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

TL;DR

This paper investigates the congruence subgroup problem for low-rank free and free metabelian groups, providing new proofs and results about the kernels of automorphism completions, revealing their structure for specific groups.

Contribution

It offers new short proofs for known results and establishes a novel result for the free metabelian group on three generators, advancing understanding of the kernel structure.

Findings

Kernel for free group on two generators is trivial.

Kernel for free metabelian group on two generators is a free profinite group on countably many generators.

Kernel for free metabelian group on three generators contains a free profinite group.

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 $n\geq4$.