Zero-one generation laws for finite simple groups

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 ∩ 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. In our main application, we apply these results to the case where r = 2 and the irreducible subvariety X = C × D, a product of two conjugacy classes of elements of finite order in G. This leads to new results on random (2, 3)-generation of finite simple groups G(q) of exceptional Lie-type: provided G(q) is not a SuzuKi group, we show that the probability that a random involution and a random element of order 3 generate G(q) tends to 1 as q → ∞. Combining this with previous results for classical groups, this shows that finite simple groups (apart from SuzuKi groups and PSp₄ (q)) are randomly (2, 3)-generated. Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie-type.