Boundary actions of CAT(0) spaces and their $C^*$-algebras

TL;DR

This paper explores the boundary actions of CAT(0) spaces, especially for right-angled Coxeter and Artin groups, establishing their $C^*$-algebra properties and providing new methods to identify $C^*$-simple groups.

Contribution

It introduces new dynamical and $C^*$-algebraic results for boundary actions of specific groups acting on CAT(0) spaces, including pure infiniteness and $C^*$-simplicity criteria.

Findings

Established pure infiniteness of reduced crossed product $C^*$-algebras for certain boundary actions.

Identified conditions for actions to be $2$-filling, leading to unital Kirchberg algebras.

Provided new methods to determine $C^*$-simplicity of generalized Baumslag-Solitar groups.

Abstract

In this paper, we study boundary actions of CAT(0) spaces from a point of view of topological dynamics and $C^*$-algebras. First, we investigate the actions of right-angled Coexter groups and right-angled Artin groups with finite defining graphs on the visual boundaries and the Nevo-Sageev boundaries of their natural assigned CAT(0) cube complexes. In particular, we establish (strongly) pure infiniteness results for reduced crossed product $C^*$-algebras of these actions through investigating the corresponding $\cat$ cube complexes and establishing necessary dynamical properties such as minimality, topological freeness and pure infiniteness of the actions. In addition, we study actions of fundamental groups of graphs of groups on the visual boundaries of their Bass-Serre trees. We show that the existence of repeatable paths essentially implies that the action is $2$-filling, from which, we also obtain a large class of unital Kirchberg algebras. Furthermore, our result also provides a new method in identifying $C^*$-simple generalized Baumslag-Solitar groups. The examples of groups obtained from our method have $n$-paradoxical towers in the sense of \cite{G-G-K-N}. This class particularly contains non-degenerated free products, Baumslag-Solitar groups and fundamental groups of $n$-circles or wedge sums of $n$-circles.