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 ${\rm Iso}(X)$ 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.