Universal length bounds for non‐simple closed geodesics on hyperbolic surfaces

We investigate the relationship, in various contexts, between a closed geodesic with self‐intersection number k (for brevity, called a k ‐geodesic) and its length. We show that for a fixed compact hyperbolic surface, the short k ‐geodesics have length comparable with the square root of k . On the other hand, if the fixed hyperbolic surface has a cusp and is not the punctured disc, then the short k ‐geodesics have length comparable with log k . The length of a k ‐geodesic on any hyperbolic surface is known to be bounded from below by a constant that goes to infinity with k . In this paper, we show that the optimal constants { M k } are comparable with log k leading to a generalization of the well‐known fact that length less than 4   log ⁡ ( 1 + 2 ) implies simple. Finally, we show that for each natural number k , there exists a hyperbolic surface where the constant M k is realized as the length of a k ‐geodesic. This was previously known for k = 1, where M 1 is the length of the figure eight on the thrice punctured sphere.