Triangle generation of finite exceptional groups of low rank

Given a triple (p1,p2,p3) of primes, the object of this paper is the study of the space Hom(Tp1,p2,p3,G) of homomorphisms from the triangle group Tp1,p2,p3 to a finite simple exceptional group G of Lie type , , G2 or . With a few exceptions, we give precise asymptotic estimates for the size of Hom(Tp1,p2,p3,G) and determine the limiting probability that a randomly chosen homomorphism from Tp1,p2,p3 to G is surjective as |G|→∞.