Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations

Petr Naryshkin; Andrea Vaccaro

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arXiv · 2507.07891

Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations

Petr Naryshkin·Andrea Vaccaro

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Abstract

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 $\sigma$-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.

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BibTeX RIS

Petr Naryshkin, Andrea Vaccaro. 2025-07-10. Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations. https://arxiv.org/abs/2507.07891

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