of the factors is dense.

[**Corollary C:**]{} [*Let $\Gamma$ be a subgroup of the mapping class group of an oriented surface of finite type and negative Euler characteristic. Assume that $\Gamma$ does not contain a normal subgroup which virtually splits as a direct product. Then $\Gamma$ is not measure equivalent to an irreducible lattice in a product of two locally compact $\sigma$-compact non-compact topological groups.*]{}

For lattices in semisimple Lie groups of higher rank, Corollary C follows from [@FM] and the beautiful work of Furman [@Fu99a]. The earlier result of Zimmer [@Z91] suffices to deduce Corollary C for the full mapping class group which admits a linear representation with infinite image. Recently, Kida [@K06] derived a much stronger rigidity result. Namely, he showed that for every countable group $\Lambda$ which is measure equivalent to the mapping class group ${{\mathcal{M}}}$ of a non-exceptional oriented surface of finite type, there is a homomorphism $\Lambda\to {{\mathcal{M}}}$ with finite kernel and finite index image.

The organization of the paper is as follows. In Section 2, we introduce our method for the construction of second bounded cohomology classes in the concrete example of the fundamental group $\Gamma$ of a *convex cocompact* complete Riemannian manifold $M$ 