Topological properties of spaces admitting free group actions (Q2892840)

The authors cite theorems by \textit{David G. Wright} [Topology 31, No. 2, 281--291 (1992; Zbl 0755.57009)] on which contractible open manifolds are covering spaces: if a one-ended open manifold \(M^n\) has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably \(\mathbb{Z}\), then \(M\) does not cover any manifold (except for itself); and that if a one-ended, simply connected, locally compact absolute neighborhood retract \(X\) with pro-monomorphic fundamental group at infinity admits an action of \(\mathbb{Z}\) by covering transformations, then the fundamental group at infinity of \(X\) is (up to pro-isomorphism) an inverse sequence of finitely generated free groups.NEWLINENEWLINEIn the present paper, this latter result is improved, by proving that \(X\) must have a stable finitely generated free fundamental group at infinity; and that if \(X\) (as above) admits a non-cocompact action of \(\mathbb{Z}\times \mathbb{Z}\) by covering transformations, then \(X\) is simply connected at infinity.NEWLINENEWLINEThe layout of the paper is as follows: In the introduction these two -- and other -- results are stated and related to the background of the problem. In section 2 some necessary definitions and background are given. In sections 3 and 4 canonical neighbourhoods of infinity in certain spaces are considered and their fundamental groups are described. In the remaining sections the proofs of the main results are given.