Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups I

Abstract

Let p 1, p 2, p 3 be primes. This is the first in a series of three articles concerned with the (p 1, p 2, p 3)-generation of the 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 quotient of the triangle group T p 1, p 2, p 3 . We present our results which are probabilistic, asymptotic and also deterministic, and provide the machinery needed to prove them when p 1, p 2, p 3 are distinct or all odd. Complete proofs are given when p 1, p 2, p 3 are odd. The second and final article investigate the cases where p 1 = 2 and p 2 = p 3, and p 1 = 2 and p 2 ≠ p 3, respectively.