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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

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.

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