The profinite completion of the fundamental group of infinite graphs of groups (Q2089818)

The theory of profinite graphs has its origin in the seminal paper by \textit{D. Gildenhuys} and \textit{L. Ribes} [J. Pure Appl. Algebra 12, 21--47 (1978; Zbl 0428.20018)] where the combinatorial group theory for profinite groups has been developed in similar lines to the characterization done by Serre and Bass of the discrete groups acting on abstract trees. This paper is dedicated to answering some questions raised by \textit{L. Ribes} [Profinite graphs and groups. Cham: Springer (2017; Zbl 1457.20001)]. Given an infinite graph of profinite groups \((\mathcal{G},\Gamma)\), authors construct a profinite graph of groups \((\overline{\mathcal{G}},\overline{\Gamma})\) such that \(\Gamma\) is densely embedded in \(\overline{\Gamma}\), the fundamental profinite group \(\Pi_{1}(\overline{\mathcal{G}},\overline{\Gamma})\) is the profinite completion of \(\pi_{1}(\mathcal{G},\Gamma)\) and the standard tree \(S(\mathcal{G},\Gamma)\) embeds densely in the standard profinite tree \(S(\overline{\mathcal{G}}, \overline{\Gamma})\) (Ribes' Open Question 6.7.1). The authors prove that a virtually free group \(G\) is subgroup conjugacy separable (Ribes' Open Question 15.11.10) and the normalizer \(N_{G}(H)\) of a finitely generated subgroup \(H\) of \(G\) is dense in \(N_{\widehat{G}}(\overline{H})\) (Ribes' Open Question 15.11.11).