Topological properties of spaces admitting a coaxial homeomorphism

TL;DR

This paper proves that spaces with a coaxial homeomorphism have a specific topological structure, extending previous results by weakening assumptions and revealing new geometric properties of their ends.

Contribution

It introduces the concept of coaxial homeomorphisms and shows that spaces admitting such actions are proper 2-equivalent to a product of a tree and a line, generalizing earlier theorems.

Findings

Spaces with coaxial homeomorphisms are proper 2-equivalent to a tree times a line.

The result applies under weaker hypotheses than previous theorems.

The end of the space resembles the suspension of a totally disconnected set.

Abstract

Wright showed that, if a 1-ended simply connected locally compact ANR Y with pro-monomorphic fundamental group at infinity admits a proper Z-action, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault strengthened that result, proving that Y also satisfies the crucial "semistability" condition. Here we get a stronger theorem with weaker hypotheses. We drop the pro-monomorphic hypothesis and simply assume that the Z-action is generated by what we call a "coaxial" homeomorphism. In the pro-monomorphic case every proper Z-action 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 a line. Even in the pro-monomorphic case this is new: it says that, from the viewpoint of fundamental group at infinity, the end of Y looks like the suspension of a totally disconnected compact set.