On fixed-point sets in the boundary of a CAT(0) space

Abstract

In this paper, we investigate the fixed-point set of an element of a CAT(0) group in its boundary. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$. Let $g\in G$ and let ${\mathcal{F}}_g$ be the fixed-point set of $g$ in the boundary $\partial X$. Then we show that ${\mathcal{F}}_g=L(Z_g)$, where $Z_g$ is the centralizer of $g$ (i.e. $Z_g=\{v\in G|gv=vg\}$) and $L(Z_g)$ is the limit set of $Z_g$ in $\partial X$. Thus we obtain that ${\mathcal{F}}_g\ne \varnothing$ if and only if the set $Z_g$ is infinite. We also show that if $g$ is a hyperbolic isometry, then $\mathcal{F}_g=\partial\Min(g)$, where $\partial\Min(g)$ is the boundary of the minimal set $\Min(g)$ of $g$. This implies that the fixed-point set ${\mathcal{F}}_g$ and the periodic-point set ${\mathcal{P}}_g$ of $g$ in $\partial X$ have suspension forms.