Bounded cohomology and isometry groups of hyperbolic spaces

Abstract

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients in the regular representation is infinite dimensional. The result holds for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually split as a direct product.