Covering Spaces of 3-manifolds

We will say that a 3-manifold is almost compact if it can be obtained from a compact manifold N by removing a closed subset of dN. Then Theorem 1 is equivalent to the assertion that the universal covering of M is almost compact. A natural way to attempt to generalize Theorem 1 is to show that other coverings of M are almost compact. It was conjectured by Simon [Si] that if M is any compact P-irreducible 3-manifold and if Mi is a covering of M with finitely generated fundamental group then Mi must be almost compact. Simon verified this conjecture for the case when TTI(MI) is the fundamental group of a boundary component of M. Jaco [J] generalized this to the case when TTI(MI) is a finitely generated peripheral subgroup of TTI(M). More recently, Thurston [Th] showed that if M admits a geometrically finite complete hyperbolic structure of infinite volume then Simon's conjecture is true. Finally, Bonahon [B] showed that any hyperbolic 3-manifold with finitely generated fundamental group is almost compact provided that 7Ti(M) is not a free product. The second result of the announcement is the following.