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Thickness, relative
hyperbolicity, and randomness in Coxeter groups
Jason Behrstock, Mark F Hagen and Alessandro SistoAppendix: Jason Behrstock, Mark F Hagen, Alessandro Sisto and Pierre-Emmanuel Caprace |
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Algebraic & Geometric Topology 17 (2017) 705–740
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Abstract |
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For right-angled Coxeter groups , we obtain a condition on that is necessary and sufficient to ensure that 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. |
Keywords
Coxeter group, divergence, relatively hyperbolic group,
thick group, random graph, Erdős–Rényi
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Mathematical Subject Classification 2010
Primary: 05C80, 20F55, 20F65
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Supplementary material |
References |
Publication
Received: 23 March 2015
Revised: 5 October 2016
Accepted: 19 October 2016
Published: 14 March 2017
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Authors |
| Jason Behrstock | |
| The Graduate Center and Lehman
College CUNY New York, NY 10016 United States |
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| Mark F Hagen | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Wilberforce Rd. Cambridge CB3 0WB United Kingdom |
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| Alessandro Sisto | |
| Departement Mathematik HG G 28 ETH Rämistrasse 101 CH-8092 Zürich Switzerland |
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| Pierre-Emmanuel Caprace | |
| Departement de Mathematiques Universite Catholique de Louvain - IRMP Chemin du Cyclotron 2, boîte L7.01.02 1348 Louvain-la-Neuve Belgium |
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