The bicorn curves on closed surfaces

TL;DR

This paper investigates bicorn curves on closed surfaces to analyze hyperbolic properties of curve graphs, establishing their hyperbolicity and improving bounds on the bounded geodesic image theorem.

Contribution

It introduces the use of bicorn curves to demonstrate hyperbolicity of curve graphs and refines bounds for the bounded geodesic image theorem.

Findings

Curve graph of any closed surface is 15-hyperbolic with one exception.

Provides tighter bounds for the bounded geodesic image theorem.

Enhances understanding of hyperbolic phenomena in surface topology.

Abstract

This paper focuses on using the theory of bicorn curves in the context of closed surfaces to understand hyperbolic phenomena of the curve graphs of those surfaces. We prove that the curve graph of any closed surface is 15-hyperbolic with one exception. Furthermore, we provide significantly tighter bounds for the bounded geodesic image theorem, originally proven by Masur--Minsky.