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

2025/10/07 by Xin Ma, Ma, Xin, Daxun Wang +3 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2510.05669

openalex publication_date 2025/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study topological dynamics on the visual boundary and several combinatorial boundaries associated to 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 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 CAT(0) cube complexes. This in particular leads to a new proof of Kar-Sageev's topological freeness result for Roller boundary actions of 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^∗-algebras.

Citations

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Cited by

• Selfless reduced free products and graph products of C^∗-algebras

Related

• Boundary actions of CAT(0) spaces and their C^*-algebras
• Comparing the Roller and B(X) boundaries of CAT(0) cube complexes
• Topological boundaries of connected graphs and Coxeter groups
• Cell-Like Equivalences and Boundaries of CAT(0) Groups
• Coarse decompositions of boundaries for CAT(0) groups