Uniform hyperbolicity of the curve graph via surgery sequences

TL;DR

This paper proves that the curve graph of any surface is Gromov-hyperbolic with a uniform constant, using techniques inspired by the hyperbolicity of the free splitting complex.

Contribution

It establishes uniform hyperbolicity of the curve graph across all surfaces, extending previous results with a new proof approach.

Findings

Curve graph is Gromov-hyperbolic with a constant independent of surface complexity.

The proof adapts methods from hyperbolicity of the free splitting complex.

Provides a unified hyperbolicity constant applicable to all surfaces.

Abstract

We prove that the curve graph $\calC^{(1)}(S)$ is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface $S$. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.