On wreath product occurring as subgroup of automata group

Abstract: A finitely generated group is said to be an automata group if it admits a faithful self-similar finite-state representation on some regular $m$-tree. We prove that if $G$ is a subgroup of an automata group, then for each finitely generated abelian group $A$, the wreath product $A \wr G$ is a subgroup of an automata group. We obtain, for example, that $C_2 \wr (C_{2} \wr \mathbb{Z})$, $\mathbb{Z} \wr (C_2 \wr \mathbb{Z})$, $C_2 \wr (\mathbb{Z} \wr \mathbb{Z})$, and $\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$ are subgroups of automata groups. In the particular case $\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$, we prove that it is a subgroup of a two-letters automata group; this solves Problem 15.19 - (b) of the Kourovka Notebook proposed by A. M. Brunner and S. Sidki in 2000 [8, 17].