PROBABILISTIC GROUP THEORY

                 Martin Liebeck (Imperial College)

A well known conjecture of John Dixon, proved about 10 years ago, states
the following. For a finite group G, let P(G) be the probability that
two elements of G, chosen uniformly at random, generate the whole of G.
Then for simple groups G, P(G) tends to 1 as |G| tends to infinity. In
other words, not only are simple groups generated by two elements, but
almost every pair of elements will generate the whole group. 

Another way of phrasing this is as follows. Let F denote the free group
on two generators, and let H = Hom(F,G) be the (finite) set of
homomorphisms from F to our finite simple group G. Then almost all
homomorphisms in H are surjective. 

One can try to generalise this in an interesting and fruitful way by
asking the same kind of question when F is a different finitely
generated group. For example, if F is the modular group PSL(2,Z) (which
is also the free product C_2 * C_3), one is asking whether simple groups
are randomly generated by two elements, one of order 2 and the other of
order 3; as another example, taking F to be the "Hurwitz triangle group"
T_{2,3,7} = <x,y,z : x2 = y3 = z7 = xyz = 1>, and G to be one of the
alternating groups A_n, we have a well known question of Graham Higman:
is A_n a Hurwitz group (i.e. an image of T_{2,3,7}) for sufficiently
large n? More to the point, is this the case "randomly" ? 

One of the most general situations for which methods seem to be
available is that in which F is a Fuchsian group (a discrete group of
isometries of the hyperbolic plane). I will discuss some of these
questions and the methods used to tackle them. These involve character
theory of finite simple groups, some algebraic group theory, and some
aspects of algebraic geometry such as geometric invariant theory. A
brief introduction to all these areas will be given.