Thickness, relative hyperbolicity, and randomness in Coxeter groups

Behrstock, Jason¹ ; Hagen, Mark² ; Sisto, Alessandro³

¹The Graduate Center and Lehman College, CUNY, New York, NY 10016, United States
²Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Rd., Cambridge, CB3 0WB, United Kingdom
³Departement Mathematik HG G 28, ETH, Rämistrasse 101, CH-8092 Zürich, Switzerland

Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 705-740

Résumé

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.

Détail

DOI : 10.2140/agt.2017.17.705

Classification : 05C80, 20F55, 20F65
Keywords: Coxeter group, divergence, relatively hyperbolic group, thick group, random graph, Erdős–Rényi

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