Inmathematics, aFuchsian group is adiscrete subgroup ofPSL(2,R). The group PSL(2,R) can be regarded equivalently as agroup of orientation-preservingisometries of thehyperbolic plane, orconformal transformations of theunit disc, or conformal transformations of theupper 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 befinitely 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 aKleinian group (a discrete subgroup ofPSL(2,C)) which is conjugate to a subgroup of PSL(2,R).

Fuchsian groups are used to createFuchsian models ofRiemann surfaces. In this case, the group may be called theFuchsian group of the surface. In some sense, Fuchsian groups do fornon-Euclidean geometry whatcrystallographic groups do forEuclidean geometry. SomeEscher graphics are based on them (for thedisc model of hyperbolic geometry).

General Fuchsian groups were first studied byHenri Poincaré (1882), who was motivated by the paper (Fuchs 1880), and therefore named them afterLazarus Fuchs.

Let${\displaystyle H=\{z\in \mathbb{C}|\mathrm{Im}z>0\}}$ be theupper half-plane. Then${\displaystyle H}$ is a model of thehyperbolic plane when endowed with the metric

${\displaystyle ds=\frac{1}{y}\sqrt{d{x}^2+d{y}^2}.}$

The groupPSL(2,R)acts on${\displaystyle H}$ bylinear fractional transformations (also known asMöbius transformations):

This action is faithful, and in fact PSL(2,R) is isomorphic to the group of allorientation-preservingisometries of${\displaystyle H}$.

A Fuchsian group${\displaystyle \mathrm{\Gamma }}$ may be defined to be a subgroup of PSL(2,R), which actsdiscontinuously on${\displaystyle H}$. That is,

• For every${\displaystyle z}$ in${\displaystyle H}$, theorbit${\displaystyle \mathrm{\Gamma }z=\{\gamma z:\gamma \in \mathrm{\Gamma }\}}$ has noaccumulation point in${\displaystyle H}$.

An equivalent definition for${\displaystyle \mathrm{\Gamma }}$ to be Fuchsian is that${\displaystyle \mathrm{\Gamma }}$ be adiscrete group, which means that:

• Every sequence${\displaystyle {\gamma }_n}$ of elements of${\displaystyle \mathrm{\Gamma }}$ converging to the identity in the usual topology of point-wise convergence is eventually constant, i.e. there exists an integer${\displaystyle \mathbb{N}}$ such that for all${\displaystyle n>\mathbb{N}}$,${\displaystyle {\gamma }_n=I}$, where${\displaystyle I}$ 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${\displaystyle H}$). Indeed, the Fuchsian group PSL(2,Z) is discrete but has accumulation points on the real number line${\displaystyle \mathrm{Im}z=0}$: elements of PSL(2,Z) will carry${\displaystyle z=0}$ to every rational number, and the rationalsQ aredense inR.

A linear fractional transformation defined by a matrix from PSL(2,C) will preserve theRiemann sphereP¹(C) =C ∪ ∞, but will send the upper-half planeH 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 aFuchsian group. Let Γ ⊂ PSL(2,C) act invariantly on a proper,open disk Δ ⊂C ∪ ∞, that is, Γ(Δ) = Δ. Then Γ isFuchsian if and only if any of the following three equivalent properties hold:

1. Γ is adiscrete group (with respect to the standard topology on PSL(2,C)).
2. Γ actsproperly discontinuously at each pointz ∈ Δ.
3. The set Δ is a subset of theregion 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-calledPicard group PSL(2,Z[i]) is discrete but does not preserve any disk in the Riemann sphere. Indeed, even themodular group PSL(2,Z), whichis a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at therational 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 aKleinian group.

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

Because of the discrete action, the orbit Γz of a pointz in the upper half-plane under the action of Γ has noaccumulation points in the upper half-plane. There may, however, be limit points on the real axis. Let Λ(Γ) be thelimit set of Γ, that is, the set of limit points of Γz forz ∈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:

AFuchsian group of the first type is a group for which the limit set is the closed real lineR ∪ ∞. This happens if the quotient spaceH/Γ has finite volume, but there are Fuchsian groups of the first kind of infinite covolume.

Otherwise, aFuchsian group is said to be of thesecond type. Equivalently, this is a group for which the limit set is aperfect set that isnowhere dense onR ∪ ∞. 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 aCantor 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 themodular group, PSL(2,Z). This is the subgroup of PSL(2,R) consisting of linear fractional transformations

wherea,b,c,d are integers. The quotient spaceH/PSL(2,Z) is themoduli space ofelliptic curves.

Other Fuchsian groups include the groups Γ(n) for each integern > 0. Here Γ(n) consists oflinear fractional transformations of the above form where the entries of the matrix

are congruent to those of the identity matrix modulon.

A co-compact example is the (ordinary, rotational)(2,3,7) triangle group, containing the Fuchsian groups of theKlein quartic and of theMacbeath surface, as well as otherHurwitz groups. More generally, any hyperbolicvon Dyck group (the index 2 subgroup of atriangle group, corresponding to orientation-preserving isometries) is a Fuchsian group.

All these areFuchsian groups of the first kind.

• Allhyperbolic andparabolic cyclic subgroups of PSL(2,R) are Fuchsian.
• Anyelliptic cyclic subgroup is Fuchsian if and only if it is finite.
• Everyabelian Fuchsian group is cyclic.
• No Fuchsian group is isomorphic toZ ×Z.
• Let Γ be a non-abelian Fuchsian group. Then thenormalizer of Γ in PSL(2,R) is Fuchsian.

Ifh is a hyperbolic element, the translation lengthL of its action in the upper half-plane is related to thetrace ofh as a 2×2 matrix by the relation

${\displaystyle |\mathrm{t}\mathrm{r}h|=2\cosh \frac{L}{2}.}$

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

• Quasi-Fuchsian group
• Non-Euclidean crystallographic group
• Schottky group

• 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 inGraduate Studies in Mathematics), America Mathematical Society, Providence, RIISBN 978-0-8218-3160-1(See Chapter 2.)
• Svetlana Katok,Fuchsian Groups (1992), University of Chicago Press, ChicagoISBN 978-0-226-42583-2
• David Mumford,Caroline Series, and David Wright,Indra's Pearls: The Vision of Felix Klein, (2002) Cambridge University PressISBN 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, CambridgeISBN 978-0-521-37674-7
• Poincaré, Henri (1882), "Théorie des groupes fuchsiens",Acta Mathematica,1, Springer Netherlands: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

• Kleinian groups
• Hyperbolic geometry
• Riemann surfaces
• Discrete groups
• Fractals

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