Fuchsian group
From Academic Kids

In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. A Fuchsian group is always a discrete group, and thus is a lattice in one of the two semisimple Lie groups PSL(2,R) or PSL(2,C).
Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In some sense, Fuchsian groups do for nonEuclidean geometry what crystallographic groups do for Euclidean geometry, but the theory is much richer. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).
Contents 
Overview
There are several equivalent definitions of a Fuchsian group that can be made, defining it to be a discrete group or as group that acts properly discontinuously; several different notions of discreteness may also apply. Furthermore, a Fuchsian group can be defined to be a subgroup of either PSL(2,R) or PSL(2,C), depending on whether one chooses to work with hyperbolic geometry on the upper halfplane or the open unit disk. In either case, a subgroup that is conjugate (in PSL(2,C)) to a Fuchsian group is also Fuchsian; although in this case the invariant domain need not be the upper halfplane or the disk, but rather a subset of the Riemann sphere <math>\hat{\mathbb {C}}=\mathbb{C}\cup\infty<math> that is isometric to the halfplane or disk.
The various different definitions can be seen to stem from a single, formal, abstract definition, which is thus stated first. This formal definition is then elucidated with simpler definitions below.
Let <math>\Gamma \in PSL(2,\mathbb{C})<math> act invariantly on a proper, open disk <math>\Delta\in \hat{\mathbb {C}}<math>, that is, <math>\Gamma(\Delta)=\Delta<math>. Then Γ is Fuchsian if and only if any of the following three properties hold:
 Γ is a discrete group (with respect to the standard topology on PSL(2,C)).
 Γ acts properly discontinuously on some point <math>z\in\Delta<math>.
 The set Δ is a subset of the region of discontinuity Ω(Γ) of Γ.
That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the Picard group PSL(2,Z[i]) on the Gaussian integers Z[i] is discrete but does not act discontinuously anywhere on the Riemann sphere. Indeed, even the modular group PSL(2,Z), which is a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the rational numbers. Similarly, the idea that Δ is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a Kleinian group.
When the invariant domain Δ is the upper half plane, then every Fuchsian group leaving the upper halfplane invariant is a subgroup of PSL(2,R). Every Fuchsian group is conjugate (in PSL(2,C)) to a discrete subgroup of PSL(2,R), which leads to the most common and simplest definition, given below. Alternately, any Fuchsian group can be made conjugate to one which leaves the open unit disk invariant.
Fuchsian groups on the upper halfplane
Let H = {z in C : Im(z) > 0} be the upper halfplane. Then H is a model for hyperbolic plane geometry, when given the element of arc length
 <math>ds=\frac{\sqrt{dx^2+dy^2}}{y}<math>
The group PSL(2,R) acts on H by linear fractional transformations:
 <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}<math>
This action is faithful, and in fact PSL(2,R) is isomorphic to the group of all orientationpreserving isometries of H.
A Fuchsian group Γ may be defined to be a subgroup of PSL(2,R), which acts discontinuously on H. That is,
 For every z in H, the orbit Γz = {γz : γ in Γ} has no accumulation point in H.
An equivalent definition for Γ to be Fuchsian is that Γ be discrete, in the following sense:
 Every sequence {γ_{n}} of elements of Γ converging to the identity in the usual topology of pointwise convergence is eventually constant, i.e. there exists an integer N such that for all n > N, γ_{n} = I, where I is the identity matrix.
Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the Riemann sphere. Indeed, the Fuchsian group PSL(2,Z) has accumulation points on the real number line Im z=0; elements of PSL(2,Z) will carry z=0 to every rational number; the rationals Q are dense in R.
Examples
By far the most prominent 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
 <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}<math>
where a,b,c,d are integers. The quotient space H/PSL(2,Z) is the moduli space of elliptic curves.
Other famous Fuchsian groups include the groups <math>\Gamma<math>(n) for each integer n > 0. Here <math>\Gamma<math>(n) consists of linear fractional transformations of the above form where the entries of the matrix
 <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}<math>
are congruent to those of the identity matrix modulo n.
All these are Fuchsian groups of the first kind, meaning that the quotient of H by these groups has finite volume.
References
 Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0226425835
 Hershel M. Farkas, Irwin Kra, Theta Constants, Riemann Surfaces and the Modular Group, American Mathematical Society, Providence RI, ISBN 0821813927 (See section 1.6)