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

Abstract

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