Length of Filling Pairs on Punctured Surface

TL;DR

This paper investigates the minimal length of filling pairs of curves on once-punctured hyperbolic surfaces, establishing a lower bound based solely on the surface's topology.

Contribution

It provides the first known lower bound for the length of filling pairs on punctured hyperbolic surfaces, depending only on topological features.

Findings

Established a lower bound for filling pair lengths

Bound depends only on surface topology

Advances understanding of geometric properties of punctured surfaces

Abstract

A pair $(\alpha, \beta)$ of simple closed curves on a surface $S_{g,n}$ of genus $g$ and with $n$ punctures is called a filling pair if the complement of the union of the curves is a disjoint union of topological disks and once punctured disks. In this article, we study the length of filling pairs on once-punctured hyperbolic surfaces. In particular, we find a lower bound of the length of filling pairs which depends only on the topology of the surface.