Fuchs group

The Fuchs group means certain subgroups of the . Fuchsian groups play an important role , especially in the theory of modular forms . The term Fuchs's group goes back to the Berlin mathematician Lazarus Immanuel Fuchs and was probably first used by Henri Poincaré . ${\displaystyle PSL(2;\mathbb {R} )}$

definition

A Fuchs group is a discrete subgroup of , i. H. in other words, that it consists of orientation- preserving isometrics of the upper complex half-plane. ${\displaystyle PSL(2;\mathbb {R} )}$

example

Probably the best-known example of a Fuchs group is the module group . Other well-known examples are congruence subsets . Note that for any number field with a wholeness ring the group is never Fuchs's because it is dense in . ${\displaystyle PSL(2;\mathbb {Z} )}$ ${\displaystyle K\neq \mathbb {Q} }$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle PSL(2;{\mathcal {O}}_{K})}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle \mathbb {R} }$

Types of Fuchs groups

A distinction is made between Fuchsian groups of the first and second kind. A decisive difference between these two types of Fuchsian groups is the geometric structure of their fundamental areas . A finitely generated Fuchs group is a Fuchs group of the first kind if and only if the hyperbolic volume of its fundamental domain is finite.

See also

• Quasifuchs group
• Klein group

literature

• Svetlana Katok : Fuchsian Groups . The University of Chicago Press, Chicago IL et al. 1992, ISBN 0-226-42583-5 .
• Donal O'Shea: Poincaré's conjecture. The story of a math adventure . S. Fischer, Frankfurt am Main 2007, ISBN 978-3-10-054020-1 .
• Toshitsune Miyake: Modular Forms . Springer, Berlin et al. 1989, ISBN 3-540-50268-8 .