The relational complexity of linear groups acting on subspaces

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