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

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 res...

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Bibliographic Details
Published in:Journal of algebra 2018-04, Vol.500, p.171-192
Main Authors: Ben-Ezra, David El-Chai, Lubotzky, Alexander
Format: Article
Language:English
Subjects:
Automorphism groups
Congruence subgroup problem
Free groups
Free metabelian groups
Profinite groups
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Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2017.01.001