Three manifold groups, Kähler groups and complex surfaces

be an exact sequence of finitely presented groups, where Q is infinite and not virtually cyclic, and is the fundamental group of some closed 3-manifold. If G is Kähler, we show that Q contains as a finite index subgroup either a finite index subgroup of the 3-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu in [DS09] by taking N to be the trivial group. If G is the fundamental group of a compact complex surface, we show that Q must contain the fundamental group of a Seifert-fibered three manifold as a finite index sub-group, and G contains as a finite index subgroup the fundamental group of an elliptic fibration. We also give an example showing that the relation of quasi-isometry does not preserve Kähler groups. This gives a negative answer to a question of Gromov which asks whethe