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?
Best Answer
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.