Recognizing a relatively hyperbolic group by its Dehn fillings

TL;DR

This paper proves a rigidity result showing that non-elementary relatively hyperbolic groups are uniquely determined by their sufficiently many isomorphic Dehn fillings, solving the isomorphism problem in this class.

Contribution

It establishes a rigidity theorem linking Dehn fillings to group isomorphism for a broad class of relatively hyperbolic groups.

Findings

Groups with sufficiently many isomorphic Dehn fillings are isomorphic.

Provides a solution to the isomorphism problem for certain relatively hyperbolic groups.

Extends the understanding of Dehn fillings in group theory.

Abstract

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.