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.

中文翻译：

---

非协约群作用和 - 无穷远的半稳定性

如果 $G$ 在几何上作用于简单连通且局部紧致的 ANR $Y$ 具有 $Y$ 中任意两条适当射线的性质，则有限呈现的 1 端群 $G$ 具有 {\it 无穷远半稳定基本群}正确同伦。$Y$ 的这个属性捕捉到了比“1-端”更强的无穷远连通性概念，并且实际上是 $G$ 的一个特征，与选择无关。它是有限表示群同伦研究的一个基本性质。虽然许多重要的群类已被证明在无穷大处具有半稳定的基本群，但近 40 年来，每个 $G$ 是否都具有这种性质的问题一直是公认的悬而未决的问题。在本文中，我们通过考虑一个群 $J$ 在这样一个 $Y$ 上的正确 {\it but non-cocompact} 动作来解决这个问题。这个 $J$ 通常是几何作用超群 $G$ 中无穷索引的一个子群；例如，$J$ 可能是无限循环的或其他一些半稳定特性已知的子群。我们将$G$ 的半稳定性质分为$J$-部分和“垂直于$J$”部分，并分析这两个部分如何组合在一起。除其他外，这种分析导致（在配套论文中）证明以前被认为可能是反例的一类群体实际上具有半稳定性。我们分析这两个部分如何组合在一起。除其他外，这种分析导致（在配套论文中）证明以前被认为可能是反例的一类群体实际上具有半稳定性。我们分析这两个部分如何组合在一起。除其他外，这种分析导致（在配套论文中）证明以前被认为可能是反例的一类群体实际上具有半稳定性。