ar X iv : m at h / 05 07 09 7 v 2 [ m at h . G R ] 1 3 Ju l 2 00 5 BOUNDED COHOMOLOGY AND ISOMETRY GROUPS OF HYPERBOLIC SPACES

Let X be an arbitrary hyperbolic geodesic metric space and let Γ be a countable subgroup of the isometry group Iso(X) of X. We show that if Γ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups H 2 b (Γ, R), H 2 b (Γ, ℓ p (Γ)) (1 ≤ p < ∞) are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type which is not virtually abelian nor virtually splits as a direct product.