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 $m(s_{0}, t) \geq 3$ for each $t \in T$ and $m(s_{0}, t_{0}) = \infty$ for some $t_{0} \in T$, then every orbit $W\alpha$ is dense in the boundary $\partial \Sigma (W, S)$ of the Coxeter system $(W, S)$, hence $\partial \Sigma (W, S)$ is minimal, where $m(s_{0}, t)$ is the order of $s_{0}t$ in $W$.