Volume and Euler classes in bounded cohomology of transformation groups

Abstract: Let $M$ be an oriented smooth manifold, and $\operatorname{Homeo}(M,\omega)$ the group of measure preserving homeomorphisms of $M$, where $\omega$ is a finite measure induced by a volume form. In this paper we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0(M,\omega)$ and $\operatorname{Homeo}(M,\omega)$ respectively, and in several cases prove their non-triviality. More precisely, we define: - Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0(M,\omega))$ where $M$ is a hyperbolic manifold of dimension $n$. - Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}(S,\omega))$ where $S$ is a closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic $S$ and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds, and hence they are non-trivial.