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.