  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.
