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.