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 π1(R). The fundamental group can be shown to be isomorphic to some subgroup Γ of the group of real Möbius transformations 
, 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 
, where H is the upper half-plane.
The Fuchsian model of R is the quotient space 
. R. Note that Rh 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 
be a faithful representation of G, and let ρ(G) be discrete. Then define the set
- A(G) = {ρ:ρ defined as above }
and add to this set a topology of pointwise convergence, so that A(G) is an algebraic topology.
The Nielsen isomorphism theorem: For any 
there exists a homeomorphism h of the upper half-plane H such that 
for all 
.
[edit] See also
An analogous construction for 3D manifolds is the Kleinian model.
