  Let $M$ be a complete connected Riemannian manifold of finite volume. In this
paper we present a new method of constructing classes in bounded cohomology of
transformation groups such as $Homeo_0(M,\mu)$, $Diff_0(M,vol)$ and
$Symp_0(M,\omega)$ (in case $M$ is symplectic). As an application we show that,
under certain conditions on $\pi_1(M)$, the $3^{rd}$ bounded cohomology of
these groups is infinite dimensional.
