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Topological properties of spaces admitting a coaxial homeomorphism

Ross Geoghegan, Craig Guilbault and Michael Mihalik

Algebraic & Geometric Topology 20 (2020) 601–642

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 –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 –action is generated by what we call a “coaxial” homeomorphism. In the pro-monomorphic case every –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 . 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.

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coaxial homeomorphism, semistability, fundamental group at infinity
Mathematical Subject Classification 2010
Primary: 20F65, 57M07, 57S30
Secondary: 57M10
Received: 29 October 2017
Revised: 19 October 2018
Accepted: 14 March 2019
Published: 23 April 2020
Ross Geoghegan
Department of Mathematical Sciences
Binghamton University
Binghamton, NY
United States
Craig Guilbault
Department of Mathematical Sciences
University of Wisconsin-Milwaukee
Milwaukee, WI
United States
Michael Mihalik
Department of Mathematics
Vanderbilt University
Nashville, TN
United States