Topological properties of spaces admitting a coaxial homeomorphism

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.