On self-similarity of wreath products of abelian groups

Abstract

We prove that in a self-similar wreath product of abelian groups $G=B$ wr $X$, if $X$ is torsion-free then $B$ is torsion of finite exponent. Therefore, in particular, the group $\mathbb{Z}$ wr $\mathbb{Z}$ cannot be self-similar. Furthemore, we prove that if $L$ is a self-similar abelian group then $L^{\omega }$ wr $C_2$ is also self-similar.