A homological characterization of topological amenability

TL;DR

This paper introduces a homological approach to characterize topological amenability and exactness of groups, generalizing previous work and providing new insights into group actions on compact spaces.

Contribution

It develops a homological framework for topological amenability and links it to exactness of groups, extending Block and Weinberger's characterization.

Findings

Homological characterization of topological amenability for group actions

Homological criterion for group exactness via Stone-ech compactification

Answering Higson's question on group exactness

Abstract

Generalizing Block and Weinberger's characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of $G$ on its Stone-\vCech compactification we obtain a homological characterization of exactness of the group, answering a question of Nigel Higson.