Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The profinite completion of the fundamental group of infinite graphs of
  groups   Let $(\mathcal{G},\Gamma)$ be an abstract graph of finite groups. If $\Gamma$
is finite, we can construct a profinite graph of groups in a natural way
$(\hat{\mathcal{G}},\Gamma)$, where $\hat{\mathcal{G}}(m)$ is the profinite
completion of $\mathcal{G}(m)$ for all $m \in \Gamma$. The main reason for this
is that $\Gamma$ is finite, so it is already profinite. In this paper we deal
with the infinite case, by constructing a profinite graph $\overline{\Gamma}$
where $\Gamma$ is densely embedded and then defining a profinite graph of
groups $(\widehat{\mathcal{G}},\overline{\Gamma})$. We also prove that the
fundamental group $\Pi_1(\widehat{\mathcal{G}},\overline{\Gamma})$ is the
profinite completion of $\Pi_1^{abs}(\mathcal{G},\Gamma)$. 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
$\overline{N_{R}(H)}=N_{\hat{R}}(\overline{H})$ 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.

24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph