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 is a special case of a lattice in a semisimple Lie group. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...
A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ...
In mathematics, a discrete group is a group G equipped with the discrete topology. ...
In colloquial usage, a lattice is a structure of crossed laths with open spaces left between them. ...
In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In some sense, Fuchsian groups do for non-Euclidean 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). In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. ...
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...
The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...
A crystallographic group is a topologically discrete subgroup of the group of isometries of some geometric space (typically, not necessarily a Euclidean space) with a compact fundamental domain. ...
In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...
Self portrait, 1943¹ Maurits Cornelis Escher (Leeuwarden, June 17, 1898 - Laren, March 27, 1972) was a Dutch artist most known for his woodcuts, lithographs and mezzotints, which tend to feature impossible constructions, explorations of infinity, and tessellations. ...
## Definition
Let *H* = {*z* in **C** : Im(*z*) > 0} be the upper half-plane. Then *H* is a model for hyperbolic plane geometry, when given the element of arc length In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...
The group *PSL*(2,**R**) acts on *H* by linear fractional transformations: In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
In mathematics, groups are often used to describe symmetries of objects. ...
Geometry In mathematics, a Möbius transformation, also called a homographic function, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞.) It is named in honor of August Ferdinand Möbius. ...
This action is faithful, and in fact *PSL*(2,**R**) is isomorphic to the group of all orientation-preserving isometries of *H*. This article or section should be merged with Orientable manifold. ...
In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...
A Fuchsian group Γ is defined to be a **discontinuous** subgroup of *PSL*(2,**R**), which means the following: In this particular instance, an equivalent condition for Γ to be discontinuous is that Γ be **discrete**, which means the following: In mathematics, groups are often used to describe symmetries of objects. ...
In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...
- Every sequence {γ
_{n}} of elements of Γ converging to the identity in the usual topology of point-wise 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. A strong warning should be issued that the properties of discontinuity and discreteness are *not* equivalent in the general case of an arbitrary group of conformal homeomorphisms of the Riemann sphere. Similar in many ways to a Fuchsian group is a Kleinian group, a discrete subgroup of *PSL*(2,**C**) that can be used to define Kleinian models of three-dimensional hyperbolic manifolds, in direct analogy to Fuchsian models. In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ...
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where Γ is a discrete subgroup of PSL(2,C). ...
A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ...
In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. ...
## 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 In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
where a,b,c,d are integers. The quotient space H/*PSL*(2,**Z**) is the moduli space of elliptic curves. In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ...
In mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a. ...
Other famous 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. 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 0-226-42583-5 |