3rd Bounded Cohomology of Volume Preserving Transformation Groups.

Let M be a Riemannian manifold with a given volume form and hyperbolic fundamental group. We will explain how to construct coclasses in the cohomology of the group of volume preserving diffeomorphisms (or homeomorphisms) of M. As an application, we show that the 3rd bounded cohomology of those groups is highly non-trivial.