Quasi-Fuchsian space of the once-punctured torus

In this thesis we look at two generator groups of Möbius transformations where the commutator of the generators is parabolic. In particular we are interested in quasi- Fuchsian groups T whose quotient surface fl/T consists of two once-punctured tori. The set of all such quasi-Fuchsian groups, which is called the quasi-Fuchsian space of the once-punctured torus, is defined in chapter 1. In chapter 2 we introduce two sets of suitable coordinates for quasi-Fuchsian space. The first is the well-known trace parameters and the other set of coordinates involves an appropriate normalisation of conjugacy classes of quasi-Fuchsian groups. We study a special class of quasi-Fuchsian groups in chapter 3 which are those groups obtained by pairing four classical circles, each of which is tangent to its neighbours. We find the exact subset of quasi-Fuchsian space where these groups lie and investigate their limiting behaviour. In chapter 4 we return to the whole of quasi-Fuchsian space of the once-punctured torus and investigate what happens when we change the generators of such a group T. In particular we reduce the modulus of the traces of generators of T, and use this information to build up a substantial picture of quasi-Fuchsian space. In the last chapter we look at the traces of elements of one particular group, which gives rise to the Diophantine equation a2 + b2+c2 = 3a6c studied by Markoff. If we arrange a solution triple of natural numbers (a, 6, c) in ascending order, so that a < b < c, it has long been conjectured that the largest number uniquely determines the triple. We finish by proving that if c, is prime then this statement is true. We show this using only algebraic number theory, but we mention the geometric motivation which originally gave the ideas for the proof, and where it appears in the earlier chapters