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2019 The IA-congruence kernel of high rank free metabelian groups
David El-Chai Ben-Ezra
Ann. K-Theory 4(3): 383-438 (2019). DOI: 10.2140/akt.2019.4.383

Abstract

The congruence subgroup problem for a finitely generated group Γ and G A u t ( Γ ) asks whether the map Ĝ A u t ( Γ ̂ ) is injective, or more generally, what its kernel C ( G , Γ ) is. Here X ̂ denotes the profinite completion of X . In this paper we investigate C ( I A ( Φ n ) , Φ n ) , where Φ n is a free metabelian group on n 4 generators, and I A ( Φ n ) = ker ( A u t ( Φ n ) G L n ( ) ) .

We show that in this case C ( I A ( Φ n ) , Φ n ) is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happens for a free metabelian group on n = 2 or 3 generators, or for finitely generated nilpotent groups.

Citation

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David El-Chai Ben-Ezra. "The IA-congruence kernel of high rank free metabelian groups." Ann. K-Theory 4 (3) 383 - 438, 2019. https://doi.org/10.2140/akt.2019.4.383

Information

Received: 2 August 2017; Revised: 28 March 2019; Accepted: 12 April 2019; Published: 2019
First available in Project Euclid: 3 January 2020

zbMATH: 07146015
MathSciNet: MR4043464
Digital Object Identifier: 10.2140/akt.2019.4.383

Subjects:
Primary: 19B37 , 20H05
Secondary: 20E18 , 20E36

Keywords: automorphism groups , congruence subgroup problem , free metabelian groups , profinite groups

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.4 • No. 3 • 2019
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