Fuchsian model

Summary

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.

A more precise definition

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By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface   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   by a subgroup   acting properly discontinuously and freely.

In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformations is the group   acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup   such that the Riemann surface   is isomorphic to  . Such a group is called a Fuchsian group, and the isomorphism   is called a Fuchsian model for  .

Fuchsian models and Teichmüller space

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Let   be a closed hyperbolic surface and let   be a Fuchsian group so that   is a Fuchsian model for  . Let   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   is finitely generated since it is isomorphic to the fundamental group of  . Let   be a generating set: then any   is determined by the elements   and so we can identify   with a subset of   by the map  . 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   there exists a self-homeomorphism (in fact a quasiconformal map)   of the upper half-plane   such that   for all  .

The proof is very simple: choose an homeomorphism   and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since   is compact.

This result can be seen as the equivalence between two models for Teichmüller space of  : the set of discrete faithful representations of the fundamental group   into   modulo conjugacy and the set of marked Riemann surfaces   where   is a quasiconformal homeomorphism modulo a natural equivalence relation.

See also

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References

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