Thickness, relative hyperbolicity, and randomness in Coxeter groups

For right-angled Coxeter groups $W_{Γ}$ , we obtain a condition on $Γ$ that is necessary and sufficient to ensure that $W_{Γ}$ 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.

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Keywords

Coxeter group, divergence, relatively hyperbolic group, thick group, random graph, Erdős–Rényi

Mathematical Subject Classification 2010

Primary: 05C80, 20F55, 20F65

Supplementary material

C++ code

References

Publication

Received: 23 March 2015

Revised: 5 October 2016

Accepted: 19 October 2016

Published: 14 March 2017

Authors

Jason Behrstock
The Graduate Center and Lehman College CUNY New York, NY 10016 United States

Mark F Hagen
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Rd. Cambridge CB3 0WB United Kingdom

Alessandro Sisto
Departement Mathematik HG G 28 ETH Rämistrasse 101 CH-8092 Zürich Switzerland

Jason Behrstock


Mark F Hagen


Alessandro Sisto


Pierre-Emmanuel Caprace
Departement de Mathematiques Universite Catholique de Louvain - IRMP Chemin du Cyclotron 2, boîte L7.01.02 1348 Louvain-la-Neuve Belgium

Volume 17, issue 2 (2017)

Jason Behrstock, Mark F Hagen and Alessandro Sisto

Appendix: Jason Behrstock, Mark F Hagen, Alessandro Sisto and Pierre-Emmanuel Caprace

Abstract