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special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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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Dahmani, F., Groves, D. The isomorphism problem for toral relatively hyperbolic groups. Publ.math.IHES 107, 211–290 (2008). https://doi.org/10.1007/s10240-008-0014-3
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DOI: https://doi.org/10.1007/s10240-008-0014-3