Book Review: Buildings of spherical type and finite $BN$-pairs

Abstract

These notes are a slightly revised and extended version of mim- graphed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968. Their main purpose is to present the solution of the following two problems: (A) Determination of the buildings of rank >; and irreducible, spherical type, other than ~ and H ("of spherical type" means "with finite Weyl 4 group", about the excluded types H, cf. the addenda on p. 274). Roughly speaking, those buildings all turn out to be associated to simple algebraic or classical groups (cf. 6. ;, 6. 1;, 8. 4. ;, 8. 22, 9. 1, 10. 2). An easy application provides the enumeration of all finite groups with BN-pairs of irreducible type and rank >;, up to normal subgroups contained in B (cf. 11. 7). (B) Determination of all isomorphisms between buildings of rank > 2 and spherical type associated to algebraic or classical simple groups and, in particular, description of the full automorphism groups of such buildings (cf. 5. 8, 5. 9, 5. 10, 6. 6, 6. 1;, 8. 6, 9. ;, 10. 4). Except for the appendices, the notes are rather strictly oriented toward these goals.--Provided by publisher. Complexes -- Coxeter complexes -- Buildings -- Reduction -- The building of a semi-simple algebraic group -- Buildings of type An, Dn, En -- Buildings of type Cn. I. Polar spaces -- Buildings of type Cn. II. Projective embeddings of polar spaces -- Buildings of type Cn. III. Non-embeddable polar spaces -- Buildings of type F4 -- Finite BN-pairs of irreducible type and rank >/= 3 -- Appendix 1. Shadows -- Appendix 2. Generators and relations.