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 an oriented closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$ -manifolds; hence, they are non-trivial.