Bounded cohomology and isometry groups of hyperbolic spaces

Abstract

Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group $\mathrm{Iso}(X)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_b^2(\Gamma ,\mathbb{R})$, $H_b^2(\Gamma ,{\ell }^p(\Gamma ))$ $(1<p<\infty )$ are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually splits as a direct product.