Fuchsian Model

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 \pi_1(R). The fundamental group can be shown to be isomorphic to some subgroup Γ of the group of real Mobius transformations SL(2,\mathbb{R}), 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.

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 f:\mathbb{H}\rightarrow R, where H is the upper half plane... The Fuchsian model of R is the quotient space R^h = \mathbb{H} / \Gamma. R. Note that R^h is a complete 2D hyperbolic manifold.

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 \rho:G\rightarrow PSL(2,\mathbb{R}) be a faithful representation of G, and let \rho(G) be discrete. Then define the set
A(G)=\{ \rho : \rho \mbox{ 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 \rho\in A(G) there exists a homeomorphism h of the upper half-plane H such that h \circ \gamma \circ h^{-1} = \rho(\gamma) for all \gamma \in G.

Related topics

An analogous construction for 3D manifolds is the Kleinian model.

 

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