# Fuchsian model

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.

## A more precise definition

By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface ${\displaystyle R}$ 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 ${\displaystyle \mathbb {H} }$ by a subgroup ${\displaystyle \Gamma }$ acting properly discontinuously and freely.

In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformation is the group ${\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}$ acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup ${\displaystyle \Gamma \subset \mathrm {PSL} _{2}(\mathbb {R} )}$ such that the Riemann surface ${\displaystyle \Gamma \backslash \mathbb {H} }$ is isomorphic to ${\displaystyle \mathbb {R} }$. Such a group is called a Fuchsian group, and the isomorphism ${\displaystyle R\cong \Gamma \backslash \mathbb {H} }$ is called a Fuchsian model for ${\displaystyle \mathbb {H} }$.

## Fuchsian models and Teichmüller space

Let ${\displaystyle R}$ be a closed hyperbolic surface and let ${\displaystyle \Gamma }$ be a Fuchsian group so that ${\displaystyle \Gamma \backslash \mathbb {H} }$ is a Fuchsian model for ${\displaystyle R}$. Let

${\displaystyle A(\Gamma )=\{\rho :\Gamma \to \mathrm {PSL} _{2}(\mathbb {R} ):\rho {\mbox{ is faithful and discrete }}\}}$

and endow this set with the topology of pointwise convergence (sometimes called "algebraic convergence"). In this particular case this topology can most easily be defined as follows: the group ${\displaystyle \Gamma }$ is finitely generated since it is isomorphic to the fundamental group of ${\displaystyle R}$. Let ${\displaystyle g_{1},\ldots ,g_{r}}$ be a generating set: then any ${\displaystyle \rho \in A(\Gamma )}$ is determined by the elements ${\displaystyle \rho (g_{1}),\ldots ,\rho (g_{r})}$ and so we can identify ${\displaystyle A(G)}$ with a subset of ${\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )^{r}}$ by the map ${\displaystyle \rho \mapsto (\rho (g_{1}),\ldots ,\rho (g_{r}))}$. 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:

For any ${\displaystyle \rho \in A(G)}$ there exists a self-homeomorphism (in fact a quasiconformal map) ${\displaystyle h}$ of the upper half-plane ${\displaystyle \mathbb {H} }$ such that ${\displaystyle h\circ \gamma \circ h^{-1}=\rho (\gamma )}$ for all ${\displaystyle \gamma \in G}$.

The proof is very simple: choose an homeomorphism ${\displaystyle R\to \rho (\Gamma )\backslash \mathbb {H} }$ and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since ${\displaystyle R}$ is compact.

This result can be seen as the equivalence between two models for Teichmüller space of ${\displaystyle R}$: the set of discrete faithful representations of the fundamental group ${\displaystyle \pi _{1}(R)}$ into ${\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}$ modulo conjugacy and the set of marked Riemann surfaces ${\displaystyle (X,f)}$ where ${\displaystyle f:R\to X}$ is a quasiconformal homeomorphism modulo a natural equivalence relation.

## References

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