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

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

No reviews yet.

No comments yet.