Dense subsets of boundaries of CAT(0) groups

Tetsuya Hosaka

paper

Dense subsets of boundaries of CAT(0) groups

arXiv:math/0510511

Abstract

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$ and suppose that there exists an element $g_{0} \in G$ such that (1) $Z_{g_{0}}$ is finite, (2) $X ∖ F_{g_{0}}$ is not connnected, and (3) each component of $X ∖ F_{g_{0}}$ is convex and not $g_{0}$ -invariant, where $Z_{g_{0}}$ is the centralizer of $g_{0}$ and $F_{g_{0}}$ is the fixed-point set of $g_{0}$ in $X$ (that is, $Z_{g_{0}} = {h \in G ∣ g_{0} h = h g_{0}}$ and $F_{g_{0}} = {x \in X ∣ g_{0} x = x}$ ). Then we show that each orbit $G α$ is dense in the boundary $\partial X$ (i.e.\ $\partial X$ is minimal) and the set ${g^{\infty} ∣ g \in G, o (g) = \infty}$ is also dense in the boundary $\partial X$ . We obtain an application for dense subsets on the boundary of a Coxeter system.