##
Geometry of the Complex of Curves I: Hyperbolicity

### Howard Masur and Yair Minsky

### 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.

A current draft of the paper is available as a
Postscript file(877K), or
(without the figures) as a
dvi file(217K).
See also the
LANL XXX Math Archives.

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