Bounded cohomology and isometry groups of hyperbolic spaces

Abstract

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 T is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups Hb2(Γ, ℝ), Hb2(Γ, ℓp(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 not containing a normal subgroup which virtually splits as a direct product. © European Mathematical Society 2008.