Non-cocompact Group Actions and $\pi_1$-Semistability at Infinity

TL;DR

This paper investigates the semistability at infinity of finitely presented groups by analyzing non-cocompact group actions, providing new insights and partial results towards the open question of universal semistability.

Contribution

It introduces a novel approach by examining non-cocompact actions of subgroups, dividing the property into parts, and proves semistability for certain previously uncertain groups.

Findings

Analysis of non-cocompact subgroup actions informs semistability.

Division of semistability into subgroup-related parts.

Proof of semistability for some groups previously considered potential counterexamples.

Abstract

A finitely presented 1-ended group $G$ has {\it 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 {\it 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.