A Fuchsian group is a subgroup of the projective special linear group that is discrete in the inherited topology. Its elements act on the upper half-plane by Möbius transformations

where and . The matrices and define the same transformation. These transformations are isometries of the hyperbolic plane preserving its manifold orientation (Voight 2021, p. 642).

Conjugation by the conformal mapping gives the equivalent action on the unit disk (Voight 2021, p. 688). If is a torsion-free group, its quotient space is a Riemann surface, and the projection is its universal cover, with deck transformation group . Every compact Riemann surface of genus arises this way (Voight 2021, pp. 642-643).

See also

Discrete Group, Fundamental Domain, Hyperbolic Plane, Möbius Transformation, Projective Special Linear Group, Riemann Surface, Universal Cover

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References

Voight, J. Quaternion Algebras. Cham, Switzerland: Springer, 2021. https://doi.org/10.1007/978-3-030-56694-4.

Cite this as:

Weisstein, Eric W. "Fuchsian Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FuchsianGroup.html

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