# Geometry of the complex of curves I: Hyperbolicity

@article{Masur1999GeometryOT,
title={Geometry of the complex of curves I: Hyperbolicity},
author={Howard A. Masur and Yair N. Minsky},
journal={Inventiones mathematicae},
year={1999},
volume={138},
pages={103-149}
}
• Published 21 April 1998
• Mathematics
• Inventiones mathematicae
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…
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## References

SHOWING 1-10 OF 69 REFERENCES
A geometric approach to the complex of curves
1. The complex of curves Geometry has played an important role in the study of Teichm uller space and its group of symmetries, the Mapping Class Group of a surface. In 13], Harvey introduced a
Automorphism of complexes of curves and of Teichmuller spaces : Interna
To every compact orientable surface one can associate following Harvey Ha Ha a combinatorial object the so called complex of curves which is analogous to Tits buildings associated to semisimple Lie
The virtual cohomological dimension of the mapping class group of an orientable surface
Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those
On a class of geodesics in Teichmiuller space
The study of the geometry of the classical Teichmiiller spaces was begun in 1959 by Kravetz [9]. The starting point was the classical theorem of TeichmUller on extremal quasiconformal maps between
The Nielsen Realization Problem
Closed, oriented surfaces of genus g > 2 admit many hyperbolic (constant Gaussian curvature -1) metrics in contrast to Mostow's rigidity theorems in higher dimensions. Only special hyperbolic
Stability of the homology of the mapping class groups of orientable surfaces
The mapping class group of F = Fgs r is F = rgs = wo(A) where A is the topological group of orientation preserving diffeomorphisms of F which are the identity on dF and fix the s punctures. When r =
On the geometry and dynamics of diffeomorphisms of surfaces
This article was widely circulated as a preprint, about 12 years ago. At that time the Bulletin did not accept research announcements, and after a couple of attempts to publish it, I gave up, and the
Bounds on least dilatations
We consider the collection of all pseudo-Anosov homeomorphisms on a surface of fixed topological type. To each such homeomorphism is associated a real-valued invariant, called its dilatation (which