Quasi-isometric rigidity of subgroups and filtered ends

Martínez-Pedroza, Eduardo; Sánchez Saldaña, Luis Jorge

doi:10.2140/agt.2022.22.3023

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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

Abstract

Let $G$ and $H$ be quasi-isometric finitely generated groups and let $P \leq G$ ; 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 $ẽ (G, P)$ 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 $P \leq G$ is qi-characteristic, then there is $Q \leq H$ such that $ẽ (G, P) = ẽ (H, Q)$ .

Keywords

filtered ends of groups, quasi-isometric rigidity, pairs of groups, geometric group theory

Mathematical Subject Classification

Primary: 20F65, 57M07

References

Publication

Received: 19 March 2021

Revised: 23 June 2021

Accepted: 14 July 2021

Published: 13 December 2022

Authors

Eduardo Martínez-Pedroza
Department of Mathematics and Statistics Memorial University of Newfoundland St John’s, NL Canada

Luis Jorge Sánchez Saldaña
Facultad de Ciencias Universidad Nacional Autónoma de México Ciudad de México Mexico

Volume 22, issue 6 (2022)