Three-manifolds and Kaehler groups

TL;DR

This paper proves that Kaehler groups that are also three-manifold groups are finite, and classifies the possible fundamental groups of three-manifolds that are also fundamental groups of non-Kaehler surfaces.

Contribution

It provides a simplified proof of a known result and classifies three-manifold groups that are also fundamental groups of non-Kaehler surfaces.

Findings

Kaehler and three-manifold groups are finite.

Groups of three-manifolds and non-Kaehler surfaces are either infinite cyclic or a product with Z/2Z.

Abstract

We give a simple proof of a result originally due to Dimca and Suciu: a group that is both Kaehler and the fundamental group of a closed three-manifold is finite. We also prove that a group that is both the fundamental group of a closed three-manifold and of a non-Kaehler compact complex surface is infinite cyclic or the direct product of an infinite cyclic group and a group of order two.