Almost simple geodesics on the triply-punctured sphere

Abstract

In this paper we study closed hyperbolic geodesics $$\gamma $$ on the triply-punctured sphere $$M = {\widehat{{\mathbb C}}}- \{0,1,\infty \}$$ that are almost simple, in the sense that the difference $$\delta = I(\gamma )-L(\gamma )$$ between the self-intersection number of $$\gamma $$ and its combinatorial (word) length is fixed. We show that for each fixed $$\delta $$ , the number of almost simple geodesics with $$L(\gamma )=L$$ is given by a quadratic polynomial $$p_\delta (L)$$ , provided $$L \ge \delta + 4$$ .