CUP PRODUCT IN BOUNDED COHOMOLOGY OF NEGATIVELY CURVED MANIFOLDS

Let M be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential 2-form xi is an element of omega 2(M) defines a bounded cocycle c xi is an element of Cb2 (M) by integrating xi over straightened 2simplices. 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 Hb2 (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 Hb center dot (M).