Gromov-hyperbolicity

The curve complex $ C(S)$ 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 $ C(S)$ is simply connected and homotopic to a wedge of spheres (so it cannot have a $ \mathrm{CAT}(k)$ metric, for any $ k \leq 0$).

Let $ \delta > 0$. A geodesic triangle is calls $ \boldsymbol{\delta}$-slim if each of its sides is contained in the $ \delta$-neighbourhood of the other two sides. A space is called $ \boldsymbol{\delta}$-hyperbolic if all geodesic triangles are $ \delta$-slim, and Gromov hyperbolic if it is $ \delta$-hyperbolic for some positive $ \delta$.

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

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

If we consider the Weil-Petersson metric on $ T(S)$ then we have that for low dimension $ T(S)$ is Gromov-hyperbolic, while it is not if the dimension is greater than $ 2$.

Theorem 4.7 (Brock [11])   The Weil-Petersson metric on $ T(X)$ is Gromov-hyperbolic if and only the dimension of $ T(X)$ is less than or equal to $ 2$.

The proof of the theorem is based on the previous result of Brock that says that $ T(S)$ (with the Weil-Petersson metric) and the complex $ C_P(S)$ are quasi-isometric. In the case of $ T(S)$ being of dimension $ 1$, one has that $ C_P(S)$ is equal to the curve complex $ C(S)$, which by Masur-Minsky theorem is Gromov-hyperbolic. The case of dimension $ 2$ 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 $ 1$. 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 $ T(S)$ we have the following theorem (the result is in the same paper):

Theorem 4.8   The rank of the Weil-Petersson metric on $ T(S)$ is at least $ [(\mathrm{dim}(T(S)) + 1) / 2]$.

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 $ T(S)$ is at least $ 3$, then $ T(S)$ 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 $ \epsilon > 0$ there is no infinite collection of pairwise disjoint balls of radius $ \epsilon$ embedded into the space.

Theorem 4.10   If the dimension of $ T(S)$ is at least $ 2$, then the moduli space $ M(S)$ admits no complete Riemannian metric of pinched negative sectional curvature.

Pablo Ares Gastesi 2005-08-31