Geodesics with one self-intersection, and other stories

Abstract

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension of the Teichmuller space of S. Since closed geodesics with one double point fall into a finite number of orbits under the mapping class group of S, we get the same asympotic estimate for the number of such geodesics of length bounded by L. We also use our (elementary) methods to do a more precise study of geodesics with a single double point on a punctured torus, including an extension of McShane's identity to such geodesics. In the second part of the paper we study the question of when a covering of the boundary of an oriented surface S can be extended to a covering of the surface S itself, we obtain a complete answer to that question, and also to the question of when we can further require the extension to be a \emph{regular} covering of S. We also analyze the question (first raised by K. Bou-Rabee) of the minimal index of a subgroup in a surface group which does not contain a given element. We give a (conjecturally) sharp result graded by the depth of an element in the lower central series, as well as "ungraded" results.