is properly discontinuous, and the quotient is the moduli space. Fixed points The Nielsen realisation problem asks whether any finite subgroup of the mapping class group has a global fixed point (a point fixed by all group elements) in Teichmüller space. In more classical terms the question is: can every finite subgroup of be realised as a group of isometries of some complete hyperbolic metric on (or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by Steven Kerckhoff. Coordinates Fenchel–Nielsen coordinates The Fenchel–Nielsen coordinates (so named after Werner Fenchel and Jakob Nielsen) on the Teichmüller space are associated to a pants decomposition of the surface . This is a decomposition of into pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in Teichmüller space, and another real parameter called the twist which is more involved to define. In case of a closed surface of genus there are curves in a pants decomposition and we get parameters, which is the dimension of . The Fenchel–Nielsen coordinates in fact define a homeomorphism . In the case of a surface with punctures some pairs of pants are "degenerate" (they have a cusp) and give only two length and twist parameters. Again in this case the Fenchel–Nielsen coordinates define a homeomorphism . Shear coordinates If the surface admits ideal triangulations (whose vertices are exactly the punctures). By the formula for the Euler characteristic such a triangulation has triangles. An hyperbolic structure on determines a (unique up to isotopy) diffeomorphism sending every triangle to an hyperbolic ideal triangle, thus a point in . The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation. There are such parameters which can each take any value in , and the completeness of the structure corresponds to a linear equation and thus we get the right dimension . These coordinates are called shear coordinates. For closed surfaces, a pair of pants can be decomposed as the union of two ideal triangles (it can be seen as an incomplete hyperbolic metric on the three-holed sphere). Thus we also get shear coordinates on . Earthquakes A simple earthquake path in Teichmüller space is a path determined by varying a single shear or length Fenchel–Nielsen coordinate (for a fixed ideal triangulation