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Thickness, relative hyperbolicity, and randomness in Coxeter groups

Jason Behrstock, Mark F Hagen and Alessandro Sisto

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

Algebraic & Geometric Topology 17 (2017) 705–740

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.

Coxeter group, divergence, relatively hyperbolic group, thick group, random graph, Erdős–Rényi
Mathematical Subject Classification 2010
Primary: 05C80, 20F55, 20F65
Supplementary material

C++ code

Received: 23 March 2015
Revised: 5 October 2016
Accepted: 19 October 2016
Published: 14 March 2017
Jason Behrstock
The Graduate Center and Lehman College
New York, NY 10016
United States
Mark F Hagen
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Rd.
United Kingdom
Alessandro Sisto
Departement Mathematik HG G 28
Rämistrasse 101
CH-8092 Zürich
Pierre-Emmanuel Caprace
Departement de Mathematiques
Universite Catholique de Louvain - IRMP
Chemin du Cyclotron 2, boîte L7.01.02
1348 Louvain-la-Neuve