# Normalizers in groups and in their profinite completions

### Luis Ribes

Carleton University, Ottawa, Canada### Pavel A. Zalesskiĭ

Universidade de Brasília, Brazil

## 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 $R$.

## Cite this article

Luis Ribes, Pavel A. Zalesskiĭ, Normalizers in groups and in their profinite completions. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 165–190

DOI 10.4171/RMI/773