Buildings in Bielefeld 

•  Mini-Course by Peter Abramenko – Buildings and Their Associated Groups
  Hybrid -- 24 and 29 September: 13:15 - 14:00, plus a 10-minute pause, 14:10 - 14:55 (CEST, UTC+2)

• Room  S0-123

• Lecture 1: Buildings and BN-pairs.
  I will briefly recall the definition of buildings and BN-pairs together with some examples. I will also recall why a BN-pair gives rise to a building and why a (thick) building on which a group G acts strongly transitively gives rise to a BN-pair in G. Some emphasis will be placed on projective spaces and the standard BN-pair in GL_n(F), where F is a skew field
  
  

• Lecture 2: The Moufang property.

Whereas buildings, including projective planes, needn't have any non-trivial automorphisms, a certain symmetry condition for projective planes (which is automatically satisfied for higher-dimensional projective spaces), introduced by Ruth Moufang, yields big symmetry groups with BN-pairs. I will explain how Tits generalized these classical results to spherical buildings of irreducible type and of rank > 3. I will also explain why every (thick) building which satisfies the appropriately defined Moufang property gives rise to a BN-pair.

• Lecture 3: RGD systems.

It will be discussed how the symmetry group of any (thick) Moufang building gives rise to a group theoretic structure called "RGD system" (after Tits: root group data). It will also be shown how an RGD system in a group G gives rise to a BN-pair (in fact: a twin BN-pair) in G, and thus to a (twin) building on which G acts strongly transitively. The BN-pairs of isotropic reductive groups over fields in fact result from RGD systems in these groups. As an affine example, I will briefly discuss the RGD system of SL_n(F[t, 1/t]), where F is a field.

• Lecture 4: Some group theoretic applications.
  This lecture will be a bit open-ended and depend on interests expressed by the audience (as far as I can accommodate them in one lecture). What I intend to discuss is Tits's simplicity criterion for groups with a spherical BN-pair and some finite properties of groups like SL_n(F_q[t]) or SL_n(F_q[t, 1/t]) for a finite field F_q.
  
  Registration link: Google Forms for registration