On dense orbits in the boundary of a Coxeter system

Abstract

In this paper, we study the minimality of the boundary of a Coxeter system. We show that for a Coxeter system $(W,S)$ if there exist a maximal spherical subset $T$ of $S$ and an element $s_0\in S$ such that $m(s_0,t)\ge 3$ for each $t\in T$ and $m(s_0,t_0)=\infty$ for some $t_0\in T$, then every orbit $Wα$ is dense in the boundary $\partialΣ(W,S)$ of the Coxeter system $(W,S)$, hence $\partialΣ(W,S)$ is minimal, where $m(s_0,t)$ is the order of $s_0t$ in $W$.