Appendix 2. Generators and relations

Abstract

The theorem 4.1.2 can be viewed as a kind of presentation of an arbitrary building of spherical type by amalgamation of buildings of rank one in buildings of rank 2. Here, similar results are obtained for BN-pairs (13.5, 13.32); it will be seen that the proofs are considerably simpler than that of 4.1.2. In 13.5, an arbitrary group G with BN-pair (B,N) appears as an amalgamated sum of the parabolic subgroups “of rank 2” containing B, but we shall see that, knowing a priori that B belongs to a BN-pair in G, one can also, under mild conditions, characterize G by means of the amalgamation of B in the parabolic subgroups of rank 1 containing it (13.20). Other theorems (13.11, 13.39) show the possibility of characterizing some BN-pairs (in particular, the “standard” BN pairs in classical or algebraic simple groups of rank ≥ 2) by means of a certain system of subgroups of B. A previous version of the results exposed here has been given in a seminar at the University of Chicago in 1963; several improvements (in particular, the present form of 13.11, 13.20 and 13.39) have been suggested by discussions with P. Fong.