Characterisations for uniform amenability

TL;DR

This paper explores various characterisations of uniform amenability in finitely generated groups, offering weakened conditions related to supports and operator norms, thereby broadening the understanding of this property.

Contribution

It introduces new, weakened characterisations of uniform amenability, expanding the criteria beyond existing conditions like the Hulanicki-Reiter condition.

Findings

Weakened the Hulanicki-Reiter condition for uniform amenability.

Extended characterisations to include support cardinalities.

Incorporated operator norm considerations into the criteria.

Abstract

In this paper, we provide several characterisations for uniform amenability concerning a family of finitely generated groups. More precisely, we show that the Hulanicki-Reiter condition for uniform amenability can be weakened in several directions, including cardinalities of supports and certain operator norms.