Point-primitive generalised hexagons and octagons and projective linear groups

TL;DR

This paper explores the classification of point-primitive generalised polygons, focusing on hexagons and octagons, and relates their symmetry groups to Lie type groups, proposing a strategy to prove their non-existence under certain conditions.

Contribution

It links the geometry of generalised polygons with the algebraic structure of Lie type groups, providing a new approach to classify or rule out certain configurations.

Findings

If a generalised hexagon or octagon has a point-primitive automorphism group with socle PSL_n(q), the point stabiliser acts irreducibly.

The paper proposes a strategy to prove the non-existence of such polygons under these group actions.

Abstract

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 $G \leqslant{\rm Aut}(\mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${\rm PSL}_n(q)$ 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.