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    Let G be a Kähler group admitting a short exact sequence

    where N is finitely generated.

    (i) Then Q cannot be non-nilpotent solvable.

    (ii) Suppose in addition that Q satisfies one of the following:

    (a) Q admits a discrete faithful non-elementary action on ℍn for some n ≥ 2.

    (b) Q admits a discrete faithful non-elementary minimal action on a simplicial tree with more than two ends.

    (c) Q admits a (strong-stable) cut R such that the intersection of all conjugates of R is trivial.

    Then G is virtually a surface group. It follows that if Q is infinite, not virtually cyclic, and is the fundamental group of some closed 3-manifold, then Q contains as a finite index subgroup either a finite index subgroup of the three-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 [Which 3-manifold groups are Kähler groups? J. Eur. Math. Soc.11 (2009) 521–528] by taking N to be the trivial group. If instead, G is the fundamental group of a compact complex surface, and N is finitely presented, then we show that Q must contain the fundamental group of a Seifert-fibered 3-manifold as a finite index subgroup, 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 whether Kähler groups can be characterized by their asymptotic geometry.

    AMSC: 57M50, 32Q15, 57M05, 14F35, 32J15


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