A Characterization of the Finite Moufang Hexagons by Generalized Homologies

A generalized homology of a generalized hexagon 5^ is an auto-morphism of S? fixing all points on two mutually opposite lines or fixing all lines through two mutually opposite points. We show that if 5? is finite and if it admits "many" generalized homologies, then 5? is Moufang and hence classical. 1. Introduction and notation, A (finite thick) generalized hexagon of order (s, t) is a point-line incidence geometry S? = (P, B, /) satisfying (GH 1) up to (GH 4). (GH 1) There are s + 1 points incident with each line, s > 1. (GH 2) There are t + 1 lines incident with each point, t > 1. (GH 3) Every two varieties (a variety is a point or a line) lie in a common circuit consisting of six points and six lines. (GH 4) For every circuit consisting of k points and k lines it must be that k > 6. At present there are, up to duality, only two classes known of finite generalized hexagons and they are related to the Chevalley groups G 2 {q) and 3 D 4 (q). We denote them respectively by G 2 (q) and 3 D 4 (q) (see e.g. [4]). Of course, there are two mutually dual choices for these generalized hexagons, but we fix one by saying that 3 D 4 (q) has order (q, q 3) and G 2 (q) is a subgeometry of 3 D 4 (q). We will define these hexagons below using Kantor's description (see [4]). We now introduce some further notation. Let 5? = (P, B, /) be a finite generalized hexagon. We will always assume that S? is thick. A circuit consisting of six points and six lines (as in (GH 3)) is called an apartment Let A be an apartment and x a variety of 5?. We denote the set of all varieties incident with x but distinct from the 12 varieties of A by A*(x). A chain of seven distinct consecutively incident varieties is called a root. If the middle element of a root is a point, then we call the root short, if the middle element is a line, then we call it a long root. Let 9ί = {xolx\ I-Ixβ) be a root. If a is an automorphism of 5? fixing all varieties incident with JCI , x 2 , x-$,