RANDOM AND DETERMINISTIC TRIANGLE GENERATION OF THREE-DIMENSIONAL CLASSICAL GROUPS III

Let p(1), p(2), p(3) be primes. This is the final paper in a series of three on the (p(1), p(2), p(3))-generation of the finite projective special unitary and linear groups PSU3(p(n)), PSL3(p(n)), where we say a noncyclic group is (p(1), p(2), p(3))-generated if it is a homomorphic image of the triangle group T-p1, p(2), p(3) . This article is concerned with the case where p(1)=2 and p(2)p(3). We determine for any primes p(2)p(3) the prime powers p(n) such that PSU3(p(n)) (respectively, PSL3(p(n))) is a quotient of T=T-2,T- p2, p(3) . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU3(p(n))) (respectively, Hom(T, PSL3(p(n)))) is surjective as p(n) tends to infinity.