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The isomorphism problem for toral relatively hyperbolic groups
Authors:François Dahmani  Daniel Groves
Affiliation:(1) Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse III, 31062 Toulouse, cedex 9, France;(2) MSCS UIC 322 SEO, M/C 249, 851 S. Morgan St., Chicago, IL 60607-7045, USA
Abstract:We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.
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