#### Volume 16, issue 6 (2016)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Semistability and simple connectivity at $\infty$ of finitely generated groups with a finite series of commensurated subgroups

### Michael L Mihalik

Algebraic & Geometric Topology 16 (2016) 3615–3640
##### Abstract

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 $H={Q}_{0}\prec {Q}_{1}\prec \cdots \prec {Q}_{k}\prec {Q}_{k+1}=G$ 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 $\infty$ of a finitely generated group (in analogy with that for finitely presented groups). Our main result is this: if a finitely generated group $G$ contains an infinite finitely generated subcommensurated subgroup $H$ of infinite index in $G$, then $G$ is one-ended and semistable at $\infty$. If, additionally, $G$ is recursively presented and $H$ is finitely presented and one-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 in works of G Conner and M Mihalik, B Jackson, V M Lew, M Mihalik, and J Profio. We also show that Grigorchuk’s group (a finitely generated infinite torsion group) and a finitely presented ascending HNN extension of this group are simply connected at $\infty$, generalizing the main result of a paper of L Funar and D E Otera.

##### Keywords
semistability, simple connectivity at infinity, commensurated, subcommensurated
##### Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F69, 57M10
##### Publication
Received: 6 January 2016
Revised: 15 March 2016
Accepted: 14 April 2016
Published: 15 December 2016
##### Authors
 Michael L Mihalik Department of Mathematics Vanderbilt University Nashville, TN 37240 United States