Bounded cohomology of transformation groups

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