Root graded groups of rank 2

Abstract

A Tits polygon is a bipartite graph in which the neighborhood of a vertex is endowed with an “opposition relation” satisfying certain properties. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. We introduce a version of the notion of a root graded group of rank 2 which generalizes the notion of a root datum for a spherical building (in rank 2) introduced by Tits in 1962 and show that this notion is equivalent to the notion of a Tits polygon. We apply this result to give a characterization of a large class of Tits hexagons associated with exceptional groups in arbitrary characteristic.