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.