Thickness, relative hyperbolicity, and randomness in coxeter groups

Jason Behrstock, Mark F. Hagen, Alessandro Sisto, Pierre Emmanuel Caprace

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)


For right-angled Coxeter groups WΓ, we obtain a condition on Γ that is necessary and sufficient to ensure that WΓ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, the peripheral subgroups are both parabolic (in the Coxeter group-theoretic sense) and strongly algebraically thick. We exhibit a polynomial-time algorithm that decides whether a right-angled Coxeter group is thick or relatively hyperbolic. We analyze random graphs in the Erdős–Rényi model and establish the asymptotic probability that a random right-angled Coxeter group is thick. In the joint appendix, we study Coxeter groups in full generality, and we also obtain a dichotomy whereby any such group is either strongly algebraically thick or admits a minimal relatively hyperbolic structure. In this study, we also introduce a notion we call intrinsic horosphericity, which provides a dynamical obstruction to relative hyperbolicity which generalizes thickness.

Original languageEnglish
Pages (from-to)705-740
Number of pages36
JournalAlgebraic and Geometric Topology
Issue number2
Publication statusPublished - 14 Mar 2017


  • Coxeter group
  • Divergence
  • Erdos-Renyi
  • Random graph
  • Relatively hyperbolic group
  • Thick group

ASJC Scopus subject areas

  • Geometry and Topology


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