Lower Bound for the Number of Non-simple Geodesics on a Pair of Pants

Let S be a surface and let P be a pair of pants. Geodesics on surfaces, and on pairs of pants specifically, have been studied extensively over the years. In this paper, we focus on getting a lower bound on the number of closed geodesics on P with given upper bounds on length and self-intersection number. As a direct consequence, we get a lower bound for the number of such closed geodesics on any surface S. In future papers, we will get upper bounds on this number for an arbitrary surface, and tighter upper bounds on pairs of pants P . The reason that the upper bounds are better on pairs of pants is that all geodesics there can be constructed explicitely, while geodesics on surfaces cannot.