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 Mobius 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.

Contents

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

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 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 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 .

Related topics

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

See also

• Fundamental polygon