The bicorn curves on closed surfaces

2025/12/11 by Takuya Katayama, Katayama, Takuya, Erika Kuno +1
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2512.10306

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.

Citations

• Bounded geodesic image theorem via bicorn curves
• Origami edge-paths in the curve graph
• Curves on the torus intersecting at most k times
• Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary
• Uniform hyperbolicity of the graphs of nonseparating curves via bicorn\n curves
• A note on acylindrical hyperbolicity of Mapping Class Groups
• Arcs intersecting at most once
• Uniform Hyperbolicity of the Graphs of Curves
• The geometry of the disk complex
• Quasi-homomorphisms on mapping class groups
• 3-manifolds as viewed from the curve complex

Related

• Bounded geodesic image theorem via bicorn curves
• A short proof of the bounded geodesic image theorem
• Uniform hyperbolicity of the graphs of nonseparating curves via bicorn\n curves
• Simple curves on surfaces
• From curve graphs to fine curve graphs and back