Compactifying coverings of closed 3-manifolds (Q908551)

It is shown that a closed, \(P^ 2\)-irreducible 3-manifold M such that \(\pi_ 1(M)\) contains the fundamental group of a closed surface of negative Euler characteristic has its universal cover \(\tilde M\) homeomorphic to \({\mathbb{R}}^ 3\). This can be rephrased by saying that \(\tilde M\) is almost compact - has a manifold compactification. A second result shows that if M is a closed, \(P^ 2\)-irreducible 3-manifold such that \(\pi_ 1(M)\) has a subgroup isomorphic to \({\mathbb{Z}}\times {\mathbb{Z}}\) then the covering of M corresponding to this subgroup is almost compact.