Bounded cohomology of transformation groups

Abstract

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 Homeo 0 (M, ω) and Homeo(M, ω) respectively, and in several cases prove their non-triviality. More precisely, we define:where M is a hyperbolic manifold of dimension n.• Euler classes in H 2 b (Homeo(S, ω)) 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.