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Quasi-isometric rigidity of subgroups and filtered ends

Eduardo Martínez-Pedroza and Luis Jorge Sánchez Saldaña

Algebraic & Geometric Topology 22 (2022) 3023–3057
DOI: 10.2140/agt.2022.22.3023

Let G and H be quasi-isometric finitely generated groups and let P G; is there a subgroup Q (or a collection of subgroups) of H whose left cosets coarsely reflect the geometry of the left cosets of P in G? We explore sufficient conditions for a positive answer.

We consider pairs of the form (G,𝒫), where G is a finitely generated group and 𝒫 a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-characteristic collection. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we list in the article some of them and provide other examples.

We first prove that if G and H are finitely generated quasi-isometric groups and 𝒫 is a qi-characteristic collection of subgroups of G, then there is a collection of subgroups  𝒬 of H such that (G,𝒫) and (H,𝒬) are quasi-isometric pairs.

Secondly, we study the number of filtered ends (G,P) of a pair of groups, a notion introduced by Bowditch, and provides an application of our main result: if G and H are quasi-isometric groups and P G is qi-characteristic, then there is Q H such that (G,P) = (H,Q).

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filtered ends of groups, quasi-isometric rigidity, pairs of groups, geometric group theory
Mathematical Subject Classification
Primary: 20F65, 57M07
Received: 19 March 2021
Revised: 23 June 2021
Accepted: 14 July 2021
Published: 13 December 2022
Eduardo Martínez-Pedroza
Department of Mathematics and Statistics
Memorial University of Newfoundland
St John’s, NL
Luis Jorge Sánchez Saldaña
Facultad de Ciencias
Universidad Nacional Autónoma de México
Ciudad de México