Shortest filling geodesics on hyperbolic surfaces

TL;DR

This paper determines the shortest filling geodesic on a hyperbolic surface of genus g, showing it is a right-angled regular polygon and providing a specific geodesic that achieves this minimum.

Contribution

It identifies the minimal length filling geodesic on hyperbolic surfaces and characterizes it explicitly as a right-angled regular polygon, a novel geometric result.

Findings

Minimal length filling geodesic is a right-angled regular (8g-4)-gon

Explicit geodesic realizing the minimal length is provided

The minimal length is characterized within the moduli space

Abstract

In this paper, we obtain the minimal length of a filling (multi-)geodesic on a genus $g$ hyperbolic surface in the moduli space of hyperbolic surfaces and show that it is realized by the geodesic whose complement is a right-angled regular $(8g-4)$-gon. A single geodesic realizing this minimum is provided.