2020/10/24 by Mattheus Aguiar, Aguiar, Mattheus, Pavel Zalesski +1
Computer Science · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2010.12720
openalex publication_date 2020/10/24 · openalex created_date 2022/08/13 · openalex updated_date 2026/07/28
Let (G,Γ) be an abstract graph of finite groups. If Γ is finite, we can construct a profinite graph of groups in a natural way (G,Γ), where G(m) is the profinite completion of G(m) for all m ∈ Γ. The main reason for this is that Γ is finite, so it is already profinite. In this paper we deal with the infinite case, by constructing a profinite graph Γ where Γ is densely embedded and then defining a profinite graph of groups (\widehatG,Γ). We also prove that the fundamental group Π1(\widehatG,Γ) is the profinite completion of Π1abs(G,Γ). This answers Open Question 6.7.1 of the book Profinite Graphs and Groups, published by Luis Ribes in 2017. Later we generalise the main theorem of a paper by Luis Ribes and the second author, proving that if R is a virtually free abstract group and H is a finitely generated subgroup of R, then NR(H)=N_R(H) answering Open Question 15.11.10 of the book of Ribes. Finally, we generalise the main theorem of a paper by Sheila Chagas and the second author, showing that every virtually free group is subgroup conjugacy separable. This answers Open Question 15.11.11 of the same book of Ribes.