The Congruence Subgroup Problem for finitely generated Nilpotent Groups

2020/05/07 by David El-Chai Ben-Ezra, Alexander Lubotzky, Ben-Ezra, David El-Chai +1
Mathematics · #11H56 #20E18 #20E36 #20F18 #20F40 #FOS: Mathematics #Group Theory (math.GR) #Primary: 19B37 #Secondary: 20H05 #math.GR #msc:11H56 #msc:19B37 #msc:20E18 #msc:20E36 #msc:20F18 #msc:20F40 #msc:20H05

paper · pdf · doi:10.48550/arxiv.2005.03263

18 pages

arxiv created 2020/05/07 · arxiv updated 2020/05/08

Abstract

The congruence subgroup problem for a finitely generated group Γ and G≤ Aut(Γ) asks whether the map G→ Aut(Γ) is injective, or more generally, what is its kernel C(G,Γ)? Here X denotes the profinite completion of X. In the case G=Aut(Γ) we denote C(Γ)=C(Aut(Γ),Γ). Let Γ be a finitely generated group, Γ=Γ/[Γ,Γ], and Γ^*=Γ/tor(Γ)≅ℤ^(d). Denote Aut^*(Γ)=\textrmIm(Aut(Γ)→ Aut(Γ^*))≤ GL_d(ℤ). In this paper we show that when Γ is nilpotent, there is a canonical isomorphism C(Γ)≃ C(Aut^*(Γ),Γ^*). In other words, C(Γ) is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut^*(Γ). In particular, in the case where Γ=Ψ_n,c is a finitely generated free nilpotent group of class c on n elements, we get that C(Ψ_n,c)=C(ℤ⁽ⁿ⁾)=\e\ whenever n≥3, and C(Ψ_2,c)=C(ℤ⁽²⁾)=F_ω = the free profinite group on countable number of generators.

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