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Non-cocompact Group Actions and $\unicode[STIX]{x1D70B}_{1}$-Semistability at Infinity

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  26 June 2019

Ross Geoghegan ,
Craig Guilbault  and
Michael Mihalik
Ross Geoghegan
Affiliation:
Department of Mathematics, State University of New York-Binghamton, BinghamtonNY, USA Email: ross@math.binghamton.edu
Craig Guilbault
Affiliation:
Department of Mathematics, University of Wisconsin-Milwaukee, MilwaukeeWI, USA Email: craigg@uwm.edu
Michael Mihalik
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN, USA Email: michael.l.mihalik@vanderbilt.edu
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Abstract

A finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

Keywords

finitely presented groupsemistability

MSC classification

Primary: 20F69: Asymptotic properties of groups
Secondary: 20F65: Geometric group theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 5 , October 2020 , pp. 1275 - 1303
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author C. G. was supported by Simons Foundation Grants 207264 & 427244, CRG.

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