BN-pairs of Finite Morley Rank

Katrin Tent

We report on recent progress on the Cherlin-Zil’ber Conjecture, viz. that a simple group of finite Morley rank is an algebraic group over an algebraically closed field. The conjecture implies in particular that any such group must have a definable ’split’ BN-pair. The following is a partial analog of the famous paper by Fong and Seitz on finite groups with split BN-pairs:

Theorem: [T] Let G be a simple group with a definable BN-pair of rank 2 where B = U.T for T = B ∩N and a normal subgroup U of B with Z(U) ≠ 1. It was shown in [TVM2] that the Weyl group W = N/B∩N has cardinality 2n with n = 3, 4, 6, 8 or 12. If G has finite Morley rank then furthermore the following holds:

(i) If n = 3, then G is definably isomorphic to P SL_{3}(K) for some algebraically closed field K. (ii) If U is nilpotent, then n ≠ 12. (iii) If Z(U) contains a B-minimal subgroup A with RM(A) ≥ RM(P_{i}/B) for both parabolic subgroups P_{1} and P_{2}, then n = 3, 4 or 6 and G is definably isomorphic to P SL_{3}(K), P Sp_{4}(K) or G_{2}(K) for some algebraically closed field K.

Note that the conditions in (iii) are necessarily satisfied if the Cherlin-Zil’ber Conjecture is true.

[T] K. Tent, Split BN-pairs of finite Morley rank, preprint.

[TVM1] K. Tent, H. Van Maldeghem, Cherlin’s Conjecture for buildings with panels of Morley rank 1, preprint.

[TVM2] K. Tent, H. Van Maldeghem, On irreducible (B,N)-pairs of rank 2, preprint.

Address: Mathematisches Institut, Universität Würzburg, 97074 Würzburg, Germany

E-mail: tent@mathematik.uni-wuerzburg.de