Universal covers and 3-manifolds.

*(English)*Zbl 0976.57003Let 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.

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.

Reviewer: Juan Antonio Pérez (Zacatecas)

##### 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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\textit{F. F. Lasheras}, J. Pure Appl. Algebra 151, No. 2, 163--172 (2000; Zbl 0976.57003)

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