Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves

TL;DR

The paper proves that graphs of nonseparating curves on oriented finite type surfaces are uniformly hyperbolic, extending to infinite type surfaces, using homology-based arguments to identify nonseparating curves.

Contribution

It introduces new homology-based methods to certify nonseparating curves, extending hyperbolicity results to infinite type surfaces.

Findings

Graphs of nonseparating curves are uniformly hyperbolic for finite type surfaces.

The approach extends hyperbolicity results to infinite type surfaces with finite positive genus.

Homology arguments effectively certify nonseparating curves.

Abstract

We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonseparating. As demonstrated by Aramayona-Valdez, this proves also that the graph of nonseparating curves for any oriented infinite type surface with finite positive genus is hyperbolic.