Self-similar products of groups

Abstract

We address the problem of determining the class of self-similar groups, and in particular its closure under restricted direct products. We show that the group ${\mathbb{Z}}^{(\omega )}$ is self-similar, that $G^{(\omega )}⋊C_2$ is self-similar whenever $G$ is, and that permutational wreath products of a finite abelian group with a self-similar group are self-similar.