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

TL;DR

This paper introduces the concept of simple connectivity at infinity for finitely generated groups and proves that groups with certain subcommensurated subgroups are 1-ended, semistable, and possibly simply connected at infinity.

Contribution

It generalizes existing results by establishing conditions under which finitely generated groups are semistable and simply connected at infinity based on their subcommensurated subgroups.

Findings

Groups with infinite, finitely generated, subcommensurated subgroups are 1-ended and semistable at infinity.

If the subcommensurated subgroup is also finitely presented and 1-ended, then the group is simply connected at infinity.

The results extend previous theorems by including subcommensurated subgroups, not just normal subgroups.

Abstract

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}.