Thickness, Relative Hyperbolicity, and Randomness in Coxeter Groups

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 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 intrinsic horosphericity which provides a dynamical obstruction to relative hyperbolicity which generalizes thickness.