Split BN-pairs of finite Morley rank

Abstract

Let G be a simple group of finite Morley rank with a definable 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 [9] (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.Ifn=3, thenGis interpretably isomorphic toPSL3(K) for some algebraically closed fieldK.Theorem 2.SupposeZ(U) contains someB-minimal subgroupA⩽Z(U) withRM(A)⩾RM(Pi/B) for both parabolic subgroupsP1andP2. Thenn=3,4 or 6 andGis interpretably isomorphic toPSL3(K), PSp4(K) orG2(K) for some algebraically closed fieldK.Theorem 3.IfUis nilpotent andn≠8, thenGis interpretably isomorphic to eitherPSL3(K), PSp4(K) orG2(K) for some algebraically closed fieldK.