# Fuchsian model

In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group ${\displaystyle \pi _{1}(R)}$. The fundamental group can be shown to be isomorphic to some subgroup Γ of the group of real Möbius transformations ${\displaystyle SL(2,{\mathbb {R} })}$, 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. The latter remark is true mostly of the creator of this page. Meanwhile, Matsuzaki reserves the term Fuchsian model for the Fuchsian group, never the surface itself.

## 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 ${\displaystyle f:{\mathbb {H} }\rightarrow R}$, where H is the upper half-plane.

The Fuchsian model of R is the quotient space ${\displaystyle R^{h}={\mathbb {H} }/\Gamma }$. R. Note that ${\displaystyle R^{h}}$ 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 closedTemplate:Dn hyperbolic surface. Let G be the Fuchsian group of R and let ${\displaystyle \rho :G\rightarrow PSL(2,{\mathbb {R} })}$ be a faithful representation of G, and let ${\displaystyle \rho (G)}$ be discrete. Then define the set

${\displaystyle A(G)=\{\rho :\rho {\mbox{ defined as above }}\}}$

and add to this set a topology of pointwise convergence, so that A(G) is an algebraic topology.

The Nielsen isomorphism theorem: For any ${\displaystyle \rho \in A(G)}$ there exists a homeomorphism h of the upper half-plane H such that ${\displaystyle h\circ \gamma \circ h^{-1}=\rho (\gamma )}$ for all ${\displaystyle \gamma \in G}$.

Most of the material here is copied, not very accurately, out of the book below (see page 12).

## References

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