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special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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 -manifolds, for . 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{c}ois 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}, mrnumber = {2434694}, zbl = {1207.20038}, language = {en}, url = {http://archive.numdam.org/articles/10.1007/s10240-008-0014-3/} }
TY - JOUR AU - Dahmani, François AU - Groves, Daniel TI - The isomorphism problem for toral relatively hyperbolic groups JO - Publications Mathématiques de l'IHÉS PY - 2008 SP - 211 EP - 290 VL - 107 PB - Institut des Hautes Études Scientifiques UR - http://archive.numdam.org/articles/10.1007/s10240-008-0014-3/ DO - 10.1007/s10240-008-0014-3 LA - en ID - PMIHES_2008__107__211_0 ER -
%0 Journal Article %A Dahmani, François %A Groves, Daniel %T The isomorphism problem for toral relatively hyperbolic groups %J Publications Mathématiques de l'IHÉS %D 2008 %P 211-290 %V 107 %I Institut des Hautes Études Scientifiques %U http://archive.numdam.org/articles/10.1007/s10240-008-0014-3/ %R 10.1007/s10240-008-0014-3 %G en %F PMIHES_2008__107__211_0
Dahmani, François; Groves, Daniel. The isomorphism problem for toral relatively hyperbolic groups. Publications Mathématiques de l'IHÉS, Volume 107 (2008), pp. 211-290. doi : 10.1007/s10240-008-0014-3. http://archive.numdam.org/articles/10.1007/s10240-008-0014-3/
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