Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras

Abstract

In this paper, we study topological dynamics on the visual boundary and several combinatorial boundaries associated to $\operatorname{CAT}(0)$ spaces. Through verifying the freeness of Myrberg points on the boundaries, we prove that a large class of these boundary actions are topologically free strong boundary actions. These include certain visual boundary actions obtained from proper isometric actions of groups on proper $\operatorname{CAT}(0)$ spaces with rank-one elements, horofunction boundary actions from actions of irreducible finitely generated infinite non-affine Coxeter groups on the Caylay graphs, and Roller-type boundary actions from certain group actions on irreducible $\operatorname{CAT}(0)$ cube complexes. This in particular leads to a new proof of Kar-Sageev's topological freeness result for Roller boundary actions of $\operatorname{CAT}(0)$ cube complexes and generalizes Klisee's topological freeness result on horofunction boundaries from hyperbolic and right angled Coxeter groups to the general case. As applications to $C^*$-algebras, our work yields new examples of $C^*$-selfless groups and of exact, purely infinite, simple reduced crossed product $C^\ast$-algebras.