The Congruence Subgroup Problem for finitely generated Nilpotent Groups

Abstract

The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphism $C\left(Γ\right)\simeq C(Aut^{*}(Γ),Γ^{*})$. In other words, $C\left(Γ\right)$ 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 $C(Ψ_{n,c})=C(\mathbb{Z}^{(n)})=\{e\}$ whenever $n\geq3$, and $C(Ψ_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_ω$ = the free profinite group on countable number of generators.