Pairings, duality, amenability and bounded cohomology

TL;DR

This paper explores the connections between different homological characterizations of amenability, providing new insights and a proof of non-vanishing for bounded cohomology in free groups.

Contribution

It offers a new perspective on amenability via bounded cohomology and uniformly finite homology, clarifying their relationship and applying these ideas to free groups.

Findings

Established a relationship between Johnson's and Block-Weinberger's characterizations of amenability.

Provided a new proof of non-vanishing of bounded cohomology for free groups.

Enhanced understanding of the interaction between bounded cohomology and uniformly finite homology.

Abstract

We give a new perspective on the homological characterisations of amenability given by Johnson in the context of bounded cohomology and by Block and Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterisations. We apply these ideas to give a new proof of non- vanishing for the bounded cohomology of a free group.