Let be a finitely generated virtually free group (a finite extension of a free group) and let be a finitely generated subgroup of . Denote by the profinite completion of and let be the closure of in . It is proved that the normalizer of in is the closure in of . The proof is based on the fact that is the fundamental group of a graph of finite groups over a finite graph and on the study of the minimal -invariant subtrees of the universal covering graph of that graph of groups. As a consequence we prove results of the following type: let be a group that is an extension of a free group by finite solvable group, and let ; then and are conjugate in if their images are conjugate in every finite quotient of .
Cite this article
Luis Ribes, Pavel A. Zalesskiĭ, Normalizers in groups and in their profinite completions. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 165–190DOI 10.4171/RMI/773