Almost 1-1 extensions of Furstenberg-Weiss type and test for amenability
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911992072110080 |
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| author | Cortez, María Isabel Gómez, Jaime |
| author_facet | Cortez, María Isabel Gómez, Jaime |
| contents | Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of $(Z,G)$ and a Borel equivariant map $ψ:Y\to X$ that induces an affine bijection $ψ^*$ between $M(Y,G)$ and $M(X,G)$, the spaces of invariant probability measures of $(Y,G)$ and $(X,G)$, respectively. If $Y$ is a Cantor set, then $(Y,G)$ and $(X,G)$ are Borel isomorphic, i.e., $ψ^*$ is also a homeomorphism. As an application, we show that the family of Toeplitz subshifts is a test for amenability for residually finite groups, i.e., a residually finite group $G$ is amenable if and only if every Toeplitz $G$-subshift has invariant probability measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost 1-1 extensions of Furstenberg-Weiss type and test for amenability Cortez, María Isabel Gómez, Jaime Dynamical Systems Group Theory 37B05, 37B10, 20E26 Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of $(Z,G)$ and a Borel equivariant map $ψ:Y\to X$ that induces an affine bijection $ψ^*$ between $M(Y,G)$ and $M(X,G)$, the spaces of invariant probability measures of $(Y,G)$ and $(X,G)$, respectively. If $Y$ is a Cantor set, then $(Y,G)$ and $(X,G)$ are Borel isomorphic, i.e., $ψ^*$ is also a homeomorphism. As an application, we show that the family of Toeplitz subshifts is a test for amenability for residually finite groups, i.e., a residually finite group $G$ is amenable if and only if every Toeplitz $G$-subshift has invariant probability measures. |
| title | Almost 1-1 extensions of Furstenberg-Weiss type and test for amenability |
| topic | Dynamical Systems Group Theory 37B05, 37B10, 20E26 |
| url | https://arxiv.org/abs/2403.06982 |