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 $π 1 (R)$ . The fundamental group can be shown to be isomorphic to some subgroup Γ of the group of real Möbius 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.

1 A more precise definition
2 Nielsen isomorphism theorem
3 See also
4 See also

[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 $f:\mathbb{H}\rightarrow R$ , 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.

[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 $\rho:G\rightarrow PSL(2,\mathbb{R})$ 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 $\rho\in A(G)$ there exists a homeomorphism h of the upper half-plane H such that $h \circ \gamma \circ h^{-1} = \rho(\gamma)$ for all $\gamma \in G$ .