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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
##### Keywords:
fake surface; universal cover; homotopy type
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##### References:
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