The Geometry and Combinatorics of Closed Geodesics on Hyperbolic Surfaces

In this thesis, we obtain combinatorial algorithms that determine the minimal number of self-intersections necessary for a free homotopy class $[\gamma]$ on an orientable surface, using algebraic input. Using this same input, we describe another algorithm which determines whether or not a minimally intersecting curve in $[\gamma]$ is \textit{filling}, that is, whether or not the complement is a disjoint union of disks or punctured disks. Next, we use these algorithms as inspiration for proving the existence of filling curves which self-intersect $2g-1$ times, which is the minimal number of intersections possible. The combinatorial viewpoint that is developed can then be used to obtain geometric information about the curves, which is the subject of the last chapter. Among other things, we obtain a sharp lower bound on the length of a filling curve with the minimal number of self-intersections on a surface of genus g