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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.

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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