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Deciding isomorphy using Dehn fillings, the splitting case

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Inventiones mathematicae Aims and scope

Le dual reste loin, solitaire et plaintif, Cherchant l’isomorphie et la trouvant rebelle

André Weil.

Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. CNRS, Institut Fourier, Univ. Grenoble Alpes, 38000, Grenoble, France

    François Dahmani

  2. Department of Mathematics and Statistics, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, E3B 5A3, Canada

    Nicholas Touikan

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  1. François Dahmani
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  2. Nicholas Touikan
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Correspondence to Nicholas Touikan.

Additional information

During the preparation of this article, the first author was supported by the ANR Grant 2011-BS01-013-02, and the Institut Universitaire de France, and the second author was supported by an NSERC PDF, ANR-2010-BLAN-116-01 GGAA, and a Fields postdoctoral fellowship.

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Dahmani, F., Touikan, N. Deciding isomorphy using Dehn fillings, the splitting case. Invent. math. 215, 81–169 (2019). https://doi.org/10.1007/s00222-018-0824-y

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  • DOI: https://doi.org/10.1007/s00222-018-0824-y

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