Universal covers and 3-manifolds

In this paper, we show that if a finitely presented group G is the fundamental group of a finite fake surface in which the link of any vertex is not homeomorphic to the 1-skeleton of a tetrahedron, then there is a finite 2-complex K with π1(K) G and whose universal cover has the proper homotopy type of a 3-manifold. As a consequence, the cohomology group H2(G;ZG) is free abelian.