Commensurated subgroups, semistability and simple connectivity at infinity

Abstract

A subgroup $Q$ of a group $G$ is commensurated if the commensurator of $Q$ in $G$ is the entire group $G$. Our main result is that a finitely generated group $G$ containing an infinite, finitely generated, commensurated subgroup $H$ of infinite index in $G$ is one-ended and semistable at $\infty$. Furthermore, if $Q$ and $G$ are finitely presented and either $Q$ is one-ended or the pair $\left(G,Q\right)$ has one filtered end, then $G$ is simply connected at $\infty$. A normal subgroup of a group is commensurated, so this result is a generalization of M Mihalik’s result [Trans. Amer. Math. Soc. 277 (1983) 307–321] and of B Jackson’s result [Topology 21 (1982) 71–81]. As a corollary, we give an alternate proof of V M Lew’s theorem that a finitely generated group $G$ containing an infinite, finitely generated, subnormal subgroup of infinite index is semistable at $\infty$. So several previously known semistability and simple connectivity at $\infty$ results for group extensions follow from the results in this paper. If $\varphi :H\to H$ is a monomorphism of a finitely generated group and $\varphi \left(H\right)$ has finite index in $H$, then $H$ is commensurated in the corresponding ascending HNN extension, which in turn is semistable at $\infty$.