Normalizers in groups and in their profinite completions

Abstract

Let $R$ be a finitely generated virtually free group (a finite extension of a free group) and let $H$ be a finitely generated subgroup of $R$. Denote by $\hat{R}$ the profinite completion of $R$ and let $\bar{H}$ be the closure of $H$ in $\hat{R}$. It is proved that the normalizer $N_{\hat{R}}(\bar{H})$ of $\bar{H}$ in $\hat{R}$ is the closure in $\hat{R}$ of $N_R(H)$. The proof is based on the fact that $R$ is the fundamental group of a graph of finite groups over a finite graph and on the study of the minimal $H$-invariant subtrees of the universal covering graph of that graph of groups. As a consequence we prove results of the following type: let $R$ be a group that is an extension of a free group by finite solvable group, and let $x,y\in R$; then $x$ and $y$ are conjugate in $R$ if their images are conjugate in every finite quotient of $R$.