The shortest non-simple closed geodesics on hyperbolic surfaces

TL;DR

This paper determines the minimal length of closed geodesics with many self-intersections on hyperbolic surfaces, showing they all lie on an ideal pair of pants with a specific length formula.

Contribution

It establishes a precise length formula for the shortest non-simple closed geodesics with many self-intersections on hyperbolic surfaces, identifying their geometric location.

Findings

Shortest geodesics with at least k self-intersections lie on an ideal pair of pants.

Length of these geodesics is 2 * arccosh(2k+1).

Results hold for sufficiently large k.

Abstract

This article explores closed geodesics on hyperbolic surfaces. We show that, for sufficiently large $k$, the shortest closed geodesics with at least $k$ self-intersections, taken among all hyperbolic surfaces, all lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.