Abstract
Buildings are combinatorial and geometric structures that were introduced by Jacques Tits for the study of semi-simple algebraic groups. When a building has the so-called Moufang property, it corresponds to the building having a rich automorphism group. An interesting subgroup of the automorphism group of a Moufang-building is the little projective group, which is the group generated by all root elations. In joined work with Hendrik Van Maldeghem, we found geometrical constructions for root elations of Moufang-buildings of finite diameter that give more insight into their fixpoint structure. We also saw a connection between the little projective groups and the groups of special projectivities of these buildings. At the moment we are working on determining the special and general groups of projectivities for all Moufang-buildings of finite diameter together with Jeroen Schillewaert. In my talk I would like to give some insight into our research, explain what elations and projectivities are and state some group theoretic consequences.
Talk
Known things:
- All thick, irreducible spherical buildings of rank at least 3 are Moufang building
- If a building is Moufang it has a rich automorphism group
- They are classified
Main Question: Can we find a geometric proof for Moufang-ness
Definitions
A building of type is a simplicial complex fulfillig some properties. Essentially the easierst buildings are given by Coxeter complexes. Glueing them together, we get thick buildings.
If an Apartment is cut in half, bothhalves are called roots. (Seen in the last talk as well) (Apartments being the coxeter complexes).
The root group is the stabilizer fixing the root pointwise. If the root group acts on the apartments transitively, they are called Moufang and if all roots are Moufang the building is called such.
Classification
The coxeter complexes to are known as classical type and they result in projective spaces. The others are known as exceptional spaces and result in polar spaces. (whatever that is)
Inside the polar spaces we do some kind of classical geometry
For a root we call the boundary a wall. This allows us to define “opposite”-ness. Two points are opposite if they are contained in all walls. A gallery is a path through adjacent chambers.
The paper was Busch, Schillewaert, Van Maldeghem
About
- Explained in a way that everyone understands
- Explained symbols
- Put the most confusing slide at the beginning such that it is familiar later on