Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms Am of an m-ary tree Tm in [2], 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 Am. The construction of A is based on the definition of a free monoid ∆ of rank s of partial diagonal monomorphisms of Am, which is used to determine the structure of CAm(A), the centralizer of A in Am. Indeed, we prove A ∗ = CAm(∆(A)) = ∆(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 Tm 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 ∆-invariant for s = 2. Finally, we focus on self-similar cyclic groups of automorphisms of Tm and compute their centralizers when m = 4.