Let
and
be quasi-isometric finitely generated groups and let
; is there a subgroup
(or a collection
of subgroups) of
whose left cosets coarsely reflect the geometry of the left cosets of
in
? We
explore sufficient conditions for a positive answer.
We consider pairs of the form
,
where
is a finitely
generated group and
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
and
are finitely generated quasi-isometric groups and
is a qi-characteristic collection of subgroups of
, then there is a
collection of subgroups
of
such
that
and
are quasi-isometric pairs.
Secondly, we study the number of filtered ends
of a pair of
groups, a notion introduced by Bowditch, and provides an application of our main result:
if
and
are quasi-isometric groups
and
is qi-characteristic,
then there is
such that
.
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