Fuchsian model
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In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group <math>\pi_1(R)</math>. The fundamental group can be shown to be isomorphic to some subgroup Γ of the group of real Möbius transformations <math>SL(2,\mathbb{R})</math>, this subgroup being a Fuchsian group. The quotient space H/Γ is then a Fuchsian model for the Riemann surface R. Many authors use the terms Fuchsian group and Fuchsian model interchangeably, letting the one stand for the other.
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[edit] A more precise definition
To be more precise, every Riemann surface has a universal covering map that is either the Riemann sphere, the complex plane or the upper half-plane. Given a covering map <math>f:\mathbb{H}\rightarrow R</math>, where H is the upper half-plane.
The Fuchsian model of R is the quotient space <math>R^h = \mathbb{H} / \Gamma</math>. R. Note that <math>R^h</math> is a complete 2D hyperbolic manifold.
[edit] Nielsen isomorphism theorem
The Nielsen isomorphism theorem basically states that the algebraic topology of a closed Riemann surface is the same as its geometry. More precisely, let R be a closed hyperbolic surface. Let G be the Fuchsian group of R and let <math>\rho:G\rightarrow PSL(2,\mathbb{R})</math> be a faithful representation of G, and let <math>\rho(G)</math> be discrete. Then define the set
- <math>A(G)=\{ \rho : \rho \mbox{ defined as above }\}</math>
and add to this set a topology of pointwise convergence, so that A(G) is an algebraic topology.
The Nielsen isomorphism theorem: For any <math>\rho\in A(G)</math> there exists a homeomorphism h of the upper half-plane H such that <math>h \circ \gamma \circ h^{-1} = \rho(\gamma)</math> for all <math>\gamma \in G</math>.
[edit] See also
An analogous construction for 3D manifolds is the Kleinian model.
