Fuchsian group

Under Fuchsian group is understood to certain subgroups of. Fuchsian groups play in particular in the theory of modular forms an important role. The term Fuchsian group goes back to the Berlin mathematician Lazarus Immanuel Fuchs and was probably first used by Henri Poincaré.

Definition

A Fuchsian group is a discrete subgroup of, ie in that it consists in other words of orientation maintaining isometric of the upper half complex plane.

Example

Probably the best known example of a Fuchsian group is the modular group. Other well-known examples are Kongruenzuntergruppen. Note that for any number field with ring the whole group is never Fuchssch because is dense in.

Classification

A distinction is Fuchsian groups of first and second order. A crucial difference between these two types of Fuchsian groups is the geometric structure of their fundamental domains. A finitely generated Fuchsian group is a Fuchsian group if and only the first kind when the hyperbolic volume of its fundamental region is finite.