Tits Polygons

Tits Polygons Synopsis

"We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a "rank 2" presentation for the group of F-rational points of an arbitrary exceptional simple group of F-rank at least 4 and to determine defining relations for the group of F-rational points of an an arbitrary group of Frank 1 and absolute type D4, E6, E7 or E8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic"--.