A short note on short pants

TL;DR

This paper provides a concise proof of an improved linear upper bound on the length of curves in pants decompositions of hyperbolic surfaces, refining previous bounds that depended on the surface's genus.

Contribution

It offers a shorter proof that slightly improves the known linear bounds on curve lengths in pants decompositions of hyperbolic surfaces.

Findings

Established a new linear upper bound for curve lengths

Provided a shorter proof of existing bounds

Improved the constants in the bounds

Abstract

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Sepp\"al\"a. The goal of this note is to give a short proof of an linear upper bound which slightly improves the best known bounds.