Gromov-hyperbolicity

The curve complex can be made into a metric space by making each simplex isometric to the standard simplex in the corresponding Euclidean space. J.L. Harer [23] proved that is simply connected and homotopic to a wedge of spheres (so it cannot have a metric, for any ).

Let . A geodesic triangle is calls -slim if each of its sides is contained in the -neighbourhood of the other two sides. A space is called -hyperbolic if all geodesic triangles are -slim, and Gromov hyperbolic if it is -hyperbolic for some positive .

Theorem 4.5 (Masur and Minsky [34])   The space is Gromov-hyperbolic.

Theorem 4.6 (Masur and Wolf [35])   with the Teichmüller metric is not Gromov-hyperbolic.

If we consider the Weil-Petersson metric on then we have that for low dimension is Gromov-hyperbolic, while it is not if the dimension is greater than .

Theorem 4.7 (Brock [11])   The Weil-Petersson metric on is Gromov-hyperbolic if and only the dimension of is less than or equal to .

The proof of the theorem is based on the previous result of Brock that says that (with the Weil-Petersson metric) and the complex are quasi-isometric. In the case of being of dimension , one has that is equal to the curve complex , which by Masur-Minsky theorem is Gromov-hyperbolic. The case of dimension is proved by method of relative hyperbolicity [17]. For the higher dimensional cases one uses a result of Gromov that says that a Gromov-hyperbolic space has rank . The rank of a space is the maximal dimension of a quasi-flat (quasi-isometric image of an Euclidean space) in the space; in the case of we have the following theorem (the result is in the same paper):

Theorem 4.8   The rank of the Weil-Petersson metric on is at least .

There are another couple of interesting results in the same paper regarding properties of metrics on Teichmüller spaces. They are as follows.

Theorem 4.9   If the dimension of is at least , then admits no proper, geodesically complete, Gromov-hyperbolic, equivariant under the mapping class group, path metric with finite covolume.

A proper metric space is a metric space where the closed bounded balls are compact. A geodesically complete metric space has finite volume if for every there is no infinite collection of pairwise disjoint balls of radius embedded into the space.

Theorem 4.10   If the dimension of is at least , then the moduli space admits no complete Riemannian metric of pinched negative sectional curvature.

Pablo Ares Gastesi 2005-08-31