Triangle groups and PSL2(q)

Abstract

We consider hyperbolic triangle groups of the form T = Tp1, p2, p3, where p1, p2, p3 are prime numbers. Let p be a prime number and n be a positive integer. We give a necessary and sufficient condition for L2(pn) to be the image of a given hyperbolic triangle group, where L2(pn) denotes the projective special linear group PSL2(pn). It follows that, given a prime number p, there exists a unique positive integer n such that L2(pn) is the image of a given hyperbolic triangle group. Finally, given a hyperbolic triangle group T, we determine the asymptotic probability that a randomly chosen homomorphism φ : T → L2(pn) is surjective, as pn tends to infinity.