Semistability and Simple Connectivity at Infinity of Finitely Generated Groups with a Finite Series of Commensurated Subgroups

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 $H=Q_0\prec Q_1\prec \mathrm{.}\mathrm{.}\mathrm{.}\prec Q_k\prec Q_{k+1}=G$ 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 infinity of a finitely generated group (in analogy with that for finitely presented groups). Our main result is: If a finitely generated group $G$ contains an infinite, finitely generated, subcommensurated subgroup $H$, of infinite index in $G$, then $G$ is 1-ended and semistable at $\mathrm{\infty }$. If additionally, $H$ is finitely presented and 1-ended, then $G$ is simply connected at $\mathrm{\infty }$. A normal subgroup of a group is commensurated, so this result is a strict generalization of a number of results, including the main theorems of G. Conner and M. Mihalik \cite{CM}, B. Jackson \cite{J}, V. M. Lew \cite{L}, M. Mihalik \cite{M1}and \cite{M2}, and J. Profio \cite{P}.