Cup product in bounded cohomology of negatively curved manifolds

Abstract

Let M M be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential 2 2 -form ξ ∈ Ω 2 ( M ) \xi \in \Omega ^2(M) defines a bounded cocycle c ξ ∈ C b 2 ( M ) c_\xi \in C_b^2(M) by integrating ξ \xi over straightened 2 2 -simplices. In particular Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] proved that, when M M is a closed hyperbolic surface, Ω 2 ( M ) \Omega ^2(M) injects this way in H b 2 ( M ) H_b^2(M) as an infinite dimensional subspace. We show that the cup product of any class of the form [ c ξ ] [c_\xi ] , where ξ \xi is an exact differential 2-form, and any other bounded cohomology class is trivial in H b ∙ ( M ) H_b^{\bullet }(M) .