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

TL;DR

This paper investigates the conditions under which the boundaries of CAT(0) groups and Coxeter groups form scrambled sets, revealing new insights into their topological and geometric properties.

Contribution

It provides a characterization of when boundaries of CAT(0) and Coxeter groups are scrambled sets, advancing understanding of their boundary dynamics.

Findings

Boundaries of certain CAT(0) groups are scrambled sets.

Conditions for Coxeter group boundaries to be scrambled sets are identified.

The study links boundary properties to group actions and geometric structures.

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.