Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909829215289344 |
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| author | Ma, Xin Wang, Daxun Yang, Wenyuan |
| author_facet | Ma, Xin Wang, Daxun Yang, Wenyuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras Ma, Xin Wang, Daxun Yang, Wenyuan Group Theory Dynamical Systems Geometric Topology Operator Algebras 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. |
| title | Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras |
| topic | Group Theory Dynamical Systems Geometric Topology Operator Algebras |
| url | https://arxiv.org/abs/2510.05669 |