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| In [[mathematics]], a '''Fuchsian group''' is a discrete subgroup of [[PSL(2,R)|PSL(2,'''R''')]]. The group PSL(2,'''R''') can be regarded as a [[group (mathematics)|group]] of [[isometry|isometries]] of the [[Hyperbolic geometry|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 [[PSL2(C)|PSL(2,'''C''')]]) that is conjugate to a subgroup of PSL(2,'''R''').
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| Fuchsian groups are used to create [[Fuchsian model]]s of [[Riemann surface]]s. 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 group]]s do for [[Euclidean geometry]]. Some [[M. C. Escher|Escher]] graphics are based on them (for the ''disc model'' of hyperbolic geometry).
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| General Fuchsian groups were first studied by {{Harvard citation text|Poincaré|1882}}, who was motivated by the paper {{Harvard citation|Fuchs|1880}} and therefore named them after [[Lazarus Fuchs]].
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| == Fuchsian groups on the upper half-plane ==
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| Let '''H''' = {''z'' in '''C''' : Im(''z'') > 0} be the [[upper half-plane]]. Then '''H''' is a model of the [[Hyperbolic space|hyperbolic plane]] when given the element of arc length
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| :<math>ds=\frac{1}{y}\sqrt{dx^2+dy^2}.</math>
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| The group [[PSL2(R)|PSL(2,'''R''')]] [[group action|acts]] on '''H''' by [[Möbius transformation|linear fractional transformations]]:
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| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}.</math>
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| This action is faithful, and in fact PSL(2,'''R''') is isomorphic to the group of all [[orientable|orientation-preserving]] [[isometry|isometries]] of '''H'''.
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| A Fuchsian group Γ may be defined to be a subgroup of PSL(2,'''R'''), which acts '''discontinuously''' on '''H'''. That is,
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| * For every ''z'' in '''H''', the [[group action|orbit]] Γ''z'' = {γ''z'' : γ in Γ} has no [[accumulation point]] in '''H'''.
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| An equivalent definition for Γ to be Fuchsian is that Γ be '''[[discrete group]]''', in the following sense:
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| * Every sequence {γ<sub>''n''</sub>} 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<sub>''n''</sub> = I, where I is the identity matrix.
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| 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 set|dense]] in '''R'''.
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| ==General definition==
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| A linear fractional transformation defined by a matrix from PSL(2,'''C''') will preserve the [[Riemann sphere]] '''P'''<sup>1</sup>('''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.
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| This motivates the following definition of a '''Fuchsian group'''. Let Γ ⊂ PSL(2,'''C''') act invariantly on a proper, [[open set|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:
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| # Γ is a [[discrete group]] (with respect to the standard topology on PSL(2,'''C''')).
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| # Γ acts [[properly discontinuously]] at each point ''z'' ∈ Δ.
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| # The set Δ is a subset of the [[region of discontinuity]] Ω(Γ) of Γ.
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| 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]].
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| It is most usual to take the invariant domain Δ to be either the [[open unit disk]] or the [[upper half-plane]].
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| ==Limit sets==
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| 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 point]]s 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:
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| 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.
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| 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]].
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| 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.
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| == Examples ==
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| 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
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| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}</math>
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| where ''a'', ''b'', ''c'', ''d'' are integers. The quotient space '''H'''/PSL(2,'''Z''') is the [[moduli space]] of [[elliptic curve]]s.
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| 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
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| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>
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| are congruent to those of the identity matrix modulo ''n''.
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| 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 group]]s. More generally, any hyperbolic [[von Dyck group]] (the index 2 subgroup of a [[triangle group]], corresponding to orientation-preserving isometries) is a Fuchsian group.
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| All these are '''Fuchsian groups of the first kind'''.
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| * All [[fractional linear transform|hyperbolic]] and [[fractional linear transform|parabolic]] cyclic subgroups of PSL(2,'''R''') are Fuchsian.
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| * Any [[fractional linear transform|elliptic]] cyclic subgroup is Fuchsian if and only if it is finite.
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| * Every [[Abelian Group|abelian]] Fuchsian group is cyclic.
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| * No Fuchsian group is isomorphic to '''Z''' × '''Z'''.
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| * Let Γ be a non-abelian Fuchsian group. Then the [[normalizer]] of Γ in PSL(2,'''R''') is Fuchsian.
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| ==Metric properties==
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| 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
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| :<math> |\mathrm{tr}\; h| = 2\cosh \frac{L}{2}.</math> | |
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| A similar relation holds for the [[systolic geometry|systole]] of the corresponding Riemann surface, if the Fuchsian group is torsion-free and co-compact.
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| == See also ==
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| * [[Quasi-Fuchsian group]]
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| * [[Non-Euclidean crystallographic group]]
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| * [[Schottky group]]
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| == References ==
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| * {{Citation | last1=Fuchs | first1=László | title=Ueber eine Klasse von Funktionen mehrerer Variablen, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen mit rationalen Coeffizienten entstehen | url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0089 | year=1880 | journal= J. Reine Angew. Math. | volume=89 | pages=151–169}}
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| * 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)''
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| * [[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).''
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| * [[Svetlana Katok]], ''Fuchsian Groups'' (1992), University of Chicago Press, Chicago ISBN 978-0-226-42583-2
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| * David Mumford, Caroline Series and David Wright, ''[[Indra's Pearls (book)|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.)''
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| * 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
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| * {{Citation | last1=Poincaré | first1=Henri | author1-link=Henri Poincaré | title=Théorie des groupes fuchsiens | publisher=Springer Netherlands | doi=10.1007/BF02592124 | jfm=14.0338.01 | year=1882 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=1 | pages=1–62}}
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| * {{Springer|id=F/f041890|first=E. B. |last=Vinberg}}
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| [[Category:Kleinian groups]]
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| [[Category:Hyperbolic geometry]]
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| [[Category:Riemann surfaces]]
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| [[Category:Discrete groups]]
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| [[Category:Fractals]]
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