Book contents
- Frontmatter
- Contents
- Preface
- List of speakers and talks
- Basics on buildings
- An introduction to generalized polygons
- Buildings and classical groups
- Twin buildings
- Twin trees and twin buildings
- Simple groups of finite Morley rank of even type
- BN-pairs and groups of finite Morley rank
- CM-trivial stable groups
- Amalgames de Hrushovski: Une tentative de classification
- Rank and homogeneous structures
- Constructions of semilinear towers of Steiner systems
- Introduction to the Lascar Group
Twin buildings
Published online by Cambridge University Press: 12 January 2010
- Frontmatter
- Contents
- Preface
- List of speakers and talks
- Basics on buildings
- An introduction to generalized polygons
- Buildings and classical groups
- Twin buildings
- Twin trees and twin buildings
- Simple groups of finite Morley rank of even type
- BN-pairs and groups of finite Morley rank
- CM-trivial stable groups
- Amalgames de Hrushovski: Une tentative de classification
- Rank and homogeneous structures
- Constructions of semilinear towers of Steiner systems
- Introduction to the Lascar Group
Summary
INTRODUCTION
The goal of this paper is to give a survey of the main ideas in the classification of 2-spherical twin buildings. A program for such a classification had been outlined by J. Tits in. There are two main conjectures in this program: Conjecture 1 concerns uniqueness, Conjecture 2 concerns existence. Conjecture 1 was proved in under a certain condition (co) which is ‘almost always’ satisfied; this result relies heavily on results proved in. It turned out that Conjecture 2 of has to be slightly modified; as already suspected in loc. cit. it was ‘too optimistic’. There is not yet a direct proof of this modified conjecture. Its validity is just a consequence of the classification which has been recently accomplished by the author. The classification is achieved by giving several construction procedures of twin buildings.
The interesting fact about these constructions is that they provide a link with several concepts developed by Tits in the ‘prehistory’ of buildings and that they shed new light on the classification of spherical buildings achieved in and on the recent classification of Moufang polygons to appear in.
We start for this reason with some historical remarks. After introducing some notation and giving some examples, we describe the main ideas in the classification of spherical buildings. In Section 6 we provide information about the origin and the main ideas of the theory of twin buildings.
- Type
- Chapter
- Information
- Tits Buildings and the Model Theory of Groups , pp. 103 - 118Publisher: Cambridge University PressPrint publication year: 2002
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