## Institute for Mathematical Sciences

## Preprint ims96-11

** H. Masur and Y. Minsky**
* Geometry of the complex of curves I: Hyperbolicity*

Abstract: 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.
(revised version of January 1998)

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