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Universal covers and 3-manifolds. (English) Zbl 0976.57003
Let us denote by \(\mathcal C\) the class of those finitely presented groups which are the fundamental groups of a finite fake surface, with no vertices whose link is homeomorphic to the 1-skeleton of a tetrahedron. The author shows in this paper that the class \(\mathcal C\) does not contain non-trivial perfect groups, and the main theorem of this paper is the following:
Theorem. If \(G\in{\mathcal C}\), there is a finite 2-complex \(K\) with \(\pi_1(K)\cong G\) and whose universal cover \(\tilde{K}\) has the proper homotopy type of a 3-manifold.
As a remarkable consequence of this theorem, the author proves that \(H^2(G;\mathbb Z G)\) is free abelian, which, according to Geoghegan and Mihalik, also occurs when \(G\) is semistable at infinity.

MSC:
57M07 Topological methods in group theory
20F65 Geometric group theory
57M10 Covering spaces and low-dimensional topology
57M20 Two-dimensional complexes (manifolds) (MSC2010)
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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References:
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