Three manifold groups, Kaehler groups and complex surfaces
  Let $ 1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ 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 Kaehler, we show that Q is either 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 by taking N to be the trivial group,   If G is the fundamental group of a compact complex surface, we show that Q must be the fundamental group of a Seifert-fibered space and G the fundamental group of an elliptic fibration.   We also give an example showing that the relation of quasi-isometry does not preserve Kaehler groups. This gives a negative answer to a question of Gromov which asks whether Kaehler groups can be characterized by their asymptotic geometry. 
Indranil Biswas, Mahan Mj and Harish Seshadri
math.GT math.AG math.DG math.GR