Uryson width and pants decompositions of hyperbolic surfaces

TL;DR

This paper establishes a bound on the size of pants decompositions of hyperbolic surfaces, showing each curve can be contained in a small diameter ball proportional to the square root of the genus plus the number of cusps.

Contribution

It provides a universal bound on the diameter of curves in pants decompositions of hyperbolic surfaces, linking geometric decomposition to surface complexity.

Findings

Each curve in the pants decomposition is contained in a ball of diameter at most C√(g + n).

The bound is universal, independent of the specific surface.

The result connects geometric and topological properties of hyperbolic surfaces.

Abstract

Suppose that $M$ is a hyperbolic surface of genus $g$ and with $n$ cusps. Then we can find a pants decomposition of $M$ composed of simple closed geodesics so that each curve is contained in a ball of diameter at most $C\sqrt{g + n}$, where $C$ is a universal constant.