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), acting freely on S3.
Cite this article
Alexander I. Suciu, Alexandru Dimca, Which 3-manifold groups are Kähler groups?. J. Eur. Math. Soc. 11 (2009), no. 3, pp. 521–528DOI 10.4171/JEMS/158