We extend results on transitive self-similar abelian subgroups of the group of automorphisms of an -ary tree by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has orbits. We prove that such a group embeds in a self-similar abelian group which is also a maximal abelian subgroup of . The construction of is based on the definition of a free monoid of rank of partial diagonal monomorphisms of . Precisely, , where denotes the product of the projections of in its action on the different orbits of maximal subtrees of , and bar denotes the topological closure. Furthermore, we prove that if is non-trivial, then , the centralizer of in . When 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 . In the final section, we introduce for , a generalized adding machine , an automorphism of , and show that its centralizer in to be a split extension of by . We also describe important submodules of .
Cite this article
Alex C. Dantas, Tulio M. G. Santos, Said N. Sidki, Self-similar abelian groups and their centralizers. Groups Geom. Dyn. 17 (2023), no. 2, pp. 577–599DOI 10.4171/GGD/710