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 Hⁿ_b ( Homeo₀(M, ω)), where M is a hyperbolic manifold of dimension n. • Euler classes in H²_b ( 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.