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

Abstract

In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group $G$ acts geometrically (i.e. properly and cocompactly by isometries) on a CAT(0) space $X$. (Such group $G$ is called a {\it CAT(0) group}.) Then the group $G$ acts by homeomorphisms on the boundary $\partial X$ of $X$ and we can define a metric $d_{\partial X}$ on the boundary $\partial X$. The boundary $\partial X$ is called a {\it scrambled set} if for any $\alpha,\beta\in\partial X$ with $\alpha\neq\beta$, (1) $\limsup\{d_{\partial X}(g\alpha,g\beta) | g\in G\}>0$ and (2) $\liminf\{d_{\partial X}(g\alpha,g\beta) | g\in G\}=0$. We investigate when are boundaries of CAT(0) groups (and Coxeter groups) scrambled sets.