Complements of closed geodesics
I will describe some nice connections between closed
geodesics on surfaces, knot theory, continued fractions and hyperbolic
three-manifolds. Using a certain gadget called a "template" for the
modular surface, found by Ghys, it is possible to obtain an upper
bound for the volume of a geodesic (or its complement in the unit
tangent bundle) in terms of its length. This is joint work with Maxime
Bergeron and Lior Silberman.