Three manifold groups, Kaehler groups and complex surfaces

TL;DR

This paper classifies certain 3-manifold groups that can appear as quotients of Kaehler and complex surface groups, revealing restrictions on their structure and providing new proofs and counterexamples related to geometric group theory.

Contribution

It establishes new restrictions on 3-manifold groups arising from Kaehler and complex surface groups, and shows that quasi-isometry does not preserve Kaehler properties.

Findings

Q is either the 3D Heisenberg group or a surface times circle when G is Kaehler.

Q must be a Seifert-fibered space group if G is a complex surface group.

Quasi-isometry does not preserve the Kaehler property, answering Gromov's question negatively.

Abstract

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.