  For $M$ a compact Riemannian manifold Brandenbursky and Marcinkowski
constructed a transfer map $H_b^*(\pi_1(M))\to H_b^*(Homeo_{vol,0}(M))$ and
used it to show that for certain $M$ the space
$\overline{EH}_b^3(Homeo_{vol,0}(M))$ is infinite-dimensional. Kimura adapted
the argument to $Diff_{vol}(D^2,\partial D^2)$. We extend both results to the
higher degrees $\overline{EH}_b^{2n}$, $n\geq 1$. We also show that for certain
$M$ the ordinary cohomology $H^*(Homeo_{vol,0}(M))$ is non-trivial in all
degrees. In our computations we view the transfer map as being induced by a
coupling of groups.
