Which 3-manifold groups are K\"ahler groups?

TL;DR

This paper completely characterizes which 3-manifold groups can also be K"ahler groups, showing that only finite groups, specifically those in O(4), qualify.

Contribution

It provides a complete classification of 3-manifold groups that are also K"ahler groups, resolving a question posed by Goldman and Donaldson.

Findings

Only finite 3-manifold groups can be K"ahler groups.

Such groups are precisely the finite subgroups of O(4).

The result confirms the conjecture that infinite 3-manifold groups cannot be K"ahler.

Abstract

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact K\"ahler manifold, then G must be finite, and thus belongs to the well-known list of finite subgroups of O(4).