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 <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.

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.

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>.

Related topics

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

See also

• Fundamental polygon