Self-similar abelian groups and their centralizers

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Autor(es): dc.contributorUniversidade de Brasília, Departamento de Matemática-
Autor(es): dc.contributorInstituto Federal Goiano-
Autor(es): dc.contributorUniversidade de Brasília, Departamento de Matemática-
Autor(es): dc.creatorDantas, Alex Carrazedo-
Autor(es): dc.creatorSantos, Tulio Marcio Gentil dos-
Autor(es): dc.creatorSidki, Said Najati-
Data de aceite: dc.date.accessioned2025-03-18T17:24:14Z-
Data de disponibilização: dc.date.available2025-03-18T17:24:14Z-
Data de envio: dc.date.issued2025-02-04-
Data de envio: dc.date.issued2025-02-04-
Data de envio: dc.date.issued2023-01-27-
Fonte completa do material: dc.identifierhttp://repositorio.unb.br/handle/10482/51468-
Fonte: dc.identifier.urihttp://educapes.capes.gov.br/handle/capes/927871-
Descrição: dc.descriptionWe 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)*.-
Descrição: dc.descriptionInstituto de Ciências Exatas (IE)-
Descrição: dc.descriptionDepartamento de Matemática (IE MAT)-
Descrição: dc.descriptionPrograma de Pós-Graduação em Matemática-
Formato: dc.formatapplication/pdf-
Idioma: dc.languageen-
Publicador: dc.publisherEMS Press-
Direitos: dc.rightsAcesso Aberto-
Direitos: dc.rightsThis work is licensed under a CC BY 4.0 license-
Palavras-chave: dc.subjectGrupos abelianos-
Palavras-chave: dc.subjectCentralizador de grupo abeliano autossimilar-
Título: dc.titleSelf-similar abelian groups and their centralizers-
Tipo de arquivo: dc.typelivro digital-
Aparece nas coleções:Repositório Institucional – UNB - Rep. 1

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