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.