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 $\omega+1$.