1998/04/30 by Howard Masur, Howard A. Masur, Yair N. Minsky · 8 citations
Computer Science · Mathematics · #Context (archaeology) #Curvature #Geometric and Algebraic Topology #Geometry #Homotopy #Hyperbolic space #Intersection (aeronautics) #Isotopy #Mapping class group #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Riemann surface #Space (punctuation) #Surface (topology) #Teichmüller space #Topological and Geometric Data Analysis #math.DS #math.GT #msc:30F60 #msc:32G15 #msc:57M07 #msc:57M50
paper · pdf · doi:10.1007/s002220050343
published as Invent. Math. 138 (1999), 103-149. · Revised version of IMS preprint. 36 pages, 6 Figures
arxiv created 1998/08/11 · openalex publication_date 1999/10/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface. We also show that the action of pseudo-Anosov mapping classes on the complex is hyperbolic, with a uniform bound on translation distance.