of a surface). The name comes from seeing the ideal triangles or the pants as tectonic plates and the shear as plate motion. More generally one can do earthquakes along geodesic laminations. A theorem of Thurston then states that two points in Teichmüller space are joined by a unique earthquake path. Analytic theory Quasiconformal mappings A quasiconformal mapping between two Riemann surfaces is a homeomorphism which deforms the conformal structure in a bounded manner over the surface. More precisely it is differentiable almost everywhere and there is a constant , called the dilatation, such that where are the derivatives in a conformal coordinate and its conjugate . There are quasi-conformal mappings in every isotopy class and so an alternative definition for The Teichmüller space is as follows. Fix a Riemann surface diffeomorphic to , and Teichmüller space is in natural bijection with the marked surfaces where is a quasiconformal mapping, up to the same equivalence relation as above. Quadratic differentials and the Bers embedding With the definition above, if there is a natural map from Teichmüller space to the space of -equivariant solutions to the Beltrami differential equation. These give rise, via the Schwarzian derivative, to quadratic differentials on . The space of those is a complex space of complex dimension , and the image of Teichmüller space is an open set. This map is called the Bers embedding. A quadratic differential on can be represented by a translation surface conformal to . Teichmüller mappings Teichmüller's theorem states that between two marked Riemann surfaces and there is always a unique quasiconformal mapping in the isotopy class of which has minimal dilatation. This map is called a Teichmüller mapping. In the geometric picture this means that for every two diffeomorphic Riemann surfaces and diffeomorphism there exists two polygons representing and an affine map sending one to the other, which has smallest dilatation among all quasiconformal maps . Metrics The Teichmüller metric If and the Teichmüller mapping between them has dilatation then the Teichmüller distance between them is by definition . This indeed defines a distance on which induces its topology, and for which it is complete. This is the metric most commonly used for the study of the metric geometry of Teichmüller space. In particular it is of interest to geometric group theorists. There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the