Quasi-isometric rigidity of subgroups and filtered ends

Let $G$ and $H$ be quasi-isometric finitely generated groups let $P\leq G$; is there a subgroup $Q$ (or collection of subgroups) whose left cosets coarsely reflect the geometry $P$ in $G$? We explore sufficient conditions for positive answer. The article consider pairs form $(G,\mathcal{P})$ where group $\mathcal{P}$ finite subgroups, notion quasi-isometry pairs, quasi-isometrically characteristic subgroups. A qi-characteristic if it belongs to collection. Distinct classes collections subgroups have been studied literature on rigidity, we list some them provide other examples. first part proves: are $G$, then $\mathcal{Q}$ such that $ (G, \mathcal{P})$ $(H, \mathcal{Q})$ pairs. second studies number filtered ends $\tilde e P)$ pair groups, introduced by Bowditch, provides an application our main result: G$ qi-characterstic, $Q\leq H$ P) = \tilde (H, Q)$.