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The Kurosh Subgroup Theorem for Profinite Groups

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UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIÊNCIAS EXATAS DEPARTAMENTO DE PÓS-GRADUAÇÃO EM MATEMÁTICA The Kurosh Subgroup Theorem for profinite groups Mattheus Pereira da Silva Aguiar Belo Horizonte - MG July 2019 UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIÊNCIAS EXATAS DEPARTAMENTO DE PÓS-GRADUAÇÃO EM MATEMÁTICA Mattheus Pereira da Silva Aguiar Supervisor: PhD. John William MacQuarrie The Kurosh Subgroup Theorem for profinite groups Thesis submitted to the examination board assigned by the Graduate Program in Mathematics of the Institute of Exact Sciences - ICEX of the Federal University of Minas Gerais, as a partial requirement to obtain a Master Degree in Mathematics. Belo Horizonte - MG July 2019 “First, have a definite, clear practical ideal; a goal, an objective. Second, have the necessary means to achieve your ends; wisdom, money, materials, and methods. Third, adjust all your means to that end.”. Aristotle ABSTRACT A profinite graph Γ is a profinite space with a graph structure, i.e., a closed set named the vertex set and two continuous incidence maps d0; d1 ∶ Γ → V (Γ). A profinite group G can act on a profinite graph and GΓ is called the quotient graph by the action of G. If G acts freely, the map ζ ∶ Γ → GΓ is called a Galois covering of the profinite graph Γ and the group G the associated group of this Galois covering. We can also define a universal Galois covering, where the associated group with this covering is the fundamental group of the profinite graph Γ, C denoted by π1 (Γ). We give an original proof of the Nielsen-Schreier Theorem for free profinite groups over a finite space, which states that every open profinite subgroup of a free profinite group on a finite space is a free profinite subgroup on a finite space using these aforementioned structures. We also define a sheaf of pro-C groups, the free pro-C product of a sheaf and use it to define a graph of pro-C groups and the fundamental group of a graph of pro-C groups in a similar manner we did previously. The analogue of the universal Galois covering is named the C standard graph, denoted by S (G; Γ). Finally we show that the free product of pro-C groups can be seen as the fundamental group of a graph of pro-C groups and use this fact to prove the Kurosh Subgroup Theorem for profinite groups, the main result of this thesis. Keywords: Profinite graphs, Profinite groups, Galois coverings, Fundamental group, Nielsen- Schreier, Graph of pro-C groups, Kurosh. RESUMO EXPANDIDO A teoria de Bass-Serre para grafos abstratos foi inaugurada no livro ’Arbres, amalgames, SL2’ (’Trees’ na versão em inglês), escrito por Jean-Pierre Serre em colaboração com Hyman Bass (1977). A motivação original do Serre era entender a estrutura de certos grupos algébricos cujas construções Bruhat-Tits são árvores. No entanto, a teoria rapidamente se tornou uma ferramenta padrão para a teoria geométrica de grupos e a topologia geométrica, particularmente no estudo de 3-variedades. Um trabalho posterior de Bass contribuiu substancialmente para a formalização e o desenvolvimento das ferramentas básicas e atualmente o termo teoria de Bass-Serre é amplamente empregado para descrever a disciplina. Essa teoria busca explorar e generalizar as propriedades de duas construções já bem conhecidas da teoria grupos: produto livre com amalgamação e extensões HNN. No entanto, ao contrário dos estudos algébricos tradicionais dessas duas estruturas, a teoria de Bass-Serre utiliza a linguagem geométrica de espaços de recobrimento e grupos fundamentais. O análogo profinito iniciou em Gildenhuys e Ribes (1978) onde o nome booleano foi utilizado. O objetivo era construir um paralelo entre a teoria de Bass-Serre de grupos abstratos agindo sobre árvores abstratas para grupos profinitos e aplicações em grupos abstratos. O C- recobrimento galoisiano universal de um grafo conexo também foi definido nesse artigo de Gildenhuys e Ribes (1978). Haran (1987) e Mel’nikov (1989) expandiram essas ideias de maneira independente e desenvolveram abordagens gerais para produtos livres de grupos profinitos indexados por um espaço profinito. O objetivo deles era ser capaz de descrever a estrutura de pelo menos certos subgrupos fechados de produtos pro-p livres de grupos pro-p, demonstrando uma versão profinita do Teorema do Subgrupo de Kurosh, que é o teorema principal dessa dissertação. Os artigos de Haran e Mel’nikov obtém resultados similares. Nesse texto, adotamos a elegante versão de Mel’nikov. As primeiras seções do capítulo 4 são baseadas em Zalesskii e Mel’nikov (1989). O primeiro capítulo dessa dissertação estabelece esse contexto histórico do desenvolvimento da teoria de grafos profinitos. O segundo inicia com a definição de limite inverso com algumas de suas propriedades, o