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

TL;DR

The paper addresses how topological dynamics on visual, Roller, and horofunction boundaries of CAT(0) spaces influence C*-algebraic properties of the acting groups. By leveraging rank-one contracting isometries and the notion of Myrberg points, it proves that boundary actions are minimal, topologically free, and strong boundary actions in broad settings, including Coxeter groups and cube complexes. These dynamical results yield new examples of $C^*$-selfless groups and exact, purely infinite, simple reduced crossed product algebras, with Kirchberg-algebra classifications (UCT) in many cases. The work deepens the link between geometric group theory and operator algebras, providing a versatile toolkit for deriving regularity properties of associated C*-algebras from boundary dynamics.

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.

Figures (7)

• Figure 1: A comparison triangle for $\Delta(p,q,r)$
• Figure 2: A neighborhood basis for the cone topology.
• Figure 3: Admissible path
• Figure 4: A $\operatorname{CAT}(0)$ cube complex $X$. The vertex $\mathfrak{h_1}$ is an essential hyperplane that separates $X$ into two half-spaces: the left part ${ \hbox{{\cr \hidewidth\reflectbox{$\m@th\vec{}\mkern4mu$}\hidewidth\cr {} $\m@th\mathfrak{h}$\cr }}}_1$ and the right part $\vec{\mathfrak{h}}_1$. On the other hand, the hyperplane $\mathfrak{h_2}$ is not essential.
• Figure 5: Parallel $3$-cliques (left) and antipodal cliques (right) in a paraclique graph. Antipodal cliques are not necessarily of same size.

Theorems & Definitions (216)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['allboundaryaresame']}
• Remark 1.5
• Lemma 1.6: Lemma \ref{['NorthSouthDynamicsCoxeter']}
• Remark 1.7
• Theorem 1.8: Proposition \ref{['homeomorphicMyrbergset']}
• Corollary 1.9: Corollary \ref{['pure inf irreducible Coxeter']}
• Definition 2.1