Bounded cohomology with coefficients in uniformly convex Banach spaces (Q290749)

Summary: 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 \(H^2_b(\Gamma;\rho)\) 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.