Diagram geometry

A diagram is like a blueprint describing a possible combination of $ 2 $- dimensional incidence structures in a higher-dimensional geometry (cf. Incidence system). In this context, a geometry of rank $ n $ is a connected graph $ \Gamma $ equipped with an $ n $- partition $ \Theta $ such that every maximal clique of $ \Gamma $ meets every class of $ \Theta $. The vertices and the adjacency relation of $ \Gamma $ are the elements and the incidence relation of the geometry, the cliques of $ \Gamma $ are called flags, the neighbourhood of a non-maximal flag is its residue, the classes of $ \Theta $ 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 $ \{ i, j \} $ should belong to. For instance, a simple stroke

a double stroke

and the "null stroke"

are normally used to denote, respectively, the class of projective planes (cf. Projective plane), the class of generalized quadrangles [a5] (cf. also Quadrangle) and the class of generalized digons (generalized digons are the trivial geometry of rank $ 2 $, where any two elements of different type are incident). The following two diagrams are drawn according to these conventions.

1)

2)

They both represent geometries of rank $ 3 $, where the residues of type $ \{ 1,2 \} $ are projective planes and the residues of type $ \{ 1,3 \} $ are generalized digons. However, the residues of type $ \{ 2,3 \} $ are projective planes in 1) and generalized quadrangles in 2). The following is a typical problem in diagram geometry: given a diagram of such-and-such type, classify the geometries that fit with it, or at least the finite or flag-transitive such geometries (a geometry $ \Gamma $ is said to be flag-transitive if $ { \mathop{\rm Aut} } ( \Gamma ) $ acts transitively on the set of maximal flags of $ \Gamma $). In the "best" cases the answer is as follows: the geometries under consideration belong to a certain well-known family or, possibly, arise in connection with certain small or sporadic groups. For instance, it is not difficult to prove that $ 3 $- dimensional projective geometries are the only examples for the above diagram 1). On the other hand, the following has been proved quite recently [a7]: If $ \Gamma $ is a flag-transitive example for the diagram 2) and if all rank- $ 1 $ residues of $ \Gamma $ are finite of size $ \geq 2 $, then $ \Gamma $ is either a classical polar space [a6] (arising from a non-degenerate alternating, quadratic or Hermitian form of Witt index $ 3 $) or a uniquely determined very small geometry with $ { \mathop{\rm Aut} } ( \Gamma ) \cong { \mathop{\rm Alt} } ( 7 ) $.

See [a3] for a survey of related theorems.

References

[a1]	F. Buekenhout, "Diagrams for geometries and groups" J. Combin. Th. A , 27 (1979) pp. 121–151
[a2]	F. Buekenhout, "Foundations of incidence geometry" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995) pp. 63–105
[a3]	F. Buekenhout, A. Pasini, "Finite diagram geometries extending buildings" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995) pp. 1143–1254
[a4]	A. Pasini, "Diagram geometries" , Oxford Univ. Press (1994)
[a5]	S.E. Payne, J.A. Thas, "Finite generalized quadrangles" , Pitman (1984)
[a6]	J. Tits, "Buildings of spherical type and finite BN-pairs" , Lecture Notes in Mathematics , 386 , Springer (1974)
[a7]	S. Yoshiara, "The flag-transitive $C_3$ geometries of finite order" J. Algebraic Combinatorics (??)