Summer School 

on 

Finite Groups and Related Geometrical Structures


UDINE
August 16th-26th 2005


 
INTRODUCTION TO SPHERICAL 
AND AFFINE BUILDINGS
by RICHARD M. WEISS
Syllabus: The course has four parts:

A. Introduction to Coxeter groups. A few basic facts about Coxeter groups are needed before we can talk about buildings.

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

C. Spherical buildings. These are the buildings whose apartments are finite. We will study root groups, generalized polygons, the opposite map, the Moufang property, root data and related algebraic structures. We will look at examples coming from classical and exceptional groups.

D. Affine buildings. These are the buildings whose apartments have a natural representation as a tiling of a Euclidean space: We will study sectors, the building at infinity (which is spherical) and root data with valuation.

Two of Jacques Tits' most remarkable accomplishments are his classification of spherical buildings and his classification (some of it in collaboration with F. Bruhat) of affine buildings (in sufficiently high rank). One goal of this course is to leave the students with some understanding of these two results.

References: 

[1] K. Brown, Buildings, Springer, 1989. 

[2] M. Ronan, Lectures on Buildings, Academic Press, New York 1989.

[3] J.Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics 386, Springer, 1974. 

[4] R. M. Weiss, The Structure of Spherical Buildings, Princeton University Press, 2003.

[5] R. M. Weiss, The Structure of  Affine Buildings, notes in preparation.
 
 
 
 
 
 
 
 

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