Title:

Triangle groups and finite simple groups

This thesis contains a study of the spaces of homomorphisms from hyperbolic triangle groups to finite groups of Lie type which leads to a number of deterministic, asymptotic,and probabilistic results on the (p1, p2, p3)generation problem for finite groups of Lie type. Let G₀ = L(pn) be a finite simple group of Lie type over the finite field Fpn and let T = Tp1,p2,p3 be the hyperbolic triangle group (x,y : xp1 = yp2 = (xy)p3 = 1) where p1, p2, p3 are prime numbers satisfying the hyperbolic condition 1/p1 + 1/p2 + 1/p3 < 1. In general, the size of Hom(T,G₀) is a polynomial in q, where q = pn, whose degree gives the dimension of Hom(T,G), where G is the corresponding algebraic group, seen as a variety. Computing the precise size of Hom(T,G₀) or giving an asymptotic estimate leads to a number of applications. One can for example investigate whether or not there is an epimorphism in Hom(T,G₀). This is equivalent to determining whether or not G₀ is a (p1, p2, p3)group. Asymptotically, one might be interested in determining the probability that a random homomorphism in Hom(T,G₀) is an epimorphism as G₀→∞ . Given a prime number p, one can also ask wether there are finitely, or infinitely many positive integers n such that L(pn) is a (p1, p2, p3)group. We solve these problems for the following families of finite simple groups of Lie type of small rank: the classical groups PSL2(q), PSL3(q), PSU3(q) and the exceptional groups 2B2(q), 2G2(q), G2(q), 3D4(q). The methods involve the character theory and the subgroup structure of these groups. Following the concept of linear rigidity of a triple of elements in GLn(Fp), used in inverse Galois theory, we introduce the concept for a hyperbolic triple of primes to be rigid in a simple algebraic group G. The triple (p1, p2, p3) is rigid in G if the sum of the dimensions of the subvarieties of elements of order p1, p2, p3 in G is equal to 2 dim G. This is the minimum required for G(pn) to have a generating triple of elements of these orders. We formulate a conjecture that if (p1, p2, p3) is a rigid triple then given a prime p there are only finitely many positive integers n such that L(pn) is a (p1, p2, p3)group. We prove this conjecture for the classical groups PSL2(q), PSL3(q), and PSU3(q) and show that it is consistent with the substantial results in the literature about Hurwitz groups (i.e. when (p1, p2, p3) = (2, 3, 7)). We also classify the rigid hyperbolic triples of primes in algebraic groups, and in doing so we obtain some new families of nonHurwitz groups.
