Dense subsets of boundaries of CAT(0) groups

TL;DR

This paper investigates conditions under which the boundary of a CAT(0) space acted upon by a group is densely filled by orbits and certain limit points, revealing minimality and density properties.

Contribution

It establishes new conditions involving elements with finite centralizers that ensure the boundary of CAT(0) groups is densely covered by orbits and limit points.

Findings

Boundaries are minimal under specified conditions.

Orbits of the group are dense in the boundary.

Limit points of infinite order elements are dense in the boundary.

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) $X\setminus F_{g_0}$ is not connnected, and (3) each component of $X\setminus F_{g_0}$ 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, $Z_{g_0}=\{h\in G| g_0h=hg_0\}$ and $F_{g_0}=\{x\in X| g_0x=x\}$). Then we show that each orbit $G \alpha$ 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.