The Congruence Subgroup Problem for the Free Metabelian group on $n\geq4$ generators

Abstract: The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether the map $\hat{Aut\left(\Gamma\right)}\to Aut(\hat{\Gamma})$ is injective, or more generally, what is its kernel $C\left(\Gamma\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C\left(\mathbb{Z}^{n}\right)=\left{ 1\right}$ for every $n\geq3$, but $C\left(\mathbb{Z}^{2}\right)=\hat{F}_{\omega}$, where $\hat{F}{\omega}$ is the free profinite group on countably many generators. Considering $\Phi{n}$, the free metabelian group on $n$ generators, it was also proven that $C\left(\Phi_{2}\right)=\hat{F}{\omega}$ and $C\left(\Phi{3}\right)\supseteq\hat{F}{\omega}$. In this paper we prove that $C\left(\Phi{n}\right)$ for $n\geq4$ is abelian. So, while the dichotomy in the abelian case is between $n=2$ and $n\geq3$, in the metabelian case it is between $n=2,3$ and $n\geq4$.