Inventiones mathematicae

, Volume 215, Issue 1, pp 81–169 | Cite as

Deciding isomorphy using Dehn fillings, the splitting case

  • François Dahmani
  • Nicholas TouikanEmail author


We solve Dehn’s isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and, if so, whether a global isomorphism can be assembled. For the class of finitely generated nilpotent groups, we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups.



Both authors wish to warmly thank Dan Segal, who kindly explained how to use a key feature of polycyclic groups, proved in [48], and Vincent Guirardel, who, among other discussions, showed us how to simplify our original argument for Sect. 7.2.2. The authors are also extremely grateful for the anonymous referee’s numerous and insightful comments, suggestions, corrections, and warnings. The majority of these have lead to improvements to the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, Institut FourierUniv. Grenoble AlpesGrenobleFrance
  2. 2.Department of Mathematics and StatisticsUniversity of New BrunswickFrederictonCanada

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