Stationary boundaries on the space of amenable subgroups and C*-simplicity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910085044764672 |
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| author | Cascioli, Anna Vio, Martín Gilabert Silva, Eduardo |
| author_facet | Cascioli, Anna Vio, Martín Gilabert Silva, Eduardo |
| contents | We give a sufficient condition for a countable group $G$ to possess a probability measure $μ$ that admits a non-trivial $μ$-boundary modeled in the space $\mathrm{Sub}_{\mathrm{am}}(G)$ of amenable subgroups of $G$. In particular, for such $μ$ the space $\mathrm{Sub}_{\mathrm{am}}(G)$ is not uniquely $μ$-stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group $G$ is C*-simple if and only if there exists $μ\in \mathrm{Prob}(G)$ such that $\mathrm{Sub}_{\mathrm{am}}(G)$ is uniquely $μ$-stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group $F$, whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial $μ$-boundary modeled on $\mathrm{Sub}_{\mathrm{am}}(G)$ is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28228 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stationary boundaries on the space of amenable subgroups and C*-simplicity Cascioli, Anna Vio, Martín Gilabert Silva, Eduardo Group Theory Dynamical Systems Operator Algebras We give a sufficient condition for a countable group $G$ to possess a probability measure $μ$ that admits a non-trivial $μ$-boundary modeled in the space $\mathrm{Sub}_{\mathrm{am}}(G)$ of amenable subgroups of $G$. In particular, for such $μ$ the space $\mathrm{Sub}_{\mathrm{am}}(G)$ is not uniquely $μ$-stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group $G$ is C*-simple if and only if there exists $μ\in \mathrm{Prob}(G)$ such that $\mathrm{Sub}_{\mathrm{am}}(G)$ is uniquely $μ$-stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group $F$, whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial $μ$-boundary modeled on $\mathrm{Sub}_{\mathrm{am}}(G)$ is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries. |
| title | Stationary boundaries on the space of amenable subgroups and C*-simplicity |
| topic | Group Theory Dynamical Systems Operator Algebras |
| url | https://arxiv.org/abs/2603.28228 |