Realizations of countable Borel equivalence relations
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866911093334474752 |
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| author | Frisch, Joshua Kechris, Alexander Shinko, Forte Vidnyánszky, Zoltán |
| author_facet | Frisch, Joshua Kechris, Alexander Shinko, Forte Vidnyánszky, Zoltán |
| contents | We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as $K_σ$ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_12486 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Realizations of countable Borel equivalence relations Frisch, Joshua Kechris, Alexander Shinko, Forte Vidnyánszky, Zoltán Logic Dynamical Systems 03E15, 54H05 We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as $K_σ$ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness. |
| title | Realizations of countable Borel equivalence relations |
| topic | Logic Dynamical Systems 03E15, 54H05 |
| url | https://arxiv.org/abs/2109.12486 |