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2016 Semistability and simple connectivity at $\infty$ of finitely generated groups with a finite series of commensurated subgroups
Michael Mihalik
Algebr. Geom. Topol. 16(6): 3615-3640 (2016). DOI: 10.2140/agt.2016.16.3615

Abstract

A subgroup H of a group G is commensurated in G if for each g G, gHg1 H has finite index in both H and gHg1. If there is a sequence of subgroups H = Q0 Q1 Qk Qk+1 = G where Qi is commensurated in Qi+1 for all i, then Q0 is subcommensurated in G. In this paper we introduce the notion of the simple connectivity at 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 . If, additionally, G is recursively presented and H is finitely presented and one-ended, then G is simply connected at . 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 , generalizing the main result of a paper of L Funar and D E Otera.

Citation

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Michael Mihalik. "Semistability and simple connectivity at $\infty$ of finitely generated groups with a finite series of commensurated subgroups." Algebr. Geom. Topol. 16 (6) 3615 - 3640, 2016. https://doi.org/10.2140/agt.2016.16.3615

Information

Received: 6 January 2016; Revised: 15 March 2016; Accepted: 14 April 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06666111
MathSciNet: MR3584269
Digital Object Identifier: 10.2140/agt.2016.16.3615

Subjects:
Primary: 20F65
Secondary: 20F69 , 57M10

Keywords: commensurated , semistability , simple connectivity at infinity , subcommensurated

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.16 • No. 6 • 2016
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