Bounding the size of certain rank 3 geometries with designs as rank 2 residues
This paper contributes to the classification of flag transitive (finite) diagram geometries. When the rank two residues are of ``classical type'' (generalized polygons, circles, tilde-geometry, affine planes, Petersen graph), then a lot of information is known. The paper under review initiates the study of diagram geometries with certain designs as residue. Basically, the authors give necessary conditions on the parameters of such rank three geometries to conclude that the diameter is bounded by some constant (usually some number not greater than six), and hence the geometry to be finite. Some special diagrams are considered and results for flag transitive such geometries are presented. For one type of rank three diagrams (a triangle two edges of which carry the label \(L\) (for linear spaces, here of order \(s\)) and one edge is labelled \(\Delta (s,2)\) (for the complement of the \((s+2) \times 2\)-grid)), the authors classify all corresponding flag transitive geometries using coset-enumeration. Other examples are also given, but not very explicitly, and the authors ask for a more geometric description of these geometries. Here is one. Their first example is constructed with \(U_3(3)\) and three copies of a subgroup isomorphic to \(L_3(2)\). Clearly this geometry can be constructed as follows: the elements of type 1, 2, 3 are the ideal planes of the \(G_2(2)\)-hexagon \(H(2)\); two elements of different type are incident if they meet in exactly three points.
scientific article; zbMATH DE number 932538
scientific article