Generalized hexagons and octagons with primitive automorphism groups

The classification of all finite flag-transitive generalized polygons is a long-standing important open problem in finite geometry. Generalized polygons, introduced by Tits, are among the most notable and prominent examples of discrete geometries, they have a lot of applications and are the building bricks of the Tits buildings. Generalized hexagons and octagons form an elusive class of generalized polygons. The automorphism groups of the only known examples are certain almost simple groups of Lie type acting transitively on the flags and primitively on both the point and the line spaces. In a joint work with Hendrik Van Maldeghem we showd that if the action of the automorphism group is flag-transitive and it is primitive on the points and on the lines, then it must be an almost simple group of Lie type.