# Hyperbolicity constants for pants and relative pants graphs

## TL;DR

This paper investigates the hyperbolicity constants of pants and relative pants graphs for specific surfaces, providing effective estimates and insights into how these constants vary with surface complexity, advancing understanding of their geometric properties.

## Contribution

It introduces effective estimates for hyperbolicity constants of pants graphs for certain surfaces, linking these constants to surface complexity and advancing geometric understanding.

## Key findings

- Hyperbolicity constant for the five-punctured sphere calculated
- Hyperbolicity constant for the twice punctured torus determined
- Relative pants graph for complexity 3 surfaces analyzed

## Abstract

The pants graph has proved to be influential in understanding 3-manifolds concretely. This stems from a quasi-isometry between the pants graph and the Teichm\"uller space with the Weil-Petersson metric. Currently, all estimates on the quasi-isometry constants are dependent on the surface in an undiscovered way. This paper starts effectivising some constants which begins the understanding how relevant constants change based on the surface. We do this by studying the hyperbolicity constant of the pants graph for the five-punctured sphere and the twice punctured torus. The hyperbolicity constant of the relative pants graph for complexity 3 surfaces is also calculated. Note, for higher complexity surfaces, the pants graph is not hyperbolic or even strongly relatively hyperbolic.