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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 $P\le G$; 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 $\left(G,\mathsc{𝒫}\right)$, where $G$ is a finitely generated group and $\mathsc{𝒫}$ 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 $\mathsc{𝒫}$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups  $\mathsc{𝒬}$ of $H$ such that $\left(G,\mathsc{𝒫}\right)$ and $\left(H,\mathsc{𝒬}\right)$ are quasi-isometric pairs.

Secondly, we study the number of filtered ends $ẽ\left(G,P\right)$ 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 $P\le G$ is qi-characteristic, then there is $Q\le H$ such that $ẽ\left(G,P\right)=ẽ\left(H,Q\right)$.

##### Keywords
filtered ends of groups, quasi-isometric rigidity, pairs of groups, geometric group theory
##### Mathematical Subject Classification
Primary: 20F65, 57M07