Higher-degree bounded cohomology of transformation groups

Abstract: 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.