Banca de DEFESA: Mattheus Pereira da Silva Aguiar

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : Mattheus Pereira da Silva Aguiar
DATE: 14/07/2023
TIME: 14:00
LOCAL: Departamento de Matemática
TITLE:

Decomposições de grupos profinitos e aplicações

KEY WORDS:

Combinatorial Group Theory; profinite groups; infinite graphs of groups; Stallings'
decomposition.

PAGES: 154
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

In this thesis, we study splittings of profinite groups as HNN-extensions and amalgamated free products. In fact, these constructions can be considered as particular cases of a profinite fundamental group of a graph of groups, which we denote by $\Pi_1(\GA,\G)$. Hence, if a given profinite group $G$ has a splitting $G=\Pi_1(\GA,\G)$ for some profinite graph of groups $(\GA,\G)$, we obtain not only properties of the group $G$ but also properties of the graph of groups $(\GA,\G)$. 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 $(\overline{\GA},\overline{\G})$ such that $\G$ embeds in $\overline{\G}$ and the profinite completion of $G$ splits as $\widehat{G}=\Pi_1(\overline{\GA},\overline{\G})$. 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. The second is related to subgroup conjugacy separability of virtually free groups, generalizing the main Theorem of a paper by Chagas and Zalesski. 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 celebrated 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 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 $H=\Pi_1(\HA,\D)$, and we suppose conjugacy classes of vertex groups are $G$-invariant then $G$ also splits as $G=\Pi_1(\GA,\G)$. 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 that $|E(\G)| \leq |E(\D)|$. With this construction in hand, we provide three applications. First, we show that if $G$ is a finitely generated pro-$p$ group having an open normal subgroup $H$ acting on a pro-$p$ tree $T$, with $\{H_v \mid v \in V(T)\}$ being $G$-invariant, then $G$ splits as $G=\Pi_1(\GA,\G)$. We also prove that generalized accessibility of finitely generated pro-$p$ groups is closed for commensurability. We finish the thesis by showing that our Theorem 11 holds even for Wilkes' example of a pro-$p$ inaccessible group.

COMMITTEE MEMBERS:
Externo à Instituição - JOHN WILLIAM MACQUARRIE - UFMG
Presidente - 1285938 - PAVEL ZALESSKI
Interna - 1177682 - SHEILA CAMPOS CHAGAS
Externo à Instituição - SLOBODAN TANUSHEVSKI - UFF
Interno - 2554004 - THEO ALLAN DARN ZAPATA