Examples
An example of a Fuchsian group is the modular group, PSL(2,Z). This is the subgroup of PSL(2,R) consisting of linear fractional transformations
where a, b, c, d are integers. The quotient space H/PSL(2,Z) is the moduli space of elliptic curves.
Other Fuchsian groups include the groups Γ(n) for each integer n > 0. Here Γ(n) consists of linear fractional transformations of the above form where the entries of the matrix
are congruent to those of the identity matrix modulo n.
A co-compact example is the (ordinary, rotational) (2,3,7) triangle group, containing the Fuchsian groups of the Klein quartic and of the Macbeath surface, as well as other Hurwitz groups. More generally, any hyperbolic von Dyck group (the index 2 subgroup of a triangle group, corresponding to orientation-preserving isometries) is a Fuchsian group.
All these are Fuchsian groups of the first kind.
- All hyperbolic and parabolic cyclic subgroups of PSL(2,R) are Fuchsian.
- Any elliptic cyclic subgroup is Fuchsian if and only if it is finite.
- Every abelian Fuchsian group is cyclic.
- No Fuchsian group is isomorphic to Z × Z.
- Let Γ be a non-abelian Fuchsian group. Then the normalizer of Γ in PSL(2,R) is Fuchsian.
Read more about this topic: Fuchsian Group
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