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 N_G(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.