space, and this is a very active subject of research in geometric group theory. Definitions Teichmüller space from complex structures Let be an orientable smooth surface (a differentiable manifold of dimension 2). Informally the Teichmüller space of is the space of Riemann surface structures on up to isotopy. Formally it can be defined as follows. Two complex structures on are said to be equivalent if there is a diffeomorphism such that: It is holomorphic (the differential is complex linear at each point, for the structures at the source and at the target) ; it is isotopic to the identity of (there is a continuous map such that . Then is the space of equivalence classes of complex structures on for this relation. Another equivalent definition is as follows: is the space of pairs where is a Riemann surface and a diffeomorphism, and two pairs are regarded as equivalent if is isotopic to a holomorphic diffeomorphism. Such a pair is called a marked Riemann surface; the marking being the diffeomeorphism; another definition of markings is by systems of curves. There are two simple examples that are immediately computed from the Uniformization theorem: there is a unique complex structure on the sphere (see Riemann sphere) and there are two on (the complex plane and the unit disk) and in each case the group of positive diffeomorphisms is contractible. Thus the Teichmüller space of is a single point and that of contains exactly two points. A slightly more involved example is the open annulus, for which the Teichmüller space is the interval (the complex structure associated to is the Riemann surface ). The Teichmüller space of the torus and flat metrics The next example is the torus In this case any complex structure can be realised by a Riemann surface of the form (a complex elliptic curve) for a complex number where is the complex upper half-plane. Then we have a bijection: and thus the Teichmüller space of is If we identify with the Euclidean plane then each point in Teichmüller space can also be viewed as a marked flat structure on Thus the Teichmüller space is in bijection with the set of pairs where is a flat surface and is a diffeomorphism up to isotopy on . Finite type surfaces These are the surfaces for which Teichmüller space is most often studied, which include closed surfaces. A surface is of finite type