Dense subsets of the boundary of a Coxeter system

Abstract:

In this paper, we investigate dense subsets of the boundary of a Coxeter system. We show that for a Coxeter system $(W,S)$, if $W^{\{s_0\}}$ is quasi-dense in $W$ and the order $o(s_0t_0)=\infty$ for some $s_0,t_0\in S$, then there exists a point $\alpha$ in the boundary $\partial \Sigma (W,S)$ of the Coxeter system $(W,S)$ such that the orbit $W\alpha$ is dense in $\partial \Sigma (W,S)$. Here $W^{\{s_0\}}=\{w\in W | \ell (ws)<\ell (w) \ \text {for each}\ s\in S\setminus \{s_0\} \}\setminus \{1\}$. We also show that if the set $\bigcup \{W^{\{s\}} | s\in S \ \text {such that}\ o(st)=\infty \ \text {for some}\ t\in S\}$ is quasi-dense in $W$, then $\{w^\infty | w\in W \ \text {such that}\ o(w)=\infty \}$ is dense in $\partial \Sigma (W,S)$.