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The Formanek-Procesi group with base a right-angled Artin group: Lie algebra and residual nilpotence

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

Published online by Cambridge University Press:  13 May 2025

V. Metaftsis  and
A.I. Papistas
V. Metaftsis*
Affiliation:
Department of Mathematics, University of the Aegean, Karlovassi, 832 00 Samos, Greece
A.I. Papistas
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, 541 24 Thessaloniki, Greece
*
Corresponding author: V. Metaftsis; Email: vmet@aegean.gr
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Abstract

In the present work, we investigate the Lie algebra of the Formanek-Procesi group $\textrm {FP}(A_{\Gamma })$ with base group $A_{\Gamma }$ a right-angled Artin group. We show that the Lie algebra $\textrm {gr}(\textrm {FP}(A_{\Gamma }))$ has a presentation that is dictated by the group presentation. Moreover, we show that if the base group $G$ is a finitely generated residually finite $p$-group, then $\textrm { FP}(G)$ is residually nilpotent. We also show that $\textrm {FP}(A_{\Gamma })$ is a residually torsion-free nilpotent group.

Keywords

Formanek-Procesi groupLie algebraresidual nilpotenceresidually torsion free nilpotent

MSC classification

Primary: 17B01: Identities, free Lie (super)algebras
Secondary: 17B40: Automorphisms, derivations, other operators 17B70: Graded Lie (super)algebras 20F40: Associated Lie structures 20F28: Automorphism groups of groups
Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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