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Baumslag–Solitar groups and residual nilpotence

Part of: Lie algebras and Lie superalgebras Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  16 June 2023

C.E. Kofinas ,
V. Metaftsis  and
A.I. Papistas
C.E. Kofinas
Affiliation:
Department of Mathematics, University of the Aegean, Karlovassi, Samos, Greece (kkofinas@aegean.gr; vmet@aegean.gr)
V. Metaftsis
Affiliation:
Department of Mathematics, University of the Aegean, Karlovassi, Samos, Greece (kkofinas@aegean.gr; vmet@aegean.gr)
A.I. Papistas
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece (apapist@math.auth.gr)
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Abstract

Let G be a Baumslag–Solitar group. We calculate the intersection $\gamma_{\omega}(G)$ of all terms of the lower central series of G. Using this, we show that $[\gamma_{\omega}(G),G]=\gamma_{\omega}(G)$, thus answering a question of Bardakov and Neschadim [1]. For any $c \in \mathbb{N}$, with $c \geq 2$, we show, by using Lie algebra methods, that the quotient group $\gamma_{c}(G)/\gamma_{c+1}(G)$ of the lower central series of G is finite.

Keywords

Baumslag-Solitar groupsresidually nilpotentlower central seriesLie algebra

MSC classification

Primary: 20F14: Derived series, central series, and generalizations 20F40: Associated Lie structures 20F05: Generators, relations, and presentations 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 17B70: Graded Lie (super)algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 2 , May 2023 , pp. 532 - 547
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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