Every finitely generated group is weakly exact

TL;DR

This paper demonstrates that all finitely generated groups possess weak invariant expectations, leading to new insights into their cohomological properties, fixed point behaviors, and a novel concept of weak exactness for Banach algebras.

Contribution

It introduces the concept of weak invariant expectations for finitely generated groups and explores their implications in cohomology, fixed point theory, and Banach algebra theory.

Findings

Hopf G-modules are relatively injective

Bounded cohomology groups vanish in positive degrees

Fixed point theorem for group actions on $\, ext{ m l}_ ext{ m ext{infty}}$-spaces

Abstract

We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf $G$-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf $G$-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on $\ell_{\infty}$-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.