The relational complexity of a subgroup 𝐺 of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Sym</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Sym({Ω}) is a measure of the way in which the orbits of 𝐺 on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">Ω</m:mi> <m:mi>k</m:mi> </m:msup> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> PSLₙ(F) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PGL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> PGLₙ(F) , for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> Fⁿ . We also bound the relational complexity of all groups lying between <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> PSLₙ(q) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">P</m:mi> <m:mo></m:mo> <m:mi mathvariant="normal">Γ</m:mi> <m:mo></m:mo> <m:msub> <m:mi mathvariant="normal">L</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Pₙ(q) , and generalise these results to the action on 𝑚-spaces for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> m≥ 1 .
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Freedman et al. (2024) studied this question.