Recognizing a relatively hyperbolic group by its Dehn fillings

Abstract

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a noncompact hyperbolic $3$-manifold such as hyperbolic knot complements. We prove a rigidity result saying that if two nonelementary relatively hyperbolic groups without certain splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of nonelementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.