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The profinite completion of relatively hyperbolic virtually special groups

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Abstract

We give a characterization of toral relatively hyperbolic virtually special groups in terms of the profinite completion. We also prove a Tits alternative for subgroups of the profinite completion Ĝ of a relatively hyperbolic virtually compact special group G and completely describe finitely generated pro-p subgroups of Ĝ. This applies to the profinite completion of the fundamental group of a hyperbolic arithmetic manifold. We deduce that all finitely generated pro-p subgroups of the congruence kernel of a standard arithmetic lattice of SO(n, 1) are free pro-p.

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Acknowledgments

I thank Andrei Rapinchuk for many valuable discussions on arithmetic groups.

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Authors and Affiliations

  1. Department of Mathematics, University of Brasilia, 70910-900, Brasilia-DF, Brasil

    Pavel Zalesskii

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  1. Pavel Zalesskii
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Correspondence to Pavel Zalesskii.

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Partially supported by CNPq and FAPDF.

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Zalesskii, P. The profinite completion of relatively hyperbolic virtually special groups. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2584-7

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  • DOI: https://doi.org/10.1007/s11856-023-2584-7

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