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 language | English |
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Pages (from-to) | 171-192 |
Number of pages | 22 |
Journal | Journal of Algebra |
Volume | 500 |
DOIs | |
State | Published - 15 Apr 2018 |
Bibliographical note
Publisher Copyright:© 2017 Elsevier Inc.
Keywords
- Automorphism groups
- Congruence subgroup problem
- Free groups
- Free metabelian groups
- Profinite groups