Bounded cohomology and isometry groups of hyperbolic spaces

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.