Diagram Geometry

The theory of buildings, created by J. Tits three decads ago, has ooered a uniied geometric treatment of nite simple groups of Lie type, both of classical and of exceptional type. (See Tits 19] and 20] for an exposition of that theory; also Ronan 15] and Brown 1].) Diagram geometry (see 13] for an exposition) is a generalization of the theory of buildings. It has been invented by F. Buekenhout 2] in the late seventies, in order to get a framework suited for sporadic simple groups, too. The concept of diagram is the basic idea of this theory. A diagram is like a blueprint describing a possible combination of 2-dimensional incidence structures in a higher-dimensional geometry. In this context, a geometry of rank n is a connected graph ? equipped with an n-partition such that every maximal clique of ? meets every class of. The vertices and the adjacency relation of ? are the elements and the incidence relation of the geometry G = (?;), the cliques of ? are called ags, the neighborhood of a non-maximal ag is its residue, the classes of are called types and the type (respectively, rank) of a residue is the set (respectively, number) of types met by it. Given a catalogue of`nice' geometries of rank 2 and a symbol for each of those classes, one can compose diagrams of any rank by those symbols. The nodes of a diagram represent types and, by attaching a label to the stroke connecting two types i and j or by some other convention, one indicates which class the residues of type fi; j g should belong to. For instance, a simple stroke a double stroke and thènull stroke' are normally used to denote, respectively, the class of projective planes, the class of generalized quadrangles (see 14]) and the class of generalized digons (generalized digons are the trivial geometries of rank 2, where any two elements of diierent type are incident). The following two diagrams are drawn according to these conventions 1) 2) They represent geometries of rank 3, where the residues of type f1; 2g are projec-tive planes and the residues of type f1; 3g are generalized digons. However, the residues of type f2; 3g are projective planes in 1) and generalized quadrangles in 2). The geometries described by 1) are precisely the (possibly degenerate) 3-dimensional projective geometries, whereas 2) includes all rank 3 polar spaces …