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Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations

Petr Naryshkin, Andrea Vaccaro

TL;DR

This work studies when amenability implies hyperfiniteness for countable Borel equivalence relations, introducing the strengthened notions of u-amenability and hyper-u-amenability. The authors prove that if a countable Borel equivalence relation is treeable and hyper-u-amenable, then it is hyperfinite, yielding several corollaries for actions of amenable and virtually free groups, as well as answers to Weiss-type questions in the treeable setting. A central technical advance is a criterion (ua_asdim) linking u-amenability, acyclic Borel graphings of finite degree, and finite Borel asymptotic dimension, which in turn implies hyperfiniteness. The paper also shows that amenable actions of countable amenable groups on standard Borel spaces produce hyper-u-amenable orbit equivalence relations when treeable, and that Borel bounded amenable relations are hyper-u-amenable, broadening the classes where amenability yields hyperfiniteness. Collectively, these results advance the understanding of the interplay between amenability, hyperfiniteness, and the union problem in Borel dynamics, with implications for measure-hyperfinite vs hyperfinite questions and the structure of orbit equivalence relations arising from group actions.

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 $σ$-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.

Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations

TL;DR

This work studies when amenability implies hyperfiniteness for countable Borel equivalence relations, introducing the strengthened notions of u-amenability and hyper-u-amenability. The authors prove that if a countable Borel equivalence relation is treeable and hyper-u-amenable, then it is hyperfinite, yielding several corollaries for actions of amenable and virtually free groups, as well as answers to Weiss-type questions in the treeable setting. A central technical advance is a criterion (ua_asdim) linking u-amenability, acyclic Borel graphings of finite degree, and finite Borel asymptotic dimension, which in turn implies hyperfiniteness. The paper also shows that amenable actions of countable amenable groups on standard Borel spaces produce hyper-u-amenable orbit equivalence relations when treeable, and that Borel bounded amenable relations are hyper-u-amenable, broadening the classes where amenability yields hyperfiniteness. Collectively, these results advance the understanding of the interplay between amenability, hyperfiniteness, and the union problem in Borel dynamics, with implications for measure-hyperfinite vs hyperfinite questions and the structure of orbit equivalence relations arising from group actions.

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 -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.

Paper Structure

This paper contains 16 sections, 20 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Borel equivalence relations and Borel graphs
  4. Borel orientations and out-degree
  5. Borel asymptotic dimension
  6. Schreier graphs and orbit equivalence relations
  7. U-amenability and hyper-u-amenability
  8. Examples
  9. Topologically amenable actions
  10. Actions of amenable groups
  11. Borel bounded equivalence relations
  12. Graphs with partial orientations
  13. Orientation with out-degree at most 1
  14. Acyclic graphs with degree at most 2
  15. Main criterion
  16. ...and 1 more sections

Key Result

Theorem 1

Let $F$ be a countable virtually free group, let $X$ be a $\sigma$-compact Polish space, and let $F \curvearrowright X$ be a continuous free action. If $E_{F}^X$ is measure-hyperfinite---in particular, if $E_{F}^X$ is amenable---then it is hyperfinite.

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Remark 2.1
  • Definition 2.2: CJMST-D
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Definition 3.2: JKL
  • ...and 33 more