Normalizers in groups and in their profinite completions

  • Luis Ribes

    Carleton University, Ottawa, Canada
  • Pavel A. Zalesskiĭ

    Universidade de Brasília, Brazil

Abstract

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

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