quasiconformal dilatations, on , which is not symmetric. The Weil–Petersson metric Quadratic differentials on a Riemann surface are identified with the tangent space at to Teichmüller space. The Weil–Petersson metric is the Riemannian metric defined by the inner product on quadratic differentials. Compactifications There are several inequivalent compactifications of Teichmüller spaces that have been studied. Several of the earlier compactifications depend on the choice of a point in Teichmüller space so are not invariant under the modular group, which can be inconvenient. William Thurston later found a compactification without this disadvantage, which has become the most widely used compactification. Thurston compactification By looking at the hyperbolic lengths of simple closed curves for each point in Teichmüller space and taking the closure in the (infinite-dimensional) projective space, introduced a compactification whose points at infinity correspond to projective measured laminations. The compactified space is homeomorphic to a closed ball. This Thurston compactification is acted on continuously by the modular group. In particular any element of the modular group has a fixed point in Thurston's compactification, which Thurston used in his classification of elements of the modular group. Bers compactification The Bers compactification is given by taking the closure of the image of the Bers embedding of Teichmüller space, studied by . The Bers embedding depends on the choice of a point in Teichmüller space so is not invariant under the modular group, and in fact the modular group does not act continuously on the Bers compactification. Teichmüller compactification The "points at infinity" in the Teichmüller compactification consist of geodesic rays (for the Teichmüller metric) starting at a fixed basepoint. This compactification depends on the choice of basepoint so is not acted on by the modular group, and in fact Kerckhoff showed that the action of the modular group on Teichmüller space does not extend to a continuous action on this compactification. Gardiner–Masur compactification considered a compactification similar to the Thurston compactification, but using extremal length rather than hyperbolic length. The modular group acts continuously on this compactification, but they showed that their compactification has strictly more points at infinity. Large-scale geometry There has been an extensive study of the geometric properties of Teichmüller space endowed with the Teichmüller metric. Known large-scale properties include: Teichmüller space contains flat subspaces of dimension , and there are no higher-dimensional quasi-isometrically embedded flats. In particular, if or or then is not hyperbolic. On the other hand,