Self-similar abelian groups and their centralizers

Alex C. Dantas¹  ; Tulio M. G. Santos² ; Said N. Sidki¹

¹Universidade de Brasília, Brazil
²Instituto Federal Goiano, Campos Belos, Brazil

Groups, geometry, and dynamics, Tome 17 (2023) no. 2, pp. 577-599

Résumé

We extend results on transitive self-similar abelian subgroups of the group of automorphisms Am​ of an m-ary tree Tm​ by Brunner and Sidki 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​. Precisely, 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. Furthermore, we prove that if A is non-trivial, then A∗=CAm​​(Δ(A)), the centralizer of Δ(A) in Am​. 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. In the final section, we introduce for m=ns≥2, a generalized adding machine a, an automorphism of Tm​, and show that its centralizer in Am​ to be a split extension of 〈a〉∗ by As​. We also describe important Zn​[As​] submodules of 〈a〉∗.

Détail

DOI : 10.4171/ggd/710

Classification : 20-XX
Mots-clés : Groups acting on rooted m-tree, self-similar abelian groups, centralizers of abelian groups