# Almost simple geodesics on the triply-punctured sphere

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## 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 \(L \ge \delta + 4\).

## Mathematics Subject Classification

30F60 57M05## Notes

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