Hostname: page-component-669899f699-tzmfd Total loading time: 0 Render date: 2025-05-02T15:53:17.991Z Has data issue: false hasContentIssue false
  • English
  • Français

Some torsion-free solvable groups with few subquotients

Part of: Special aspects of infinite or finite groups Structure and classification 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: [email protected]
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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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

REFERENCES

Brieussel, J.. About the speed of random walks on solvable groups. ArXiv:1505.03294.Google Scholar
Brieussel, J. and Zheng, T.. Speed of random walks, isoperimetry and compression of finitely generated groups. Ann. of Math. (2) 193(1) (2021), 1–105.CrossRefGoogle Scholar
Hall, P.. Finiteness conditions for soluble groups. Proc. London Math. Soc. (3) 4 (1954), 419–436.Google Scholar
Jacoboni, L. and Kropholler, P.. Soluble groups with no ${\mathbb{Z}}\wr{\mathbb{Z}}$ sections. Ann. H. Lebesgue 3 (2020), 981–998.CrossRefGoogle Scholar
Kropholler, P. H.. On finitely generated soluble groups with no large wreath product sections. Proc. London Math. Soc. (3) 49(1) (1984), 155–169.Google Scholar
Kropholler, P. H.. A note on the cohomology of metabelian groups. Math. Proc. Camb. Phil. Soc. 98 (1985), no. 3, 437–445.CrossRefGoogle Scholar
Le Boudec, A. and Matte Bon, N.. Growth of actions of solvable groups. ArXiv:2205.11924 (2022).Google Scholar
Pittet, Ch. and Saloff–Coste, L.. Random walks on finite rank solvable groups. J. Eur. Math. Soc. (JEMS) 5(4) (2003), 313–342.CrossRefGoogle Scholar
Robinson, D.. A course in the theory of groups. second ed., Graduate Texts in Math., 80 (Springer-Verlag, New York, 1996).Google Scholar
Saloff–Coste, L.. Analysis on Riemannian co-compact covers. Surv. Differ. Geom., vol. 9, (Int. Press, Somerville, MA, 2004), pp. 351–384.CrossRefGoogle Scholar