John led the discussion last week.  I am a bit late in posting as I have been busy organising the WA Junior Olympiad which was on Saturday.

The topic this week is what is called Buekenhout geometries or diagram geometries. We also have a new name.

As we have seen so far, a projective space can be associated with the Dynkin diagram of type $A_n$ and polar spaces can be associated with the Dynkin diagram of type $B_n$. Tits’ theory of buildings generalises these notions and associates to each simple group of Lie type a geometry known as a building and this building can be encoded in the Dynkin diagram associated with the group. In his seminal 1979 paper, `Diagrams for geometries and groups’ in JCTA, Francis Buekenhout looked to generalise this further and introduced the notion of diagram geometries. These are also referred to as Buekenhout geometries.