The relational complexity of linear groups acting on subspaces

Saul Daniel Freedman; Veronica Kelsey; Colva Roney-Dougal

doi:10.1515/jgth-2023-0262

The relational complexity of linear groups acting on subspaces

Saul Daniel Freedman, Veronica Kelsey^*, Colva Roney-Dougal

^*Corresponding author for this work

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Abstract

The relational complexity of a subgroup G of Sym(Ω) is a measure of the way in which the orbits of G on Ω^k for various k determine the original action of G. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSL_n(𝔽) and PGL_n(𝔽), for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of 𝔽ⁿ. We also bound the relational complexity of all groups lying between PSL_n(q) and PΓL_n(q), and generalise these results to the action on m-spaces for m at least 1.

Original language	English
Number of pages	29
Journal	Journal of Group Theory
Volume	Ahead of Print
Early online date	14 Feb 2024
DOIs	https://doi.org/10.1515/jgth-2023-0262
Publication status	E-pub ahead of print - 14 Feb 2024

Keywords

Relational complexity
Linear groups
Subspace actions

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10.1515/jgth-2023-0262Licence: CC BY

Freedman-2024-Relational-complexity-of-linear-JGT-20230262-CCBY
Copyright © 2024 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License.
Final published version, 325 KBLicence: CC BY

Cite this

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title = "The relational complexity of linear groups acting on subspaces",

abstract = "The relational complexity of a subgroup G of Sym(Ω) is a measure of the way in which the orbits of G on Ωk for various k determine the original action of G. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSLn(𝔽) and PGLn(𝔽), for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of 𝔽n. We also bound the relational complexity of all groups lying between PSLn(q) and PΓLn(q), and generalise these results to the action on m-spaces for m at least 1.",

keywords = "Relational complexity, Linear groups, Subspace actions",

author = "Freedman, {Saul Daniel} and Veronica Kelsey and Colva Roney-Dougal",

note = "Funding: This work was supported by EPSRC grant no. EP/R014604/1, and also partially supported by a grant from the Simons Foundation. The first author was supported by the University of St Andrews (St Leonard{\textquoteright}s International Doctoral Fees Scholarship & School of Mathematics and Statistics PhD Funding Scholarship), and by EPSRC grant no. EP/W522422/1. The second author is funded by the Heilbronn Institute.",

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AU - Freedman, Saul Daniel

AU - Kelsey, Veronica

AU - Roney-Dougal, Colva

N1 - Funding: This work was supported by EPSRC grant no. EP/R014604/1, and also partially supported by a grant from the Simons Foundation. The first author was supported by the University of St Andrews (St Leonard’s International Doctoral Fees Scholarship & School of Mathematics and Statistics PhD Funding Scholarship), and by EPSRC grant no. EP/W522422/1. The second author is funded by the Heilbronn Institute.

PY - 2024/2/14

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N2 - The relational complexity of a subgroup G of Sym(Ω) is a measure of the way in which the orbits of G on Ωk for various k determine the original action of G. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSLn(𝔽) and PGLn(𝔽), for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of 𝔽n. We also bound the relational complexity of all groups lying between PSLn(q) and PΓLn(q), and generalise these results to the action on m-spaces for m at least 1.

AB - The relational complexity of a subgroup G of Sym(Ω) is a measure of the way in which the orbits of G on Ωk for various k determine the original action of G. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSLn(𝔽) and PGLn(𝔽), for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of 𝔽n. We also bound the relational complexity of all groups lying between PSLn(q) and PΓLn(q), and generalise these results to the action on m-spaces for m at least 1.

KW - Relational complexity

KW - Linear groups

KW - Subspace actions

U2 - 10.1515/jgth-2023-0262

DO - 10.1515/jgth-2023-0262

M3 - Article

SN - 1433-5883

VL - Ahead of Print

JO - Journal of Group Theory

JF - Journal of Group Theory

ER -

The relational complexity of linear groups acting on subspaces

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