Point-primitive generalised hexagons and octagons and projective linear groups

We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $\mathcal{S}$ is a finite thick generalised hexagon or octagon with $G⩽\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${\mathrm{P}\mathrm{S}\mathrm{L}}_n(q)$ where $n \geqslant 2$, then the stabiliser of a point acts irreducibly on the natural module. We describe a strategy to prove that such a generalised hexagon or octagon $\mathcal{S}$ does not exist.