Split BN-pairs of finite Morley rank

✍ Scribed by Katrin Tent

• Book ID: 104307217
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 270 KB
• Volume: 119
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(02)00058-1

✦ Synopsis

Let G be a simple group of ÿnite Morley rank with a deÿnable BN-pair of (Tits) rank 2 where B = UT for T = B ∩ N and U a normal subgroup of B with Z(U ) = 1. By (Forum Math. 13 (2001) 853) the Weyl group W = N=T has cardinality 2n with n = 3; 4; 6; 8 or 12. We prove here: Theorem 1. If n = 3, then G is interpretably isomorphic to PSL3(K) for some algebraically closed ÿeld K.

Theorem 2. Suppose Z(U ) contains some B-minimal subgroup A 6 Z(U ) with RM (A) ¿ RM (Pi=B) for both parabolic subgroups P1 and P2. Then n = 3; 4 or 6 and G is interpretably isomorphic to PSL3(K), PSp4(K) or G2(K) for some algebraically closed ÿeld K.

Theorem 3. If U is nilpotent and n = 8, then G is interpretably isomorphic to either PSL3(K), PSp4(K) or G2(K) for some algebraically closed ÿeld K.