  A subgroup $H$ of a group $G$ is $commensurated$ in $G$ if for each $g\in G$,
$gHg^{-1}\cap H$ has finite index in both $H$ and $gHg^{-1}$. If there is a
sequence of subgroups $H=Q_0\prec Q_1\prec ...\prec Q_{k}\prec Q_{k+1}=G$ 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 $\infty$.
If additionally, $H$ is finitely presented and 1-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 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}.
