The relational complexity of linear groups acting on subspaces

The relational complexity of a subgroup 𝐺 of Sym ⁢ ( Ω ) \mathrm{Sym}({\Omega}) is a measure of the way in which the orbits of 𝐺 on Ω k \Omega^{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 ) \mathrm{PSL}_{n}(\mathbb{F}) and PGL n ⁢ ( F ) \mathrm{PGL}_{n}(\mathbb{F}) , for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of F n \mathbb{F}^{n} . We also bound the relational complexity of all groups lying between PSL n ⁢ ( q ) \mathrm{PSL}_{n}(q) and P ⁢ Γ ⁢ L n ⁢ ( q ) \mathrm{P}\Gamma\mathrm{L}_{n}(q) , and generalise these results to the action on 𝑚-spaces for m ≥ 1 m\geq 1 .