The congruence subgroup problem for low rank free and free metabelian groups

David El Chai Ben-Ezra*, Alexander Lubotzky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

The congruence subgroup problem for a finitely generated group Γ asks whether Aut(Γ)ˆ→Aut(Γˆ) is injective, or more generally, what is its kernel C(Γ)? Here Xˆ denotes the profinite completion of X. In this paper we first give two new short proofs of two known results (for Γ=F2 and Φ2) and a new result for Γ=Φ3: (1) C(F2)={e} when F2 is the free group on two generators.(2) C(Φ2)=Fˆω when Φn is the free metabelian group on n generators, and Fˆω is the free profinite group on ℵ0 generators.(3) C(Φ3) contains Fˆω.Results (2) and (3) should be contrasted with an upcoming result of the first author showing that C(Φn) is abelian for n≥4.

Original languageEnglish
Pages (from-to)171-192
Number of pages22
JournalJournal of Algebra
Volume500
DOIs
StatePublished - 15 Apr 2018

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

  • Automorphism groups
  • Congruence subgroup problem
  • Free groups
  • Free metabelian groups
  • Profinite groups

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