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
BN-pairs and groups of finite Morley rank
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
Abstract
We describe how the geometry of Tits buildings and generalized polygons enters the study of groups of finite Morley rank.
Introduction
Model theory is concerned with the study and classification of first order structures and their theories. Theories which were hoped to be particularly easy to classify are those having very few models up to isomorphism in ‘most’ infinite cardinalities. In the 1970's, Zil'ber started the program to classify all uncountably categorical theories, i.e. all theories having (up to isomorphims) exactly one model in every cardinality k ≥ ℵ1 - an example being the theory of algebraically closed fields of some fixed characteristic.
It quickly turned out that in such theories, definable, and thus themselves uncountably categorical, groups entered the picture almost inevitably as so-called ‘binding groups’ between definable sets and hence it became particularly important to classify the definably simple uncountably categorical groups, i.e. groups without definable proper normal subgroups. These are exactly the simple groups of finite Morley rank.
Finite Morley Rank
The Morley rank is a model theoretic dimension on definable sets, which to some extent behaves like the algebraic dimension in the sense of algebraic geometry.
- Type
- Chapter
- Information
- Tits Buildings and the Model Theory of Groups , pp. 173 - 184Publisher: Cambridge University PressPrint publication year: 2002