A remarkable identity for lengths of curves
In this thesis we will prove the following new identity Σγ 1/(1 + exp |γ|) = 1/2, where the sum is over all closed simple geodesics γ on a punctured torus with a complete hyperbolic structure, and |γ| is the length of γ. Although it is well known that there are relations between the lengths of simple geodesics on a hyperbolic surface (for example the Fricke trace relations and the Selberg trace formula) this identity is of a wholly different character to anything in the literature. Our methods are purely geometric; that is, the techniques are based upon the work of Thurston and others on geodesic laminations rather than the analytic approach of Selberg. The first chapter is intended as an exposition of some relevant theory concerning laminations on a punctured surface. Most important of these results is that a leaf of a compact lamination cannot penetrate too deeply into a cusp region. Explicit bounds for the maximum depth are given; in the case of a torus a simple geodesic is disjoint from any cusp region whose bounding curve has length less than 4, and this bound is sharp. Another significant result is that a simple geodesic which enters a small cusp region is perpendicular to the horocyclic foliation of the cusp region. The second chapter is concerned with Gcusp, the set of ends of simple geodesics with at least one end up the cusp. A natural metric on Gcusp is introduced so that we can discuss approximation theory. We divide the geodesics of Gcusp into three classes according to the behaviour of their ends; each class also has a characterisation in terms of how well any member geodesic can be approximated. An example is given to demonstrate how this classification generalises some ideas in the classical theory of Diophantine approximation. The first class consists of geodesics with both ends up the cusp. Restricting to the punctured torus it is shown that for such a geodesic, γ, there is a portion of the cusp region surrounding each end which is disjoint from all other geodesics in Gcusp. We call such a portion a gap. The geometry of the gaps attached to γ is described and their area computed by elementary trigonometry. The area is a function of the length of the unique closed simple geodesic disjoint from γ. Next we consider the a generic geodesic in Gcusp, that is, a geodesic with a single end up the cusp and another end spiralling to a minimal compact lamination which is not a closed geodesic. We show that such a geodesic is the limit from both the right and left of other geodesics in Gcusp. Finally we give a technique for approximating a geodesic with a single end up the cusp and the other end spiralling to a closed geodesic. Essentially we repeatedly Dehn twist a suitable geodesic in Gcusp round this closed geodesic. The results of this chapter are then combined with a theorem of J. Birman and C. Series to yield the identity.