Curvature-free linear length bounds on geodesics in closed Riemannian surfaces

This paper proves that in any closed Riemannian surface $M$ with diameter $d$, the length of the $k^\text{th}$-shortest geodesic between two given points $p$ and $q$ is at most $8kd$. This bound can be tightened further to $6kd$ if $p = q$. This improves prior estimates by A. Nabutovsky and R. Rotman.Comment: 21 pages, 7 figures. To be published in Tran. Amer. Math. Soc. Minor revisions only, such as: main result restricted to the main case of 2-spheres while valid for other closed surfaces; minor corrections to the statements of Lemmas 3.3, 3.4 and 4.1, and to the labeling of Figure 6; hand-drawn figures replaced with vector graphics; cap product structure uses c in H^2, not H^