  We present a variation of quasi-isometry to approach the problem of defining
a geometric notion equivalent to commensurability. In short, this variation can
be summarized as "quasi-isometry with uniform parameters for a large enough
family of generating systems". Two similar notions (using isometries and rough
isometries instead, respectively) are presented alongside.
  This article is based mainly on a chapter of the author's doctoral thesis
(\cite{Lochmann_dissertation}).
