Dehn fillings for relatively hyperbolic groups generalize the topological
Dehn surgery on a non-compact hyperbolic 3-manifold such as a hyperbolic knot
complement. We prove a rigidity result saying that if two non-elementary
relatively hyperbolic groups without suitable 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
non-elementary relatively hyperbolic groups with residually finite parabolic
groups and with no suitable splittings.Comment: Minor modification (including typesetting). 56 page