Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911415424516096 |
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| author | Naryshkin, Petr Vaccaro, Andrea |
| author_facet | Naryshkin, Petr Vaccaro, Andrea |
| contents | We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. One of the corollaries that we get is that if a countable Borel equivalence relation is measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $σ$-compact Polish space, then it is hyperfinite. We also obtain that if a countable Borel equivalence relation is treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space, or if it is treeable, amenable and Borel bounded, then it is hyperfinite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07891 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations Naryshkin, Petr Vaccaro, Andrea Logic Dynamical Systems Group Theory We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. One of the corollaries that we get is that if a countable Borel equivalence relation is measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $σ$-compact Polish space, then it is hyperfinite. We also obtain that if a countable Borel equivalence relation is treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space, or if it is treeable, amenable and Borel bounded, then it is hyperfinite. |
| title | Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations |
| topic | Logic Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2507.07891 |