# Fuchsian model

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In [mathematics](/source/Mathematics), a **Fuchsian model** is a representation of a hyperbolic [Riemann surface](/source/Riemann_surface) *R* as a quotient of the [upper half-plane](/source/Upper_half-plane) **H** by a [Fuchsian group](/source/Fuchsian_group). Every hyperbolic Riemann surface admits such a representation. The concept is named after [Lazarus Fuchs](/source/Lazarus_Fuchs).

## A more precise definition

By the [uniformization theorem](/source/Uniformization_theorem), every Riemann surface is either [elliptic](/source/Elliptic_geometry), [parabolic](/source/Parabolic_geometry_(differential_geometry)) or [hyperbolic](/source/Hyperbolic_geometry). 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](/source/Hyperbolic_plane) \mathbb H by a subgroup \Gamma acting [properly discontinuously](/source/Properly_discontinuous_action) and [freely](/source/Free_action).

In the [Poincaré half-plane model](/source/Poincar%C3%A9_half-plane_model) for the hyperbolic plane the group of [biholomorphic transformations](/source/Biholomorphic_transformation) is the group \mathrm{PSL}_2(\mathbb R) acting by [homographies](/source/Homography), and the uniformization theorem means that there exists a [discrete](/source/Discrete_subset), [torsion-free](/source/Torsion-free_group) 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 R.

## 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 \colon \Gamma \to \mathrm{PSL}_2(\Reals)\colon \rho \text{ 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](/source/Finitely_generated_group) 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(\Gamma) 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](/source/Dehn%E2%80%93Nielsen_theorem)) then has the following statement:

*For any \rho\in A(\Gamma) there exists a self-[homeomorphism](/source/Homeomorphism) (in fact a [quasiconformal map](/source/Quasiconformal_map)) h of the upper half-plane \mathbb H such that h \circ \gamma \circ h^{-1} = \rho(\gamma) for all \gamma \in \Gamma.*

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](/source/Teichm%C3%BCller_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\colon R \to X is a quasiconformal homeomorphism modulo a natural equivalence relation.

## See also

- the [Kleinian model](/source/Kleinian_model), an analogous construction for [3-manifolds](/source/3-manifolds)
- [Fundamental polygon](/source/Fundamental_polygon)

## References

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

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