In mathematics, a **Fuchsian group** is a discrete subgroup of PSL(2,**R**). The group PSL(2,**R**) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,**R**) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,**C**)) which is conjugate to a subgroup of PSL(2,**R**).

- Fuchsian groups on the upper half-plane
- General definition
- Limit sets
- Examples
- Metric properties
- See also
- References

Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the **Fuchsian group of the surface**. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the *disc model* of hyperbolic geometry).

General Fuchsian groups were first studied by HenriPoincaré ( 1882 ), who was motivated by the paper ( Fuchs 1880 ), and therefore named them after Lazarus Fuchs.

Let **H** = {*z* in **C** : Im(*z*) > 0} be the upper half-plane. Then **H** is a model of the hyperbolic plane when endowed with the metric

The group PSL(2,**R**) acts on **H** by linear fractional transformations (also known as Möbius transformations):

This action is faithful, and in fact PSL(2,**R**) is isomorphic to the group of all orientation-preserving 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 Γ*z*= {γ*z*: γ in Γ} has no accumulation point in**H**.

An equivalent definition for Γ to be Fuchsian is that Γ be a ** discrete group **, which means that:

- 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.

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 full Riemann sphere (as opposed to **H**). Indeed, the Fuchsian group PSL(2,**Z**) is discrete but has accumulation points on the real number line Im *z* = 0: elements of PSL(2,**Z**) will carry *z* = 0 to every rational number, and the rationals **Q** are dense in **R**.

A linear fractional transformation defined by a matrix from PSL(2,**C**) will preserve the Riemann sphere **P**^{1}(**C**) = **C** ∪ ∞, but will send the upper-half plane **H** to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,**R**) to a discrete subgroup of PSL(2,**C**) preserving Δ.

This motivates the following definition of a **Fuchsian group**. Let Γ ⊂ PSL(2,**C**) act invariantly on a proper, open disk Δ ⊂ **C** ∪ ∞, that is, Γ(Δ) = Δ. Then Γ is **Fuchsian** if and only if any of the following three equivalent properties hold:

- Γ is a discrete group (with respect to the standard topology on PSL(2,
**C**)). - Γ acts properly discontinuously at each point
*z*∈ Δ. - 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 so-called **Picard group** PSL(2,**Z**[*i*]) is discrete but does not preserve any disk in 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.

It is most usual to take the invariant domain Δ to be either the open unit disk or the upper half-plane.

Because of the discrete action, the orbit Γ*z* of a point *z* in the upper half-plane under the action of Γ has no accumulation points in the upper half-plane. There may, however, be limit points on the real axis. Let Λ(Γ) be the limit set of Γ, that is, the set of limit points of Γ*z* for *z* ∈ **H**. Then Λ(Γ) ⊆ **R** ∪ ∞. The limit set may be empty, or may contain one or two points, or may contain an infinite number. In the latter case, there are two types:

A **Fuchsian group of the first type** is a group for which the limit set is the closed real line **R** ∪ ∞. This happens if the quotient space **H**/Γ has finite volume, but there are Fuchsian groups of the first kind of infinite covolume.

Otherwise, a **Fuchsian group** is said to be of the **second type**. Equivalently, this is a group for which the limit set is a perfect set that is nowhere dense on **R** ∪ ∞. Since it is nowhere dense, this implies that any limit point is arbitrarily close to an open set that is not in the limit set. In other words, the limit set is a Cantor set.

The type of a Fuchsian group need not be the same as its type when considered as a Kleinian group: in fact, all Fuchsian groups are Kleinian groups of type 2, as their limit sets (as Kleinian groups) are proper subsets of the Riemann sphere, contained in some circle.

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.

If *h* is a hyperbolic element, the translation length *L* of its action in the upper half-plane is related to the trace of *h* as a 2×2 matrix by the relation

A similar relation holds for the systole of the corresponding Riemann surface, if the Fuchsian group is torsion-free and co-compact.

In geometry and complex analysis, a **Möbius transformation** of the complex plane is a rational function of the form

In mathematics, the **modular group** is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices *A* and −*A* are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.

