  In 1992, David Wright proved a remarkable theorem about which contractible
open manifolds are covering spaces. He showed that if a one-ended open manifold
M has pro-monomorphic fundamental group at infinity which is not pro-trivial
and is not stably Z, then M does not cover any manifold (except itself). In the
non-manifold case, Wright's method showed that when a one-ended, simply
connected, locally compact ANR X with pro-monomorphic fundamental group at
infinity admits an action of Z by covering transformations then the fundamental
group at infinity of X is (up to pro-isomorphism) an inverse sequence of
finitely generated free groups. We improve upon this latter result, by showing
that X must have a stable finitely generated free fundamental group at
infinity. Simple examples show that a free group of any finite rank is
possible. We also prove that if X (as above), admits a non-cocompact action of
Z+Z by covering transformations, then X is simply connected at infinity.
Corollary: Every finitely presented one-ended group G which contains an element
of infinite order satisfies exactly one of the following: 1) G is simply
connected at infinity; 2) G is virtually a surface group; 3) The fundamental
group at infinity of G is not pro-monomorphic. Our methods also provide a quick
new proof of Wright's open manifold theorem.
