Bounded cohomology of transformation groups

Let M be a complete connected Riemannian manifold of finite volume. We present a new method of constructing classes in bounded cohomology of transformation groups such as $$\hbox {Homeo}_0(M,\mu )$$ , $$\hbox {Diff}_0(M,\hbox {vol})$$ and $$\hbox {Symp}_0(M,\omega )$$ . As an application we show that for many manifolds (in particular for hyperbolic surfaces) the third bounded cohomology of these groups is infinite dimensional.