R²-irreducible universal covering spaces of P²-irreducible open 3-manifolds

An irreducible open 3-manifold $W$ is {\bf R}$^2$-irreducible if every proper plane in $W$ splits off a halfspace. In this paper it is shown that if such a $W$ is the universal cover of a connected, {\bf P}$^2$-irreducible open 3-manifold $M$ with finitely generated fundamental group, then either $W$ is homeomorphic to {\bf R}$^3$ or the group is a free product of infinite cyclic groups and infinite closed surface groups. Given any such finitely generated group uncountably many $M$ are constructed with that fundamental group such that their universal covers are {\bf R}$^2$-irreducible, are not homeomorphic to {\bf R}$^3$, and are pairwise non-homeomorphic. These results are related to the conjecture that closed, orientable, irreducible, aspherical 3-manifolds are covered by {\bf R}$^3$.