FuchsianGroup - crowlogic/arb4j GitHub Wiki

A Fuchsian group is a discrete subgroup of the group PSL(2,R), the group of orientation-preserving isometries of the hyperbolic plane. The term "Fuchsian group" is derived from the name of the German mathematician Lazarus Fuchs, who made significant contributions to the theory of linear differential equations and complex analysis.

Fuchsian groups are fundamental objects in the study of hyperbolic geometry and have deep connections to various fields of mathematics, including number theory, topology, and the theory of RiemannSurfaces. A prototypical example of a Fuchsian group is the modular group, which is closely related to the theory of modular forms and elliptic curves.

Fuchsian groups can be thought of as the hyperbolic analogs of crystallographic groups in Euclidean geometry. While crystallographic groups describe the symmetries of periodic tilings of the Euclidean plane, Fuchsian groups describe the symmetries of discrete objects in the hyperbolic plane, such as the regular tessellations of the Poincaré disk model.

The study of Fuchsian groups often involves analyzing their action on the hyperbolic plane, as well as understanding their various subgroups and quotient spaces. These investigations have led to many important results and concepts in the theory of discrete groups and geometric group theory.