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2017 Thickness, relative hyperbolicity, and randomness in Coxeter groups
Jason Behrstock, Mark Hagen, Alessandro Sisto
Algebr. Geom. Topol. 17(2): 705-740 (2017). DOI: 10.2140/agt.2017.17.705


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.


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Jason Behrstock. Mark Hagen. Alessandro Sisto. "Thickness, relative hyperbolicity, and randomness in Coxeter groups." Algebr. Geom. Topol. 17 (2) 705 - 740, 2017.


Received: 23 March 2015; Revised: 5 October 2016; Accepted: 19 October 2016; Published: 2017
First available in Project Euclid: 19 October 2017

zbMATH: 06698196
MathSciNet: MR3623669
Digital Object Identifier: 10.2140/agt.2017.17.705

Primary: 05C80, 20F55, 20F65

Rights: Copyright © 2017 Mathematical Sciences Publishers


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Vol.17 • No. 2 • 2017
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