Minimality of the boundary of a right-angled Coxeter system

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 $\partial\Sigma(W,S)$.