Simple groups of finite Morley rank and Tits buildings

THEOREM A: If B is an infinite Moufang polygon of finite Morley rank, then B is either the projective plane, the symplectic quadrangle, or the split C ayley hexagon over some algebraically closed field. In particular, B is an algebraic polygon. It follows that any infinite simple group of finite Morley rank with a sphe rical Moufang BN-pair of Tits rank 2 is either PSL3(K),PSp(4)(K) or G(2)(K) for some algebraically closed field K. Spherical irreducible buildings of Tits rank greater than or equal to 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theore m A we show THEOREM B: If G is an infinite simple group of finite Morley rank with a sp herical Moufang BN-pair of Tits rank greater than or equal to 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraicall y closed field.