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https://en.wikipedia.org/wiki/Teichm%C3%BCller_space
Teichmuller space

It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6g-6 for a surface of genus g >= 2. In this way Teichmuller space can be viewed as the universal covering orbifold of the Riemann moduli space.

Contents
1 History
2 Definitions
2.1 Teichmuller space from complex structures
2.2 The Teichmuller space of the torus and flat metrics
2.3 Finite type surfaces
2.4 Teichmuller spaces and hyperbolic metrics
2.5 The topology on Teichmuller space
2.6 More examples of small Teichmuller spaces
2.7 Teichmuller space and conformal structures
2.8 Teichmuller spaces as representation spaces
2.9 A remark on categories
2.10 Infinite-dimensional Teichmuller spaces
3 Action of the mapping class group and relation to moduli space
3.1 The map to moduli space
3.2 Action of the mapping class group
3.3 Fixed points
4 Coordinates
4.1 Fenchel?Nielsen coordinates
4.2 Shear coordinates
4.3 Earthquakes
5 Analytic theory
5.1 Quasiconformal mappings
5.2 Quadratic differentials and the Bers embedding
5.3 Teichmuller mappings
6 Metrics
6.1 The Teichmuller metric
6.2 The Weil?Petersson metric
7 Compactifications
7.1 Thurston compactification
7.2 Bers compactification
7.3 Teichmuller compactification
7.4 Gardiner?Masur compactification
8 Large-scale geometry
9 Complex geometry
9.1 Metrics coming from the complex structure
9.2 Kahler metrics on Teichmuller space
9.3 Equivalence of metrics
10 See also
11 References
12 Sources
13 Further reading

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