Volume and Euler classes in bounded cohomology of transformation groups

Let M be an oriented smooth manifold and Homeo\!(M,ω ) the group of measure preserving homeomorphisms of M , where ω 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 Homeo₀\!(M,ω ) and Homeo₊\!(M,ω ) , respectively, and in several cases prove their non-triviality. More precisely, we define: • Volume classes in Hbⁿ(Homeo₀\!(M,ω )) , where M is a hyperbolic manifold of dimension n . • Euler classes in Hb²(Homeo₊(S,ω )) , 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.