In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.

A more precise definition

By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic which is not isomorphic to either the Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of the hyperbolic plane acting properly discontinuously and freely.

In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformations acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free.

Fuchsian models and Teichmüller space

is finitely generated. Then we give it the subspace topology.

The Nielsen isomorphism theorem (this is not standard terminology and this result is not directly related to the Dehn–Nielsen theorem) then has the following statement: is compact.

This result can be seen as the equivalence between two models for Teichmüller space is a quasiconformal homeomorphism modulo a natural equivalence relation.

See also

• the Kleinian model, an analogous construction for 3-manifolds

• Fundamental polygon

References

Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford (1998).