Generalized polygons with split Levi factors

Abstract.

¶Theorem. Suppose that the group G has an irreducible spherical BN-pair of rank 2. If for every 2-transitive subgroup L of the Levi factors of G the one-point stabilizer B splits as B = U × T with [B, B] = U, then the Tits-building associated to the BN-pair of G is Moufang.¶¶This implies in particular that a simple group G with a spherical BN-pair of rank 2 whose Levi factors are permutation equivalent to subgroups of PGL₂(K) or the Suzuki group GSz(K) is (essentially) classical.