Asymptotic property C of the wreath product $\mathbb{Z}\wr \mathbb{Z}$

Abstract

Using the relationship between transfinite asymptotic dimension and asymptotic property C, we obtain that the wreath product $\mathbb{Z}\wr \mathbb{Z}$ has asymptotic property C. Specifically, we prove that the transfinite asymptotic dimension of the wreath product $\mathbb{Z}\wr \mathbb{Z}$ does not exceed $\omega +1$.