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 $R$ 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 $\mathbb {H}$ by a subgroup $\Gamma$ acting properly discontinuously and freely.

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

Fuchsian models and Teichmüller space

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

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 $\Gamma$ is finitely generated since it is isomorphic to the fundamental group of $R$ . Let $g_{1},\ldots ,g_{r}$ be a generating set: then any $\rho \in A(\Gamma )$ is determined by the elements $\rho (g_{1}),\ldots ,\rho (g_{r})$ and so we can identify $A(G)$ with a subset of $\mathrm {PSL} _{2}(\mathbb {R} )^{r}$ by the map $\rho \mapsto (\rho (g_{1}),\ldots ,\rho (g_{r}))$ . 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 $\rho \in A(G)$ there exists a self-homeomorphism (in fact a quasiconformal map) $h$ of the upper half-plane $\mathbb {H}$ such that $h\circ \gamma \circ h^{-1}=\rho (\gamma )$ for all $\gamma \in G$ .

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

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

References

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

Fuchsian model

A more precise definition

Fuchsian models and Teichmüller space

References

See also