Kahler groups and 3-manifold groups

Let G be the fundamental group of a compact Kahler manifold and Q the
fundamental group of a compact 3-manifold. In this talk we show that if there is a surjective homomorphism from G to Q then either Q is the fundamental group of a product of a compact surface and a circle or
the 3-dimensional Heisenberg group. The proof uses the Geometrization of 3-manifolds and the Delzant-Gromov cut-Kahler theorem.