The Congruence Subgroup Problem for the Free Metabelian group on n≥4 generators

2017/01/10 by David BenEzra, Ben-Ezra, David El-Chai
Mathematics · #20E18 #20H05 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Primary: 19B37 #Secondary: 20E36 #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1701.02459

openalex publication_date 2017/01/10 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

The congruence subgroup problem for a finitely generated group Γ asks whether the map Aut(Γ)→ Aut(Γ) is injective, or more generally, what is its kernel C(Γ)? Here X denotes the profinite completion of X. It is well known that for finitely generated free abelian groups C(ℤⁿ)=\ 1\ for every n≥3, but C(ℤ²)=F_ω, where F_ω is the free profinite group on countably many generators. Considering Φ_n, the free metabelian group on n generators, it was also proven that C(Φ₂)=F_ω and C(Φ₃)⊇F_ω. In this paper we prove that C(Φ_n) for n≥4 is abelian. So, while the dichotomy in the abelian case is between n=2 and n≥3, in the metabelian case it is between n=2,3 and n≥4.

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