Higher-degree bounded cohomology of transformation groups

For $M$ a compact Riemannian manifold Brandenbursky and Marcinkowski constructed a transfer map $H_b^{\ast }({\pi }_1(M))\to H_b^{\ast }(Home{o}_{vol,0}(M))$ and used it to show that for certain $M$ the space ${\overline{EH}}_b^3(Home{o}_{vol,0}(M))$ is infinite-dimensional. Kimura adapted the argument to $Dif{f}_{vol}(D^2,\mathrm{\partial }{D}^2)$. We extend both results to the higher degrees ${\overline{EH}}_b^{2n}$, $n\ge 1$. We also show that for certain $M$ the ordinary cohomology $H^{\ast }(Home{o}_{vol,0}(M))$ is non-trivial in all degrees. In our computations we view the transfer map as being induced by a coupling of groups.