que nos possibilita definir um espaço profinito, que surge como o limite inverso de espaços topológicos finitos, cada um com a topologia discreta. Fornecidas tais definições básicas, definimos um grafo profinito, que é um espaço profinito com estrutura de grafo. O espaço de vértices é um subspaço fechado (logo também profinito) e as aplicações de incidência são aplicações contínuas (no ponto de vista topológico), do espaço todo para o espaço dos vértices. Então definimos um q-morfismo de grafos profinitos, que é uma generalização do morfismo usual de grafos abstratos, pois permite que arestas sejam mapeadas em vértices. Em seguida definimos ação de grupos profinitos em garfos profinitos e o grafo quociente pela ação de um grupo profinito G, que é o espaço das órbitas pela ação de G e como um exemplo construímos o grafo de Cayley de um grupo profinito. No terceiro capítulo introduzimos o conceito de recobrimento galoisiano e do C- recobrimento galoisiano, que é uma aplicação ζ ∶ Γ → ∆ = GΓ de um grafo profinito Γ no grafo quociente pela ação do grupo profinito G, ∆. O grupo profinito G é dito o grupo associado ao recobrimento galoisiano. Definimos então morfismos de recobrimentos galoisianos e o conceito C de C-recobrimento galoisiano universal, cujo grupo associado é o grupo fundamental π1 (Γ) do grafo conexo Γ. Em seguida estabelecemos condições para que 0-seções e 0-transversais existam e a construção dos recobrimentos galoisianos universais. Terminamos o capítulo apresentando dois exemplos importantes: 3.4.5, 3.4.6 e uma demonstração original da versão profinita do Teorema de Nielsen-Schreier (quando o espaço profinito é finito), cujo enunciado estabelece que todo subgrupo aberto de um grupo profinito livre é profinito livre em um espaço finito. Esse teorema é demonstrado em [2], do artigo de Binz, Neukirch e Wenzel [1971] e utiliza a versão abstrata. Ribes e Steinberg (2010) deram uma nova demonstração sem utilizar a versão abstrata, através de produtos entrtelaçados, mas não é tão simples quanto a apresentada nessa dissertação. A abordagem também é completamente diferente. O último capítulo contém os principais tópicos dessa dissertação: a definição de um feixe de pro-C grupos, o produto livre de um feixe de pro-C grupos, grafos de pro-C grupos e especializações, o grupo fundamental de um grafo de pro-C grupos, o grafo padrão de um grafo de pro-C grupos e o Teorema de Kurosh, que estabelece: Theorem 1 (Kurosh). Seja C uma pseudo-variedade de grupos finitos fechada para extensão. n Seja G = ∐i=1 Gi o produto pro-C livre de um número finito de grupos pro-C Gi. Se H é um subgrupo aberto de G, então n −1 H = O O (H ∩ gi,τ Gigi,τ ) F i=1 τ∈HG~Gi −1 é um produto pro-C livre de grupos H ∩ gi,τ Gigi,τ , onde, cada i = 1; ⋯; n, gi,τ varia sobre um sistema de representantes das classes laterais duplas HG~Gi, e F é um grupo pro-C livre de posto finito rF , n rF = 1 − t + Q(t − ti); i=1 onde t = [G ∶ H] e ti = SHG~GiS: A ideia da demonstração é baseada em enxergar o produto livre de um número finito de grupos pro-C como o grupo fundamental de um grafo de pro-C grupos em que todos os grupos de aresta são triviais e H age sobre o grafo padrão de pro-C grupos. ACKNOWLEDGEMENT I would first like to thank God for the life, health, faith, determination and wisdom. I would also like to thank my mother and my sister that always had been on my side encouraging, supporting and giving me strength to surpass all the challenges I had to face in this journey. I love you both. To Professor Anderson Porto that was and is a friendly encouraging, since the mathematical Olympiads, giving me the hand and the road map to achieve uncountable many realisations. Also to Professors Leonardo Gomes and Ana Cristina Vieira for the guidance and to the master and undergraduate colleagues, in special to Maria Cecíllia Alecrim, Moacir Aloísio, José Alves and Mateus Figueira for the companionship and friendship. Finally I would like to thank my thesis advisor Professor John MacQuarrie. The door to Prof. MacQuarrie’s office was always open whenever I had a question or a new idea about my research or writing. The present work was realised with the support of the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Code of financing 001. Contents 1 Introduction1 2 Profinite graphs3 2.1 Inverse limits.......................................3 2.2 Profinite graphs......................................7 2.3 Groups acting on profinite graphs........................... 16 3 The fundamental group of a profinite graph 21 3.1 Galois coverings..................................... 21 3.2 Universal Galois coverings and fundamental groups................. 26 3.3 0-transversals and 0-sections.............................. 28 3.4 Existence of universal Galois coverings and the Nielsen-Schreier Theorem for pro-C groups......................................
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