Characterizations of locally finite actions of groups on sets

TL;DR

This paper characterizes locally finite group actions on sets through equidecomposability and relates this property to the finiteness of the group's uniform Roe algebra, providing a new perspective on group actions and operator algebras.

Contribution

It establishes a novel equivalence between local finiteness of actions and non-equidecomposability with proper subsets, linking group action properties to operator algebra finiteness.

Findings

Locally finite actions are characterized by non-equidecomposability with proper subsets.

A group is locally finite if and only if its uniform Roe algebra is finite.

Provides a new criterion connecting group actions with operator algebra properties.

Abstract

We show that an action of a group on a set $X$ is locally finite if and only if $X$ is not equidecomposable with a proper subset of itself. As a consequence, a group is locally finite if and only if its uniform Roe algebra is finite.