Bounded cohomology with coefficients in uniformly convex Banach spaces

Abstract

We show that for acylindrically hyperbolic groups $\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\rho$ of $\Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H_b^2(\Gamma ;\rho )$ is infinite dimensional. The result was known for the regular representations on ${\ell }^p(\Gamma )$ with $1<p<\infty$ by a different argument. But our result is new even for a non-abelian free group in this great generality for representations, and also the case for acylindrically hyperbolic groups follows as an application.