idea behind an asymptotic cone of a metric space is to study “large scale” geometry of a metric space by viewing it from “infinite distance”. The term “asymptotic cone” was coined by Gromov in [@Gr81] to prove that a group of polynomial growth is virtually nilpotent. Later the notion of asymptotic cone was formally developed for arbitrary finitely generated groups by L. van den Dries, A.J. Wilkie in [@vdDW84]. Using asymptotic cones, one can characterize several important classes of groups. For example, groups of polynomial growth are precisely groups with all asymptotic cones locally compact \[[@Gr81; @Pau89; @D02]\].In [@Gr87] M. Gromov characterized hyperbolic groups in terms of asymptotic cones.

A finitely generated group is hyperbolic if and only if all of its asymptotic cones are $\mathbb{R}$-tree.

It was shown in [@DP99] that asymptotic cones of non-elementary hyperbolic groups are all isometric to the complete homogeneous $\mathbb{R}$-tree of valence continuum. Asymptotic cones of quasi-isometric spaces are bi-Lipschitz equivalent and hence the topology of an asymptotic cone of a finitely generated group does not depend on the choice of the generating set. For a survey of results on asymptotic cones and quasi isometric rigidity results see [@D02; @DS05]. In the paper [@OOS07], 