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.

The title has been changed. The old title was "Bounded cohomology via quasi-trees". We prove a theorem for free groups (Theorem 1.1) using actions on trees, then deal with acylindrically hyperbolic groups using a work by Hull-Osin (Corollary 1.2). In the old version we had a direct proof using quasi-trees. We move the discussion on strongly contracting geodesics to a separate paper ([3])

References in corpus (3)

• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• Isometry groups of proper CAT(0)-spaces
• Split quasicocycles

Cited by in corpus (8)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• Primeness results for von Neumann algebras associated with surface braid groups
• Bounded cohomology via partial differential equations, I
• Acylindrical hyperbolicity of groups acting on trees
• Stable commutator length on mapping class groups
• Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles
• Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic