Invariant expectations and vanishing of bounded cohomology for exact groups

TL;DR

This paper characterizes exact groups through invariant expectations and demonstrates that such groups have vanishing bounded cohomology with certain modules, linking group exactness to cohomological properties.

Contribution

It introduces a new operator called an invariant expectation to characterize exact groups and connects this to the vanishing of bounded cohomology with modules derived from the Hopf algebra structure.

Findings

Exact groups can be characterized by the existence of an invariant expectation.

Exactness implies the vanishing of bounded cohomology with specific modules.

The approach links group properties to cohomological vanishing results.

Abstract

We study exactness of groups and establish a characterization of exact groups in terms of the existence of a continuous linear operator, called an invariant expectation, whose properties make it a weak counterpart of an invariant mean on a group. We apply this operator to show that exactness of a finitely generated group $G$ implies the vanishing of the bounded cohomology of $G$ with coefficients in a new class of modules, which are defined using the Hopf algebra structure of $\ell_1(G)$.