Hyperfinite actions on countable sets and probability measure spaces

TL;DR

This paper introduces hyperfiniteness for group actions on countable sets and measure spaces, providing characterizations and answering a question about free group actions.

Contribution

It defines hyperfiniteness for permutation actions and offers geometric and analytic characterizations, extending known concepts for amenable actions.

Findings

Hyperfiniteness characterized for group actions on countable sets and measure spaces

Answer to van Douwen's question on free group actions on countable sets

New framework connecting hyperfiniteness with geometric and analytic properties

Abstract

We introduce the notion of hyperfiniteness for permutation actions of countable groups on countable sets and give a geometric and analytic characterization, similar to the known characterizations for amenable actions. We also answer a question of van Douwen on actions of the free group on two generators on countable sets.