2021/10/06 by Dantas, Alex C., Santos, Tulio M. G., Sidki, Said N.
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2110.02441
We extend results on transitive self-similar abelian subgroups of the group of automorphisms Am of an m-ary tree Tm in \citeBS, 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 Δ 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 (Δ(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 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 Δ-invariant for s=2. Finally, we focus on self-similar cyclic groups of automorphisms of Tm and compute their centralizers when m=4.