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Counting problems in graph products and relatively hyperbolic groups

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Abstract

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e., they have full density with respect to counting in balls for the word metric in the Cayley graph) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including all right-angled Artin groups and right-angled Coxeter groups.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St George St., Toronto, ON, Canada

    Ilya Gekhtman & Giulio Tiozzo

  2. Department of Mathematics, Temple University, 1805 North Broad St., Philadelphia, PA, 19122, USA

    Samuel J. Taylor

Authors
  1. Ilya Gekhtman
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  2. Samuel J. Taylor
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  3. Giulio Tiozzo
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Correspondence to Samuel J. Taylor.

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Gekhtman, I., Taylor, S.J. & Tiozzo, G. Counting problems in graph products and relatively hyperbolic groups. Isr. J. Math. 237, 311–371 (2020). https://doi.org/10.1007/s11856-020-2008-x

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  • DOI: https://doi.org/10.1007/s11856-020-2008-x

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