# Thickness, relative hyperbolicity, and randomness in Coxeter groups

Jason Behrstock, Mark F. Hagen, Alessandro Sisto, Pierre-Emmanuel Caprace

Research output: Contribution to journalArticle (Academic Journal)peer-review

13 Citations (Scopus)

## Abstract

For right-angled Coxeter groups $W_{\Gamma}$, we obtain a condition on $\Gamma$ that is necessary and sufficient to ensure that $W_{\Gamma}$ 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 Erd\'{o}s-R\'{e}nyi 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 there 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 \emph{intrinsic horosphericity} which provides a dynamical obstruction to relative hyperbolicity which generalizes thickness.
Original language English Algebraic and Geometric Topology https://doi.org/10.2140/agt.2017.17.705 Published - 14 Mar 2017

### Bibliographical note

Primary article by Behrstock, Hagen, and Sisto with an appendix by Behrstock, Caprace, Hagen, and Sisto. 31 pages, 5 figures, 1 table. All necessary C++ code can be downloaded from this ArXiv page. The same C++ code, along with instructions and control scripts, is available at http://www-personal.umich.edu/~mfhagen/thickracgcode.tar

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