Article

Publications mathématiques

, Volume 107, Issue 1, pp 211-290

First online:

The isomorphism problem for toral relatively hyperbolic groups

  • François DahmaniAffiliated withInstitut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse III Email author 
  • , Daniel GrovesAffiliated withMSCS UIC 322 SEO, M/C 249

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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special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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.