Geodesics with one self-intersection, and other stories

In this paper, we show that for any hyperbolic surface SS, the number of geodesics of length bounded above by LL in the mapping class group orbit of a fixed closed geodesic γγ with a single double point is asymptotic to Ldim(Teichmuller space of S.)Ldim(Teichmuller space of S.). Since closed geodesics with one double point fall into a finite number of Mod(S) orbits, we get the same asymptotic estimate for the number of such geodesics of length bounded by LL, and systems of curves, where one curve has a self-intersection, or there are two curves intersecting once. 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 SS can be extended to a covering of the surface SS 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 regular   covering of SS.We also analyze the question of the minimal index of a subgroup in a surface group which does not contain a given element. We show that we have a linear bound for the index of an arbitrary subgroup, a cubic bound for the index of a normal subgroup, but also poly-log bounds for each fixed level in the lower central series (using elementary arithmetic considerations) — the results hold for free groups and fundamental groups of closed surfaces.