in a simplicial classifying space of a finite index subgroup in a relatively hyperbolic group grows linearly with its index. These results generalize earlier work of the first author in the context of hyperbolic groups.