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- W4300908035 abstract "The congruence subgroup problem for a finitely generated group $Gamma$ asks whether $widehat{Autleft(Gammaright)}to Aut(hat{Gamma})$ is injective, or more generally, what is its kernel $Cleft(Gammaright)$? 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. $Cleft(F_{2}right)=left{ eright}$ when $F_{2}$ is the free group on two generators. 2. $Cleft(Phi_{2}right)=hat{F}_{omega}$ 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. $Cleft(Phi_{3}right)$ contains $hat{F}_{omega}$. Results 2. and 3. should be contrasted with an upcoming result of the first author showing that $Cleft(Phi_{n}right)$ is abelian for $ngeq4$." @default.
- W4300908035 created "2022-10-04" @default.
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- W4300908035 date "2016-08-14" @default.
- W4300908035 modified "2023-10-14" @default.
- W4300908035 title "The Congruence Subgroup Problem for low rank Free and Free Metabelian groups" @default.
- W4300908035 doi "https://doi.org/10.48550/arxiv.1608.04151" @default.
- W4300908035 hasPublicationYear "2016" @default.
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