Fuchsian group
Encyclopedia
In mathematics
, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded as a group
of 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) = PSL(2,R).2 (so that it contains orientation-reversing elements) and sometimes it is allowed to be a Kleinian group
(a discrete group
of PSL(2,C)) that is conjugate to a subgroup of PSL(2,R).
Fuchsian groups are used to create Fuchsian model
s of Riemann surface
s. 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 , who was motivated by the paper and therefore named them after Lazarus Fuchs
.
when given the element of arc length
The group PSL(2,R) acts
on H by linear fractional 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,
An equivalent definition for Γ to be Fuchsian is that Γ be discrete group
, in the following sense:
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) 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.
P1(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 properties hold:
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 number
s. 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.
of Γ, that is, the set of limit points of Γz 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
. 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.
, 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 curve
s.
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.
A similar relation holds for the systole of the corresponding Riemann surface, if the Fuchsian group is torsion-free and co-compact.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded as a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
of the hyperbolic plane
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, 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) = PSL(2,R).2 (so that it contains orientation-reversing elements) and sometimes it is allowed to be a Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
(a discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
of PSL(2,C)) that is conjugate to a subgroup of PSL(2,R).
Fuchsian groups are used to create Fuchsian model
Fuchsian model
In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group \pi_1...
s of Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s. In some sense, Fuchsian groups do for non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
what crystallographic groups do for Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. Some Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
graphics are based on them (for the disc model of hyperbolic geometry).
General Fuchsian groups were first studied by , who was motivated by the paper and therefore named them after Lazarus Fuchs
Lazarus Fuchs
Lazarus Immanuel Fuchs was a German mathematician who contributed important research in the field of linear differential equations...
.
Fuchsian groups on the upper half-plane
Let H = {z in C : Im(z) > 0} be the upper half-plane. Then H is a model of the hyperbolic planeHyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
when given the element of arc length
The group PSL(2,R) acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on H by linear fractional transformations:
This action is faithful, and in fact PSL(2,R) is isomorphic to the group of all orientation-preserving isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
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 orbitGroup actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
Γz = {γz : γ in Γ} has no accumulation point in H.
An equivalent definition for Γ to be Fuchsian is that Γ be discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
, in the following sense:
- Every sequence {γn} 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, γn = 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) 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
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in R.
General definition
A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphereRiemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
P1(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
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
disk Δ ⊂ C ∪ ∞, that is, Γ(Δ) = Δ. Then Γ is Fuchsian if and only if any of the following three properties hold:
- Γ is a discrete groupDiscrete groupIn mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
(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
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
PSL(2,Z), which is a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. 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
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
.
It is most usual to take the invariant domain Δ to be either the open unit disk or the upper half-plane.
Limit sets
Because of the discrete action, the orbit Γz 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 setLimit set
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time...
of Γ, that is, the set of limit points of Γz 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
. 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 setCantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
.
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.
Examples
An example of a Fuchsian group is the modular groupModular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
, 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
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s.
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
(2,3,7) triangle group
In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84, of its automorphism group.A note on terminology – the "...
, containing the Fuchsian groups of the Klein quartic
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...
and of the Macbeath surface
Macbeath surface
In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface....
, as well as other Hurwitz groups. More generally, any hyperbolic von Dyck group (the index 2 subgroup of a triangle group
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
, 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 abelianAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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.
Metric properties
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 relationA similar relation holds for the systole of the corresponding Riemann surface, if the Fuchsian group is torsion-free and co-compact.
See also
- Quasi-Fuchsian groupQuasi-Fuchsian groupIn the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two...
- Non-Euclidean crystallographic groupNon-Euclidean crystallographic groupIn mathematics, a non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry groups correspond to the wallpaper groups in euclidean geometry...
- Schottky groupSchottky groupIn mathematics, a Schottky group is a special sort of Kleinian group, first studied by .-Definition:Fix some point p on the Riemann sphere...