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 $s\geq 1$ orbits. We prove that such a group $A$ embeds in a self-similar abelian group $A^$ which is also a maximal abelian subgroup of $\mathcal{A}\_m$. The construction of $A^{}$ is based on the definition of a free monoid $\Delta$ of rank $s$ of partial diagonal monomorphisms of $\mathcal{A}m$. Precisely, $A^{} = \overline{\Delta(B(A))}$, 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 $A^{} = C{\mathcal{A}m} (\Delta(A))$, 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 $m=ns \geq 2$, 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 $\langle a \rangle^{}$ by $\mathcal{A}\_s$. We also describe important $\mathbb{Z}\_n \[\mathcal{A}\_s]$ submodules of $\langle a\rangle^{}$.