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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 asequence 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 simpleconnectivity at infinity of a finitely generated group (in analogy with thatfor finitely presented groups). Our main result is: If a finitely generatedgroup $G$ 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 simplyconnected at $\infty$. A normal subgroup of a group is commensurated, so thisresult is a strict generalization of a number of results, including the maintheorems 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}.

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