Abstract groups vs their profinite completions
For an abstract group R that is either free-by-finite or polycyclic-by-finite, in Chap. 13 one studies the relationship between certain constructions in R (normalizers and centralizers of a finitely generated subgroup or the intersection of finitely generated subgroups) and corresponding constructions in the profinite completion R of R. It is proved, for example, that if H is a finitely generated subgroup of R, the topological closure NR(H) (in R) of the normalizer NR(H) of H in R coincides with NR(H¯), the normalizer in R of the closure of H in R. For finitely generated subgroups H1 and H2 of R, it is proved that (Formula Presented.). In fact the results are obtained in greater generality for a free-by- C group, i.e., an extension of a free abstract group by a group in a pseudovariety of finite groups C (a collection of finite groups closed under subgroups, quotients and finite direct products), and instead of the profinite completion, one considers the pro- C completion.
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Ribes, L. (2017). Abstract groups vs their profinite completions. In Ergebnisse der Mathematik und ihrer Grenzgebiete. doi:10.1007/978-3-319-61199-0_13