Topological properties of spaces admitting free group actions

Abstract

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 Mn 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 absolute neighborhood retract 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. We deduce the following corollary in group theory: Every finitely presented one-ended group G that contains an element of infinite order satisfies exactly one of the following conditions: (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. © 2012 London Mathematical Society.