Bounded cohomology with coefficients in uniformly convex Banach spaces

We show that for acylindrically hyperbolic groups $Γ$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $ρ$ of $Γ$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(Γ;ρ)$ is infinite dimensional. The result was known for the regular representations on $\ell^p(Γ)$ 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.

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