Shortest k-Geodesics on Hyperbolic Surfaces

TL;DR

This paper establishes new bounds relating the lengths of closed geodesics with a given number of self-intersections on hyperbolic surfaces, linking geometric and topological complexities more tightly than previous estimates.

Contribution

It provides explicit upper bounds for geodesic lengths based on the shortest figure eight curve and improves bounds on maximum self-intersection numbers, refining the understanding of geodesic complexity.

Findings

Derived explicit bounds for shortest geodesics with k self-intersections.

Improved the asymptotic upper bound for maximum self-intersection number from 512 to 128.

Connected geometric length estimates with combinatorial self-intersection data.

Abstract

We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of a shortest closed geodesic with exactly $k$ self-intersections in terms of the length $L_\textswab{8}(X)$ of a shortest figure eight curve, improving Basmajian's estimate. We analyze the geometry of a shortest figure eight curve and explicitly build families of words in $\pi_1(X)$ whose geodesic representatives realize prescribed self-intersection numbers. As a consequence, we improve existing estimates on the maximal self-intersection number $I_k(X)$ of shortest geodesics with at least $k$ self-intersections, reducing the asymptotic upper bound from 512 to 128. This provides a sharper quantitative connection between the geometry and combinatorial complexity of non-simple closed geodesics on hyperbolic surfaces.