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Teichmüller space

In [mathematics] , given a [Riemann surface] * X * , the ** Teichmüller space ** of * X * , notated * TX * or Teich( * X * ), is a [complex] [manifold] whose points represent all [complex structure] s of Riemann surfaces whose underlying [topological structure] is the same as that of * X * . It is named after the German mathematician [Oswald Teichmüller] .

** Relation to moduli space **

The Teichmüller space of a surface is related to its [moduli space] , but preserves more information about the surface. More precisely, the surface * X * (or its underlying topological structure) provides a * marking * * X → Y * of each Riemann surface * Y * represented in * TX * : whereas moduli space identifies all surfaces which are isomorphic, * TX * only identifies those surfaces which are isomorphic via a [biholomorphic] map * f * that is [isotopic] to the identity (with respect to the marking, hence its need). The automorphisms of * X * , up to isotopy, form a discrete group (the ** Teichmüller modular group ** , or [mapping class group] of * X * ) that acts on * TX * . The action is as follows: if [ * g * ] is an element of the mapping class group of * X * , then [ * g * ] sends the point represented by the marking * h: X → Y * to the point with the marking * hg : X → X → Y * . The quotient of * TX * by this action is precisely the moduli space of * X * .

** Properties of ** ** TX **

The Teichmüller space of

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