The Congruence Subgroup Problem for low rank Free and Free Metabelian groups

Abstract

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 $n\ge 4$.

arXiv comment20 pages