Cup product in bounded cohomology of negatively curved manifolds

Abstract:

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