Buekenhout geometry

Summary

A Buekenhout geometry, or diagram geometry, is a set of objects of several types with a symmetric incidence relation. Every flag of pairwise incident objects extends to a chamber, which contains one object of each type. It generalizes projective spaces, generalized polygons and Tits buildings. Francis Buekenhout introduced a diagram that records, for each pair of types, which rank $2$ geometries appear. These diagrams are used to describe and classify geometries, including geometries for sporadic simple groups.