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

David El Chai Ben-Ezra; Alexander Lubotzky

doi:10.1016/j.jalgebra.2017.01.001

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

David El Chai Ben-Ezra, Alexander Lubotzky

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

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)=Fˆ_ω when Φ_n is the free metabelian group on n generators, and Fˆ_ω is the free profinite group on ℵ₀ generators.(3) C(Φ₃) contains Fˆ_ω.Results (2) and (3) should be contrasted with an upcoming result of the first author showing that C(Φ_n) is abelian for n≥4.

Original language	English
Pages (from-to)	171-192
Number of pages	22
Journal	Journal of Algebra
Volume	500
DOIs	https://doi.org/10.1016/j.jalgebra.2017.01.001
State	Published - 15 Apr 2018

Keywords

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

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1016/j.jalgebra.2017.01.001

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title = "The congruence subgroup problem for low rank free and free metabelian groups",

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.",

keywords = "Automorphism groups, Congruence subgroup problem, Free groups, Free metabelian groups, Profinite groups",

author = "Ben-Ezra, {David El Chai} and Alexander Lubotzky",

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year = "2018",

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day = "15",

doi = "10.1016/j.jalgebra.2017.01.001",

language = "الإنجليزيّة",

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T1 - The congruence subgroup problem for low rank free and free metabelian groups

AU - Ben-Ezra, David El Chai

AU - Lubotzky, Alexander

PY - 2018/4/15

Y1 - 2018/4/15

N2 - 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.

AB - 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.

KW - Automorphism groups

KW - Congruence subgroup problem

KW - Free groups

KW - Free metabelian groups

KW - Profinite groups

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DO - 10.1016/j.jalgebra.2017.01.001

M3 - مقالة

SN - 0021-8693

VL - 500

SP - 171

EP - 192

JO - Journal of Algebra

JF - Journal of Algebra

ER -

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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