ZERO-ONE GENERATION LAWS FOR FINITE SIMPLE GROUPS

摘要

Let G be a simple algebraic group over the algebraic closure of F-p (p prime), and let G(q) denote a corresponding finite group of Lie-type over F-q, where q is a power of p. Let X be an irreducible subvariety of G(r) for some r >= 2. We prove a zero-one law for the probability that G(q) is generated by a random r-tuple in X(q) = X boolean AND G(q)(r): the limit of this probability as q increases (through values of q for which X is stable under the Frobenius morphism defining G(q)) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs G(q) to be generated by an r-tuple in X(q) for two sufficiently large values of q. We also prove a version of this result where the underlying characteristic is allowed to vary.