A note on fake surfaces and universal covers

In this paper, we consider compact fake surfaces X with the property that each component of the subgraph Γ⊂X of triple edges contains at most one point whose link in X is homeomorphic to the 1-skeleton of a tetrahedron (type III). Assuming the subgroups Λi⊆π1(X) generated by the loops in Γ at the points vi of type III are of a certain kind, an application of F.F. Lasheras [Proc. Amer. Math. Soc. 128 (2000) 893–902; J. Pure Appl. Algebra 151 (2) (2000) 163–172] leads us to finding a compact polyhedron K with π1(K)≅π1(X) and whose universal cover K has the proper homotopy type of a 3-manifold. This result extends the work in F.F. Lasheras [J. Pure Appl. Algebra 151 (2) (2000) 163–172] about a question on finitely presented groups.