Normalizers in groups and in their profinite completions

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 R̂ the profinite completion of R and let H̄ be the closure of H in R̂. It is proved that the normalizer NR̂(H̄) of H̄ in R̂ is the closure in R̂ of NR(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 ∈ R; then x and y are conjugate in R if their images are conjugate in every finite quotient of R. Let R be a residually finite abstract group. Then R is embedded naturally in its profinite completion R̂ = lim ←− U∈U R/U, where U denotes the collection of all normal subgroups U of finite index in R. Given a subset X of R, denote its topological closure in R̂ by X̄. This paper is concerned with the following problem: if H is a finitely generated subgroup of R, what is the relationship between the normalizer NR(H) of H in R and the normalizer NR̂(H̄) of H̄ in R̂? Originally this question arose in [10] while studying conjugacy separability in groups that arise as iterations of amalgamated free products of certain groups. In [10] the question is answered when R is a finite extension of a polycyclic group; and then the answer is the desirable one: NR̂(H̄) is the closure in R̂ of NR(H). Perhaps not completely surprising given the nature of polycyclic groups, the proof of that result is ‘arithmetic’ and eventually it relies on number theoretic results and methods. In this paper we deal with the case when R is a finitely generated abstract virtually free group, i.e., a finite extension of a free group Φ. Such a group is residually finite. In fact our results are placed in a more general setting: we Mathematics Subject Classification (2010): 20E18, 20E08.