Characterizations of locally finite actions of groups on sets

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.

4 pages. Minor clarification made in the paragraph after Remark 2.6. To appear in the Glasgow Mathematical Journal

Cited by in corpus (6)

• Low dimensional properties of uniform Roe algebras
• Rigidity of ${\ell }^p$ Roe-type algebras
• A quasi-local characterisation of $L^p$-Roe algebras
• Structure and $K$-theory of ${\ell }^p$ uniform Roe algebras
• On the $C^{\ast }$-algebra generated by the Koopman representation of a topological full group
• Classification of Uniform Roe algebras of locally finite groups