Minimality of the boundary of a right-angled Coxeter system

Abstract

In this paper, we show that the boundary $\partial \Sigma (W,S)$ of a right-angled Coxeter system $(W,S)$ is minimal if and only if $W_{\tilde{S}}$ is irreducible, where $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $W$. We also provide several applications and remarks. In particular, we obtain that for a right-angled Coxeter system $(W,S)$, the set $\{w^{\infty }|w\in W,o(w)=\infty \}$ is dense in the boundary $\partial \Sigma (W,S)$.

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• CAT(0) groups and Coxeter groups whose boundaries are scrambled sets