Perturbation of Wreath Products and Quasi-Isometric Rigidity 1

We extend the notion of perturbation of wreath products, and by the methods similar to [3], prove that certain perturbations are still quasi-isometric to the original wreath product. As an application, we show that a property of containing free subgroup or free subsemigroup, and many other algebraic properties of groups are not quasi-isometrically rigid.