Title:	Self-similar abelian groups and their centralizers
Authors:	Dantas, Alex Carrazedo Santos, Tulio Marcio Gentil dos Sidki, Said Najati
metadata.dc.contributor.affiliation:	Universidade de Brasília, Departamento de Matemática Instituto Federal Goiano Universidade de Brasília, Departamento de Matemática
Assunto::	Grupos abelianos Centralizador de grupo abeliano autossimilar
Issue Date:	28-Jan-2023
Publisher:	EMS Press
Citation:	DANTAS, Alex C.; SANTOS, Tulio M. G.; SIDKI, Said N. Self-similar abelian groups and their centralizers. Groups Geometry, and Dynamics, [S. l.], v. 17, n. 2, p. 577–599, 2023. DOI: 10.4171/GGD/710. Disponível em: https://ems.press/journals/ggd/articles/9221800.
Abstract:	We extend results on transitive self-similar abelian subgroups of the group of automor-phisms 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)*.
metadata.dc.description.unidade:	Instituto de Ciências Exatas (IE) Departamento de Matemática (IE MAT)
metadata.dc.description.ppg:	Programa de Pós-Graduação em Matemática
Licença::	This work is licensed under a CC BY 4.0 license
DOI:	10.4171/GGD/710
Appears in Collections:	Artigos publicados em periódicos e afins

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