Uniform hyperbolicity of nonseparating curve graphs of nonorientable surfaces

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.

Figures (8)

• Figure 1: Two points $x$ and $y$ of $a\cap b$ consective along $b$ whose signs in $\mathrm{nbd}(\beta)$ are the same (left-hand side) and different (right-hand side).
• Figure 2: The way to make new curves $c_{1}$ and $c_{2}$ when the signs in $\mathrm{nbd}(\beta)$ at the intersection points $x$ and $y$ between $a$ and $b$ are the same (left) and different (right).
• Figure 3: Examples of intersections between $a$ and $c$ if $c$ is two-sided (left) and one-sided (right).
• Figure 4: Examples of intersections between $a$ and $c$ if $c$ is two-sided (left) and one-sided (right).
• Figure 5: Intersections between $c$ and $c'$ when the signs in $\mathrm{nbd}(\alpha)$ of the intersection points $y, z$ between $a$ and $b$ are the same and the signs in $\mathrm{nbd}(\alpha)$ of the intersection points $x, y$ between $a$ and $b$ are the same, and $c$ is two-sided (left-hand side) and one-sided (right-hand side).

Theorems & Definitions (26)

• Theorem 1.1
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Definition 2.5
• Lemma 3.1
• proof : Proof of Lemma \ref{['nonseparating_curve_graphs_are_quasi-isometric']}