3-MANIFOLD GROUPS, KÄHLER GROUPS AND COMPLEX SURFACES

Biswas, Indranil and Mj, Mahan and Seshadri, Harish (2012) 3-MANIFOLD GROUPS, KÄHLER GROUPS AND COMPLEX SURFACES. In: Communications in Contemporary Mathematics, 14 (6). p. 1250038.

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Abstract

Let G be a Kahler group admitting a short exact sequence 1 -> N -> G -> Q -> 1 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 H-n 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 Kahler 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 Kahler groups. This gives a negative answer to a question of Gromov which asks whether Kahler groups can be characterized by their asymptotic geometry.

Item Type:	Journal Article
Publication:	Communications in Contemporary Mathematics
Publisher:	World Scientific Publishing Company
Additional Information:	Copyright of this article belongs to World Scientific Publishing Company.
Keywords:	Kahler Group; 3-Manifolds; Complex Surface; Quasi-Isometry
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	15 Feb 2013 12:08
Last Modified:	15 Feb 2013 12:08
URI:	http://eprints.iisc.ac.in/id/eprint/45365

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