Which 3-manifold groups are Kähler groups?

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 Kähler manifold, then G must be finite, and thus belongs to the well-known list of finite subgroups of O(4).

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