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Fuchsian model

Group representation of a Riemann surface

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated March 28, 2022

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.

01A 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 transformations 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 R.

02Fuchsian 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}(\mathbb {R} )\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 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) then has the following statement:

For any \rho \in A(\Gamma ) 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 \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 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.

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Sources and credits

This article is adapted from the Wikipedia article “Fuchsian model”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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