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.
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity.Math
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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.