Vol. 4, No. 3, 2019

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The IA-congruence kernel of high rank free metabelian groups

David El-Chai Ben-Ezra

Vol. 4 (2019), No. 3, 383–438

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

We show that in this case C(IA(Φn),Φn) 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.

congruence subgroup problem, automorphism groups, profinite groups, free metabelian groups
Mathematical Subject Classification 2010
Primary: 19B37, 20H05
Secondary: 20E18, 20E36
Received: 2 August 2017
Revised: 28 March 2019
Accepted: 12 April 2019
Published: 17 December 2019
David El-Chai Ben-Ezra
Department of Mathematics
University of California
San Diego, CA
United States