The profinite completion of the fundamental group of infinite graphs of groups

Abstract: 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.