Borel asymptotic dimension and hyperfinite equivalence relations

TL;DR

This paper introduces a new Borel asymptotic dimension concept, proving that certain group actions generate hyperfinite equivalence relations, with implications for dynamics, tilings, and operator algebras.

Contribution

It extends Gromov's asymptotic dimension to Borel equivalence relations and proves hyperfiniteness for actions of polycyclic and other solvable groups.

Findings

Countable Borel equivalence relations with finite Borel asymptotic dimension are hyperfinite.

All free actions of large classes of solvable groups have finite Borel asymptotic dimension.

Applications include results in Borel chromatic numbers, tilings, and $C^*$-algebras.

Abstract

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of groups including the lamplighter group and all virtually solvable groups having finite Pr\"ufer rank. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov's notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero-dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Folner tilings, topological dynamics, and $C^*$-algebras.