Título:	Splittings of profinite groups and its applications
Autor(es):	Aguiar, Mattheus Pereira da Silva
Orientador(es):	Zalesski, Pavel
Assunto:	Grupos profinitos Teoria combinatória de grupos
Data de publicação:	8-Ago-2024
Data de defesa:	14-Jul-2023
Referência:	AGUIAR, Mattheus Pereira da Silva. Splittings of profinite groups and its applications. 2023. 123 f. Tese (Doutorado em Matemática) — Universidade de Brasília, Brasília, 2023.
Abstract:	In this thesis we study one of the main objects in profinite combinatorial group theory: splittings of profinite groups as HNN-extensions or amalgamated free products. We answer three Open Questions proposed by Luis Ribes in his 2017 book "Profinite Graphs and Groups" (see Open Questions 6.7.1, 15.11.10, and 15.11.11 of [31]). These results generalize the main Theorems of [7] and [34]. We also generalize the pro-p version of the celebrated Stallings’ decomposition theorem to splittings over infinite pro-p groups. This extends by far the Weigel-Zalesski result from 2017 and it does not have any abstract analogs. Finally we prove that generalized accessibility of finitely generated pro-p groups is closed for commensurability. Profinite amalgamated products and profinite HNN-extensions can be considered as particular cases of profinite fundamental groups of graphs of groups, which we denote by Π1pG, Γq. Hence, if a profinite group G has a splitting G “ Π1pG, Γq for some profinite graph of groups pG, Γq, we obtain not only properties of the group G but also properties of the graph of groups pG, Γq. In the first part, given an abstract group G that splits as the fundamental group of an infinite graph of groups, we construct a profinite graph of groups pG, Γq such that Γ embeds in Γ and the profinite completion of G splits as Π1pG, Γq. This answers an Open Question of Ribes. With this construction in hand, we answer two more Open Questions of Ribes. The first concerns the closure of normalizers, which generalizes the main Theorem of a paper by Ribes and Zalesski (cf. [34]). The second is related to subgroup conjugacy separability of virtually free groups, generalizing the main Theorem of a paper by Chagas and Zalesski (cf. [7]). Our strategy for solving the problems above is to describe the profinite fundamental group of a graph of groups in the language of paths. Since it behaves very well via inverse limits, it facilitates the interrelation between the abstract and the profinite settings. We continue our journey by investigating the Stallings’ decomposition Theorem. It states that the splitting of a finite index subgroup H of a finitely generated group G as an amalgamated free product or an HNN-extension over a finite group implies the same for G. The pro-p version of this result was obtained by Weigel and Zalesskii (see [45]) in 2017. We proved that, in the category of pro-p groups, splitting theorems hold beyond splittings over finite groups. In fact, if G is a finitely generated pro-p group having an open normal subgroup H that splits as H “ Π1pH, ∆q, and we suppose conjugacy classes of vertex groups are G-invariant then G also splits as G “ Π1pG, Γq (see Theorem 11). If H is a non-trivial free pro-p product we obtain, as a particular case, the aforementioned Weigel-Zalesski Theorem. The main tool behind the proof is our Limitation Theorem, which establishes a bound for EpΓq, namely |EpΓq| ď |Ep∆q|. We attach to our Limitation Theorem the following result: if G is a finitely generated pro-p group having an open normal subgroup H acting on a pro-p tree T, with tHv | v P V pTqu being G-invariant, then G splits as G “ Π1pG, Γq. With these results in hand, we provide a powerful application: generalized accessibility of finitely generated pro-p groups is closed for commensurability. We finish the thesis by showing that our Theorem 9 holds even for Wilkes’ example of a pro-p inaccessible group.
Unidade Acadêmica:	Instituto de Ciências Exatas (IE) Departamento de Matemática (IE MAT)
Informações adicionais:	Tese (doutorado) — Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2023.
Programa de pós-graduação:	Programa de Pós-Graduação em Matemática
Licença:	A concessão da licença deste item refere-se ao termo de autorização impresso assinado pelo autor com as seguintes condições: Na qualidade de titular dos direitos de autor da publicação, autorizo a Universidade de Brasília e o IBICT a disponibilizar por meio dos sites www.bce.unb.br, www.ibict.br, http://hercules.vtls.com/cgi-bin/ndltd/chameleon?lng=pt&skin=ndltd sem ressarcimento dos direitos autorais, de acordo com a Lei nº 9610/98, o texto integral da obra disponibilizada, conforme permissões assinaladas, para fins de leitura, impressão e/ou download, a título de divulgação da produção científica brasileira, a partir desta data.
Agência financiadora:	Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
Aparece nas coleções:	Teses, dissertações e produtos pós-doutorado

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