# On self-similarity of wreath products of abelian groups

### Alex C. Dantas

Universidade Tecnológica Federal do Paraná, Guarapuava, Brazil### Said N. Sidki

Universidade de Brasilia, Brazil

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## 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.

## Cite this article

Alex C. Dantas, Said N. Sidki, On self-similarity of wreath products of abelian groups. Groups Geom. Dyn. 12 (2018), no. 3, pp. 1061–1068

DOI 10.4171/GGD/462