The Congruence Subgroup Problem for finitely generated Nilpotent Groups

DOI10.1515/JGTH-2021-0039zbMATH Open1511.20091arXiv2005.03263MaRDI QIDQ6340217

Alexander Lubotzky, David El-Chai Ben-Ezra

Publication date: 7 May 2020

Abstract: The congruence subgroup problem for a finitely generated group $Gamma$ and $GleqAut(Gamma)$ asks whether the map $hatGoAut(hatGamma)$ is injective, or more generally, what is its kernel $Cleft(G,Gammaight)$? Here $hatX$ denotes the profinite completion of $X$. In the case $G=Aut(Gamma)$ we denote $Cleft(Gammaight)=Cleft(Aut(Gamma),Gammaight)$. Let $Gamma$ be a finitely generated group, $$, and $$. Denote $Au{t}^{*}(Gamma)=extrmIm(Aut(Gamma)oAut(Gamm{a}^{*}))leqG{L}_d(mathbbZ)$. In this paper we show that when $Gamma$ is nilpotent, there is a canonical isomorphism $Cleft(Gammaight)simeqC(Au{t}^{*}(Gamma),Gamm{a}^{*})$. In other words, $Cleft(Gammaight)$ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group $Au{t}^{*}(Gamma)$. In particular, in the case where $Gamma=Ps{i}_{n,c}$ is a finitely generated free nilpotent group of class $c$ on $n$ elements, we get that $C(Ps{i}_{n,c})=C(mathbb{Z}^{(n)})=e$ whenever $ngeq3$, and $C(Ps{i}_{2,c})=C(mathbb{Z}^{(2)})=hat{F}_{omega}$ = the free profinite group on countable number of generators.

Mathematics Subject Classification ID

Subgroup theorems; subgroup growth (20E07) Nilpotent groups (20F18) Automorphism groups of groups (20F28) Limits, profinite groups (20E18)

This page was built for publication: The Congruence Subgroup Problem for finitely generated Nilpotent Groups