The isomorphism problem for toral relatively hyperbolic groups
Publications Mathématiques de l'IHÉS, Tome 107 (2008) , pp. 211-290.

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special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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 $n\ge 3$. 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.

@article{PMIHES_2008__107__211_0,
author = {Dahmani, Fran\c cois and Groves, Daniel},
title = {The isomorphism problem for toral relatively hyperbolic groups},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {211--290},
publisher = {Institut des Hautes \'Etudes Scientifiques},
volume = {107},
year = {2008},
doi = {10.1007/s10240-008-0014-3},
zbl = {1207.20038},
mrnumber = {2434694},
language = {en},
url = {archive.numdam.org/item/PMIHES_2008__107__211_0/}
}
Dahmani, François; Groves, Daniel. The isomorphism problem for toral relatively hyperbolic groups. Publications Mathématiques de l'IHÉS, Tome 107 (2008) , pp. 211-290. doi : 10.1007/s10240-008-0014-3. http://archive.numdam.org/item/PMIHES_2008__107__211_0/

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