On self-similarity of wreath products of abelian groups

Abstract

We prove that a self-similar free abelian group has finite rank. We apply the result to self-similar wreath products of abelian groups $G=BwrX$. We show that if $X$ is torsion-free, then $B$ is torsion of finite exponent. Furthemore, we construct a self-similar group $G=Bwr{C}_2$ where $B$ is free abelian of infinite rank.

arXiv commentTheorem 1 from the previous version is removed as it has a faulty proof