Lower bound on the number of non-simple closed geodesics on surfaces

Abstract

We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for arbitrary surfaces follows from the lower bound on pairs of pants. This lower bound demonstrates that as the self-intersection number $K = K(L)$ goes from a constant to a quadratic function of $L$, the number of closed geodesics transitions from polynomial to exponential in $L$. We show upper bounds on the number of such geodesics in a subsequent paper.