Topological properties of spaces admitting a coaxial homeomorphism

Abstract

Wright (1992) showed that, if a $1$–ended, simply connected, locally compact ANR $Y$ with pro-monomorphic fundamental group at infinity (ie representable by an inverse sequence of monomorphisms) admits a $\mathbb{Z}$–action by covering transformations, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault (2012) strengthened that result, proving that $Y$ also satisfies the crucial semistability condition (ie representable by an inverse sequence of epimorphisms).

Here we get a stronger theorem with weaker hypotheses. We drop the “pro-monomorphic hypothesis” and simply assume that the $\mathbb{Z}$–action is generated by what we call a “coaxial” homeomorphism. In the pro-monomorphic case every $\mathbb{Z}$–action by covering transformations is generated by a coaxial homeomorphism, but coaxials occur in far greater generality (often embedded in a cocompact action). When the generator is coaxial, we obtain the sharp conclusion: $Y$ is proper $2$–equivalent to the product of a locally finite tree with $ℝ$. Even in the pro-monomorphic case this is new: it says that, from the viewpoint of the fundamental group at infinity, the “end” of $Y$ looks like the suspension of a totally disconnected compact set.