Semistability and simple connectivity at ∞ of finitely generated groups with a finite series of commensurated subgroups

Mihalik, Michael¹

¹Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 6, pp. 3615-3640

Résumé

A subgroup H of a group G is commensurated in G if for each g ∈ G, gHg−1 ∩ H has finite index in both H and gHg−1. 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.

Détail

DOI : 10.2140/agt.2016.16.3615

Classification : 20F65, 20F69, 57M10
Keywords: semistability, simple connectivity at infinity, commensurated, subcommensurated

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