Topological properties of spaces admitting free group actions

TL;DR

This paper investigates the topological properties of spaces with free group actions, showing that such spaces have stable free fundamental groups at infinity and classifying groups based on their behavior at infinity.

Contribution

The paper improves Wright's results by proving that spaces with free group actions have stable finitely generated free fundamental groups at infinity and classifies finitely presented groups with elements of infinite order.

Findings

Spaces with free group actions have stable finitely generated free fundamental groups at infinity.

If a space admits a non-cocompact Z+Z action, it is simply connected at infinity.

Finitely presented groups with an element of infinite order are either simply connected at infinity, virtually surface groups, or have non-pro-monomorphic fundamental groups at infinity.

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 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.