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I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on the fundamental group of closed manifolds.

I wanted to know what happen in dimension 3: which are the conditions on a finitely presented group to be realized as the fundamental group of a closed (connected) 3-manifolds? I know there are strong restriction in dimension 3, but is there an "if and only if" characterization of fundamental groups of closed 3-manifolds? In affirmative case, can you suggest me some reference?

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Igor's suggestion of the recent paper by Aschenbrenner, Friedl, and Wilton is probably the best place to start as it has a good treatment this problem which includes both a summary of recent advances and a litany of open problems.

If you want to work through a classification of geometric and non-hyperbolic manifold groups, Thurston's book "Three-Dimensional Geometry and Topology" (especially sections 4.3, 4.4. and 4.7) is very handy. I have also found the wikipedia article Seifert fiber spaces together with Scott's article: The geometries of 3-manifolds (errata here) to be especially useful in dealing with the groups of Seifert fiber spaces.

Finally, Groves, Manning, and Wilton provide a theoretical algorithm for determining if a group with solvable word problem is a three manifold group in this preprint.

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