Uniform hyperbolicity of nonseparating curve graphs of nonorientable surfaces

TL;DR

This paper proves that the graph of nonseparating curves on a nonorientable surface is connected and Gromov hyperbolic with a uniform constant, extending known results from orientable surfaces.

Contribution

It establishes the uniform hyperbolicity of nonseparating curve graphs for nonorientable surfaces, using bicorn curves and adapting Rasmussen's approach.

Findings

The nonseparating curve graph is connected.

The graph is Gromov hyperbolic with a uniform constant.

The hyperbolicity constant does not depend on the surface's topological type.

Abstract

Let $N$ be a connected finite type nonorientable surface with or without boundary components and punctures. We prove that the graph of nonseparating curves of $N$ is connected and Gromov hyperbolic with a constant which does not depend on the topological type of the surface by using the bicorn curves introduced by Przytycki and Sisto. The proof is based on the argument by Rasmussen on the uniform hyperbolicity of graphs of nonseparating curves for finite type orientable surfaces.