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

Given an infinite graph of profinite groups ( $${\cal G}$$ , Γ) we construct a $$\overline {\cal G} ,\overline \Gamma $$ ) such that Γ is densely embedded in the fundamental group $${\Pi _1}\left( {\overline } \right)$$ completion $${\pi {{\cal G},\Gamma and standard tree $$S\left( embeds . This answers Ribes’ question [5, Question 6.7.1]. Generalizing main results [8] [2] answer two other questions Ribes Questions 15.11.10 15.11.11] proving virtually free G subgroup conjugacy separable normalizer NG(H) finitely generated H dense $${N_{\hat G}}\left( We also give entirely new description using language paths.