Short closed geodesics with self-intersections

TL;DR

This paper studies the minimal length closed geodesics with at least k self-intersections on hyperbolic surfaces, establishing universal linear bounds on their intersection numbers and asymptotic behavior in cusped cases.

Contribution

It provides the first universal linear bounds on self-intersection numbers of minimal length geodesics with at least k self-intersections on hyperbolic surfaces.

Findings

Self-intersection numbers are bounded above by a universal linear function in k.

In surfaces with cusps, self-intersection numbers grow asymptotically like k.

The results hold for any hyperbolic surface, regardless of topology.

Abstract

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in $k$ (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like $k$ for growing $k$.