Summer School

on 

Finite Groups and Related Geometrical Structures


TOBLACH-DOBBIACO
August 28th-September 8th 2006


 
Introduction To Spherical and Affine Buildings
Richard M. Weiss


Syllabus: The course has four parts.

A. Introduction to Coxeter groups. We will present the basic facts about Coxeter groups which are needed to get started in the theory of buildings.

B. Introduction to buildings. We will introduce buildings as edge-colored graphs satisfying certain properties and study the basic structures of a building: residues, roots and apartments.

C. Spherical buildings. These are the buildings whose apartments are finite. We will study root groups, generalized polygons and the Moufang property. We will give an overview of the classification of Moufang polygons and spherical buildings of rank at least three.

D. Affine buildings (also known as Euclidean buildings). These are the buildings whose apartments have a natural representation as a tiling of a Euclidean space. We will study the building at infinity (a spherical building), "tree-preserving" isomorphisms and root data with valuation. We will give an overview of the classification of affine buildings whose building at infinity satisfies the Moufang property.

Among Jacques Tits's most remarkable accomplishments are his classifications results for spherical and affine buildings. The goal of this course is to leave the students with some appreciation of this work.  

References:

1. K. Brown, Buildings, Springer 1989.

2. F. Bruhat and J. Tits, Groupes réductifs sur un corps local, I. Données radicielles valués, Publ. Math. I.H.E.S. 41 (1972), 5-252.

3. M. Ronan, Lectures on Buildings, Academic Press, New York 1989.

4. J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 386, Springer, 1974.

5. J. Tits, Immeubles de type affine, in Buildings and the Geometry of Diagrams (Como 1984), pp. 159-190, Lecture Notes in Math. 1181, Springer, 1986.

6. J. Tits and R. M. Weiss, Moufang Polygons, Springer, 2002.

7. R. M. Weiss, The Structure of Spherical Buildings, Princeton, 2003.

8. R. M. Weiss, The Structure of Affine Buildings, in preparation.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

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