Almost simple geodesics on the triply punctured sphere

Abstract

Every closed hyperbolic geodesic $\gamma$ on the triply--punctured sphere $M=\hat{\mathbb{C}}-\{0,1,\infty \}$ has a self--intersection number $I(\gamma )\ge 1$ and a combinatorial length $L(\gamma )\ge 2$, the latter defined by the number of times $\gamma$ passes through the upper halfplane. In this paper we show that $\delta (\gamma )=I(\gamma )-L(\gamma )\ge -1$ for all closed geodesics; and that for each fixed $\delta$, the number of geodesics with invariants $(\delta ,L)$ is given exactly by a quadratic polynomial $p_{\delta }(L)$ for all $L\ge 4+\delta$.