# Self-similar abelian groups and their centralizers

### Alex C. Dantas

Universidade de Brasília, Brazil### Tulio M. G. Santos

Instituto Federal Goiano, Campos Belos, Brazil### Said N. Sidki

Universidade de Brasília, Brazil

## Abstract

We extend results on transitive self-similar abelian subgroups of the group of automorphisms $A_{m}$ of an $m$-ary tree $T_{m}$ by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has $s≥1$ orbits. We prove that such a group $A$ embeds in a self-similar abelian group $A_{∗}$ which is also a maximal abelian subgroup of $A_{m}$. The construction of $A_{∗}$ is based on the definition of a free monoid $Δ$ of rank $s$ of partial diagonal monomorphisms of $A_{m}$. Precisely, $A_{∗}=Δ(B(A)) $, where $B(A)$ denotes the product of the projections of $A$ in its action on the different $s$ orbits of maximal subtrees of $T_{m}$, and bar denotes the topological closure. Furthermore, we prove that if $A$ is non-trivial, then $A_{∗}=C_{A_{m}}(Δ(A))$, the centralizer of $Δ(A)$ in $A_{m}$. When $A$ is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also $Δ$-invariant for $s=2$. In the final section, we introduce for $m=ns≥2$, a generalized adding machine $a$, an automorphism of $T_{m}$, and show that its centralizer in $A_{m}$ to be a split extension of $⟨a⟩_{∗}$ by $A_{s}$. We also describe important $Z_{n}[A_{s}]$ submodules of $⟨a⟩_{∗}$.

## Cite this article

Alex C. Dantas, Tulio M. G. Santos, Said N. Sidki, Self-similar abelian groups and their centralizers. Groups Geom. Dyn. 17 (2023), no. 2, pp. 577–599

DOI 10.4171/GGD/710