Asymptotic dimension and hyperfiniteness of generic Cantor actions

TL;DR

This paper demonstrates that for certain groups with finite asymptotic dimension, the generic continuous action on Cantor space results in hyperfinite orbit equivalence relations, especially for free groups.

Contribution

It establishes that groups locally of finite asymptotic dimension have generic Cantor actions with hyperfinite orbit relations, answering a specific open question.

Findings

Generic actions of these groups have hyperfinite orbit equivalence relations.

The result applies notably to free groups.

It confirms a conjecture for a broad class of groups.

Abstract

We show that for a countable discrete group which is locally of finite asymptotic dimension, the generic continuous action on Cantor space has hyperfinite orbit equivalence relation. In particular, this holds for free groups, answering a question of Frisch-Kechris-Shinko-Vidny\'anszky.