Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms $\mathcal{A}_m$ of an $m$-ary tree $\mathcal{T}_m$ in \cite{BS}, 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 $Δ$ of rank $s$ of partial diagonal monomorphisms of $\mathcal{A}_m$, which is used to determine the structure of $C_{\mathcal{A}_m}(A)$, the centralizer of $A$ in $\mathcal{A}_m$. Indeed, we prove $A^*=C_{\mathcal{A}_m} (Δ(A))=\overline{ Δ({B(A)})}$, where $B(A)$ denotes the product of the projections of $A$ in its action on the different $s$ orbits of maximal subtrees of $\mathcal{T}_m$ and bar denotes the topological closure. 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 $s=2$. Finally, we focus on self-similar cyclic groups of automorphisms of $\mathcal{T}_m$ and compute their centralizers when $m=4.$