##

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