Every finitely generated group is weakly exact

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 Gmodules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on ∞-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras. © 2011 Elsevier Inc. All rights reserved