Self-similar abelian groups and their centralizers

Abstract

We extend results on transitive self-similar abelian subgroups of the group of automorphisms ${\mathcal{A}}_m$ of an $m$-ary tree ${\mathcal{T}}_m$ by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has $s\ge 1$ orbits. We prove that such a group $A$ embeds in a self-similar abelian group $A^{\ast }$ which is also a maximal abelian subgroup of ${\mathcal{A}}_m$. The construction of $A^{\ast }$ is based on the definition of a free monoid $\Delta$ of rank $s$ of partial diagonal monomorphisms of ${\mathcal{A}}_m$. Precisely, $A^{\ast }=\overline{\Delta (B(A))}$, where $B(A)$ denotes the product of the projections of $A$ in its action on the different $s$ orbits of maximal subtrees of ${\mathcal{T}}_m$, and bar denotes the topological closure. Furthermore, we prove that if $A$ is non-trivial, then $A^{\ast }=C_{{\mathcal{A}}_m}(\Delta (A))$, the centralizer of $\Delta (A)$ in ${\mathcal{A}}_m$. 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 $\Delta$-invariant for $s=2$. In the final section, we introduce for $m=ns\ge 2$, a generalized adding machine $a$, an automorphism of ${\mathcal{T}}_m$, and show that its centralizer in ${\mathcal{A}}_m$ to be a split extension of $⟨a{⟩}^{\ast }$ by ${\mathcal{A}}_s$. We also describe important ${\mathbb{Z}}_n[{\mathcal{A}}_s]$ submodules of $⟨a{⟩}^{\ast }$.