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 non-Hurwitz groups.