Thickness, relative hyperbolicity, and randomness in Coxeter groups

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 Erdós-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 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.

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

References in corpus (3)

• Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
• Global Structural Properties of Random Graphs
• Random graph products of finite groups are rational duality groups

Cited by in corpus (10)

• Global Structural Properties of Random Graphs
• Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen
• Divergence of CAT(0) Cube Complexes and Coxeter Groups
• A remark on thickness of free-by-cyclic groups
• On the coarse geometry of certain right-angled Coxeter groups
• Quasi-isometrically rigid subgroups in right-angled Coxeter groups
• Divergence of finitely presented subgroups of CAT(0) groups
• Random Artin groups
• Detecting a subclass of torsion-generated groups
• Divergence, thickness and hypergraph index for general Coxeter groups