Volume and Euler classes in bounded cohomology of transformation groups

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