In mathematics, a **congruence subgroup** of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are *even*. More generally, the notion of **congruence subgroup** can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

In mathematics, the projective special linear group **PSL(2, 7)**, isomorphic to **GL(3, 2)**, is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A_{5} with 60 elements, isomorphic to PSL(2, 5).

In mathematics, a **linear fractional transformation** is, roughly speaking, an invertible transformation of the form

In non-Euclidean geometry, the **Poincaré half-plane model** is the upper half-plane, denoted below as **H**, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

In mathematics, a topological group *G* is called a **discrete group** if there is no limit point in it. Equivalently, the group *G* is discrete if and only if its identity is isolated.

In mathematics, a **prime geodesic** on a hyperbolic surface is a **primitive** closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem.

In number theory and algebraic geometry, a **modular curve***Y*(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane **H** by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, **Z**). The term modular curve can also be used to refer to the **compactified modular curves***X*(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field of rational numbers **Q** or a cyclotomic field **Q**(ζ_{n}). The latter fact and its generalizations are of fundamental importance in number theory.

In mathematics, the **Selberg trace formula**, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group G on the space *L*^{2}(Γ\*G*) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G.

In mathematics, a **Kleinian group** is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space **H**^{3}. The latter, identifiable with PSL(2, **C**), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, **C**) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball *B*^{3} in **R**^{3}. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of **H**^{3}, PGL(2, **C**). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

In mathematics, a **Fuchsian model** is a representation of a hyperbolic Riemann surface *R* as a quotient of the upper half-plane **H** by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.

In mathematics, a **Kleinian model** is a model of a three-dimensional hyperbolic manifold *N* by the quotient space where is a discrete subgroup of PSL(2,**C**). Here, the subgroup , a Kleinian group, is defined so that it is isomorphic to the fundamental group of the surface *N*. Many authors use the terms *Kleinian group* and *Kleinian model* interchangeably, letting one stand for the other. The concept is named after Felix Klein.

In mathematics, a **fundamental polygon** can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:

In geometry, a **Schwarz triangle**, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in Schwarz (1873).

In mathematics, more particularly in the fields of dynamical systems and geometric topology, an **Anosov map** on a manifold *M* is a certain type of mapping, from *M* to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems.

In mathematics, the special linear group **SL(2, R)** or **SL _{2}(R)** is the group of 2 × 2 real matrices with determinant one:

In mathematics, the **ping-pong lemma**, or **table-tennis lemma**, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group.

In mathematics, the **trace field** of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name *field of definition*.

**Arithmetic Fuchsian groups** are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group . They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.

- Fuchs, Lazarus (1880), "Ueber eine Klasse von Funktionen mehrerer Variablen, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen mit rationalen Coeffizienten entstehen",
*J. Reine Angew. Math.*,**89**: 151–169 - Hershel M. Farkas, Irwin Kra,
*Theta Constants, Riemann Surfaces and the Modular Group*, American Mathematical Society, Providence RI, ISBN 978-0-8218-1392-8*(See section 1.6)* - Henryk Iwaniec,
*Spectral Methods of Automorphic Forms, Second Edition*, (2002) (Volume 53 in*Graduate Studies in Mathematics*), America Mathematical Society, Providence, RI ISBN 978-0-8218-3160-1*(See Chapter 2.)* - Svetlana Katok,
*Fuchsian Groups*(1992), University of Chicago Press, Chicago ISBN 978-0-226-42583-2 - David Mumford, Caroline Series, and David Wright,
*Indra's Pearls: The Vision of Felix Klein*, (2002) Cambridge University Press ISBN 978-0-521-35253-6.*(Provides an excellent exposition of theory and results, richly illustrated with diagrams.)* - Peter J. Nicholls,
*The Ergodic Theory of Discrete Groups*, (1989) London Mathematical Society Lecture Note Series 143, Cambridge University Press, Cambridge ISBN 978-0-521-37674-7 - Poincaré, Henri (1882), "Théorie des groupes fuchsiens",
*Acta Mathematica*, Springer Netherlands,**1**: 1–62, doi: 10.1007/BF02592124 , ISSN 0001-5962, JFM 14.0338.01 - Vinberg, Ernest B. (2001) [1994], "Fuchsian group",
*Encyclopedia of Mathematics*, EMS Press

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