TY - JOUR
T1 - The congruence subgroup problem for finitely generated nilpotent groups
AU - Ben-Ezra, David El Chai
AU - Lubotzky, Alexander
N1 - Publisher Copyright:
© 2021 Walter de Gruyter GmbH, Berlin/Boston 2022.
PY - 2022/5/1
Y1 - 2022/5/1
N2 - The congruence subgroup problem for a finitely generated group and for G Aut./ asks whether the map OG ! Aut. O / is injective, or more generally, what its kernel C.G; / is. Here OX denotes the profinite completion of X. In the case G D Aut./, we write C./ D C.Aut./; /. Let be a finitely generated group, N D =.; ., and D N =tor. N / . Z.d/. Define (for presented) In this paper we show that, when is nilpotent, there is a canonical isomorphism (for presented) In other words, C./ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut./. In particular, in the case where D .n;c is a finitely generated free nilpotent group of class c on nelements, we get that C.n;c/ D C.Z.n// D e whenever n 3, and C.2;c/ D C.Z.2// D FO is the free profinite group on countable number of generators.
AB - The congruence subgroup problem for a finitely generated group and for G Aut./ asks whether the map OG ! Aut. O / is injective, or more generally, what its kernel C.G; / is. Here OX denotes the profinite completion of X. In the case G D Aut./, we write C./ D C.Aut./; /. Let be a finitely generated group, N D =.; ., and D N =tor. N / . Z.d/. Define (for presented) In this paper we show that, when is nilpotent, there is a canonical isomorphism (for presented) In other words, C./ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut./. In particular, in the case where D .n;c is a finitely generated free nilpotent group of class c on nelements, we get that C.n;c/ D C.Z.n// D e whenever n 3, and C.2;c/ D C.Z.2// D FO is the free profinite group on countable number of generators.
UR - http://www.scopus.com/inward/record.url?scp=85116433659&partnerID=8YFLogxK
U2 - 10.1515/jgth-2021-0039
DO - 10.1515/jgth-2021-0039
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AN - SCOPUS:85116433659
SN - 1433-5883
VL - 25
SP - 411
EP - 432
JO - Journal of Group Theory
JF - Journal of Group Theory
IS - 3
ER -