Torsion-free relatively hyperbolic groups with non-relatively-hyperbolic peripheral subgroups are finite index rigid, and peripheral-structure-preserving isomorphisms between finite index subgroups force equal indices.
Finite index rigidity of hyperbolic groups
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abstract
We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a graph to the minimal number of cells in a simplicial classifying space for the group. As a corollary we prove that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index.
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Finite Index Rigidity of Relatively Hyperbolic Groups
Torsion-free relatively hyperbolic groups with non-relatively-hyperbolic peripheral subgroups are finite index rigid, and peripheral-structure-preserving isomorphisms between finite index subgroups force equal indices.