Semistability and simple connectivity at $\infty$ of finitely generated groups with a finite series of commensurated subgroups

Abstract

A subgroup $H$ of a group $G$ is commensurated in $G$ if for each $g\in G$, $gH{g}^{-1}\cap H$ has finite index in both $H$ and $gH{g}^{-1}$. If there is a sequence of subgroups $H=Q_0\prec Q_1\prec \cdots \prec Q_k\prec Q_{k+1}=G$ where $Q_i$ is commensurated in $Q_{i+1}$ for all $i$, then $Q_0$ is subcommensurated in $G$. In this paper we introduce the notion of the simple connectivity at $\infty$ of a finitely generated group (in analogy with that for finitely presented groups). Our main result is this: if a finitely generated group $G$ contains an infinite finitely generated subcommensurated subgroup $H$ of infinite index in $G$, then $G$ is one-ended and semistable at $\infty$. If, additionally, $G$ is recursively presented and $H$ is finitely presented and one-ended, then $G$ is simply connected at $\infty$. A normal subgroup of a group is commensurated, so this result is a strict generalization of a number of results, including the main theorems in works of G Conner and M Mihalik, B Jackson, V M Lew, M Mihalik, and J Profio. We also show that Grigorchuk’s group (a finitely generated infinite torsion group) and a finitely presented ascending HNN extension of this group are simply connected at $\infty$, generalizing the main result of a paper of L Funar and D E Otera.