The Congruence Subgroup Problem for low rank Free and Free Metabelian groups
The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether $\widehat{Aut\left(\Gamma\right)}\to Aut(\hat{\Gamma})$ is injective, or more generally, what is its kernel $C\left(\Gamma\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $\Gamma=F_{2}$ and $\Phi_{2}$) and a new result for $\Gamma=\Phi_{3}$: 1. $C\left(F_{2}\right)=\left\{ e\right\}$ when $F_{2}$ is the free group on two generators. 2. $C\left(\Phi_{2}\right)=\hat{F}_{\omega}$ when $\Phi_{n}$ is the free metabelian group on $n$ generators, and $\hat{F}_{\omega}$ is the free profinite group on $\aleph_{0}$ generators. 3. $C\left(\Phi_{3}\right)$ contains $\hat{F}_{\omega}$. Results 2. and 3. should be contrasted with an upcoming result of the first author showing that $C\left(\Phi_{n}\right)$ is abelian for $n\geq4$.