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Some torsion-free solvable groups with few subquotients

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

Published online by Cambridge University Press:  02 October 2023

ADRIEN LE BOUDEC  and
NICOLÁS MATTE BON
ADRIEN LE BOUDEC
Affiliation:
CNRS, UMPA - ENS Lyon, 46 Allée d’Italie, 69364 Lyon, France. e-mail: adrien.le-boudec@ens-lyon.fr
NICOLÁS MATTE BON
Affiliation:
CNRS, Institut Camille Jordan (ICJ, UMR CNRS 5208), Université de Lyon, 43 Blvd. du 11 Novembre 1918, 69622 Villeurbanne, France. e-mail: mattebon@math.univ-lyon1.fr
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Abstract

We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no “torsion-free version” of P. Kropholler’s theorem, which characterises solvable groups of infinite rank via their metabelian subquotients.

MSC classification

Primary: 20F16: Solvable groups, supersolvable groups
Secondary: 20E22: Extensions, wreath products, and other compositions 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 2 , March 2024 , pp. 279 - 286
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency.

References

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