Recognizing a relatively hyperbolic group by its Dehn fillings

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.

Minor modification (including typesetting). 56 pages

Cited by in corpus (8)

• Groups acting acylindrically on hyperbolic spaces
• Cohomology of group theoretic Dehn fillings I: Cohen-Lyndon type theorems
• Boundaries of Dehn fillings
• Dehn filling Dehn twists
• Cohomology of group theoretic Dehn fillings II
• Rank One Hilbert Geometries
• Finitely generated simple sharply 2-transitive groups
• Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components