Quasi-isometric rigidity of subgroups and filtered ends

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 (e) over tilde (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 (e) over tilde (G, P) = (e) over tilde (H, Q).