Quasi-isometric rigidity of subgroups and Filtered ends

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. The article consider pairs of the form $(G,\mathcal{P})$ where $G$ is a finitely generated group and $\mathcal{P}$ 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. The first part of the article proves: if $G$ and $H$ are finitely generated quasi-isometric groups and $\mathcal{P}$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups $\mathcal{Q}$ of $H$ such that $(G,\mathcal{P})$ and $(H,\mathcal{Q})$ are quasi-isometric pairs. The second part of the article studies the number of filtered ends $\tilde{e}(G,P)$ 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 $P\le G$ is qi-characterstic, then there is $Q\le H$ such that $\tilde{e}(G,P)=\tilde{e}(H,Q)$.

Version 3. 25 pages. Updated list of references. Version accepted for publication in AGT

References in corpus (1)

• Cusped spaces and quasi-isometries of relatively hyperbolic groups

Cited by in corpus (4)

• Quasi-isometry invariance of relative filling functions
• Asymptotic geometry of lamplighters over one-ended groups
• Hyperbolically embedded subgroups and quasi-isometries of pairs
• The quasi-isometry invariance of the Coset Intersection Complex