Isometry Groups of Proper Hyperbolic Spaces

Abstract.

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H² _cb (G, L^p(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).