# 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} }

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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