Fuchsian group

A discrete group of holomorphic transformations (cf. Discrete group of transformations) of an (open) disc $ K $ onto the Riemann sphere, that is, of a disc or a half-plane onto the complex plane. Most often one takes for $ K $ the upper half-plane

$$ U = \{ {z \in \mathbf C } : { \mathop{\rm Im} z > 0 } \} $$

or the unit disc

$$ D = \{ {z \in \mathbf C } : {| z | < 1 } \} . $$

In the first case the elements of a Fuchsian group are Möbius transformations (cf. Fractional-linear mapping)

$$ z \mapsto \ \frac{az + b }{cz + d } $$

with real coefficients, and a Fuchsian group is nothing other than a discrete subgroup of $ \mathop{\rm PSL} _ {2} $. In the second case the elements of a Fuchsian group are Möbius transformations with pseudo-unitary matrices.

If one regards the disc $ K $ as a conformal model of the Lobachevskii plane, then a Fuchsian group can be defined as a discrete group of motions of it preserving orientation. Fuchsian groups are a particular case of Kleinian groups (cf. Kleinian group).

Arbitrary Fuchsian groups were first studied by H. Poincaré (see [2]) in 1882 in connection with the uniformization problem. He called the groups Fuchsian in honour of L. Fuchs, whose paper [1] inspired him to introduce this concept. To describe Fuchsian groups, Poincaré applied a combinatoric-geometric method, which subsequently became one of the main methods in the theory of discrete groups of transformations. The concept of a Fuchsian group provided a basis for the theory of automorphic functions created by Poincaré and F. Klein.

A Fuchsian group that preserves some point in the closure $ \overline{K}\; $ of $ K $, or a straight line in the sense of Lobachevskii geometry, is called elementary. If $ \Gamma $ is a non-elementary Fuchsian group, then the set $ L ( \Gamma ) $ of limit points of the orbit of a point $ x \in \overline{K}\; $ lying on the circle $ \partial K $ is independent of $ x $ and is called the limit set of the group $ \Gamma $. The group $ \Gamma $ is called a Fuchsian group of the first kind if $ L ( \Gamma ) = \partial K $, and of the second kind otherwise (then $ L ( \Gamma ) $ is a nowhere-dense perfect subset of $ \partial K $).

A finitely-generated Fuchsian group is of the first kind if and only if the area (in the sense of Lobachevskii geometry) of its fundamental domain is finite. One can choose as a fundamental domain of such a group $ \Gamma $ a polygon $ P $ of the Lobachevskii plane with sides

$$ a _ {1} , b _ {1} ^ \prime , a _ {1} ^ \prime , b _ {1} \dots a _ {g} ,\ b _ {g} ^ \prime , a _ {g} ^ \prime , b _ {g} , c _ {1} ,\ c _ {1} ^ \prime \dots c _ {n} , c _ {n} ^ \prime $$

in such a way that

$$ a _ {i} = \ \alpha _ {i} ( a _ {i} ^ \prime ),\ \ b _ {i} = \ \beta _ {i} ( b _ {i} ^ \prime ),\ \ c _ {i} = \ \gamma _ {i} ( c _ {i} ^ \prime ) $$

for some elements

$$ \alpha _ {1} \dots \alpha _ {g} ,\ \ \beta _ {1} \dots \beta _ {g} ,\ \ \gamma _ {1} \dots \gamma _ {n} $$

which generate $ \Gamma $, with defining relations

$$ \prod _ {i = 1 } ^ { g } ( \alpha _ {i} \beta _ {i} \alpha _ {i} ^ {-} 1 \beta _ {i} ^ {-} 1 ) \prod _ {i = 1 } ^ { n } \gamma _ {i} = 1, $$

$$ \gamma _ {i} ^ {k _ {i} } = 1,\ i = 1 \dots n, $$

where $ k _ {i} $ is an integer $ \geq 2 $ or $ \infty $. The element $ \gamma _ {i} $ leaves the vertex $ C _ {i} $ of $ P $ common to $ c _ {i} $ and $ c _ {i} ^ \prime $ fixed. It is elliptic if $ k _ {i} < \infty $, and parabolic if $ k _ {i} = \infty $; in the latter case $ C _ {i} $ lies on $ \partial K $, that is, it is an improper point of the Lobachevskii plane. Every elliptic or parabolic element of $ \Gamma $ is conjugate to a power of some unique generator $ \gamma _ {i} $. The angles of $ P $ at the vertices $ C _ {i} $, $ i = 1 \dots n $, are equal to $ 2 \pi /k $; the sum of all remaining angles is $ 2 \pi $. The sides $ a _ {i} $ and $ a _ {i} ^ \prime $, and also $ b _ {i} $ and $ b _ {i} ^ \prime $, and $ c _ {i} $ and $ c _ {i} ^ \prime $, have the same length. Conversely, every convex polygon on the Lobachevskii plane that satisfies these conditions is the fundamental polygon of the type described above of some finitely-generated Fuchsian group of the first kind.

Any system of generators of $ \Gamma $ obtained by the method described is called standard. Under an abstract isomorphism of finitely-generated Fuchsian groups that maps the set of parabolic elements of one group onto the set of parabolic elements of the other, every standard system of generators is mapped to a standard system of generators.

The area of a fundamental domain of $ \Gamma $ is equal to $ - 2 \pi \chi ( \Gamma ) $, where

$$ \chi ( \Gamma ) = \ \chi ( g; k _ {1} \dots k _ {n} ) = \ 2 - 2g - \sum _ {i = 1 } ^ { n } \left ( 1 - { \frac{1}{k _ {i} } } \right ) . $$

The collection of numbers $ ( g; k _ {1} \dots k _ {n} ) $, where $ k _ {1} \dots k _ {n} $ are taken to be unordered, is a topological invariant of $ \Gamma $, regarded as a group of homeomorphisms of the disc, and is called its signature. The only restriction on the signature is the condition

$$ \tag{* } \chi ( g; k _ {1} \dots k _ {n} ) < 0. $$

For a subgroup $ \Delta $ of finite index in a Fuchsian group $ \Gamma $ the Riemann–Hurwitz formula

$$ \chi ( \Delta ) = \ \chi ( \Gamma ) [ \Gamma : \Delta ] $$

holds. In every Fuchsian group there is a subgroup of finite index that has no elements of finite order.

The quotient space $ K/ \Gamma $ is compactified by adding a finite number of points corresponding to the improper vertices of a fundamental polygon. There is a unique complex structure on the compactified space $ S $ for which the quotient mapping $ p: K \rightarrow S $ is holomorphic. Here $ S $ is a Riemann surface of genus $ g $, and $ p $ is a regular branched covering with branching indices $ k _ {1} \dots k _ {n} $. Conversely, the uniformization theorem asserts that for any compact Riemann surface $ S $ with given points $ x _ {1} \dots x _ {n} $ and for any $ k _ {1} \dots k _ {n} $( $ k _ {i} $ is an integer $ \geq 2 $ or $ \infty $) satisfying the condition (*), there is a regular holomorphic branched covering $ p: K \rightarrow S $ that is branched over precisely the points $ x _ {1} \dots x _ {n} $ with branching indices $ k _ {1} \dots k _ {n} $, respectively. The covering $ p $ is uniquely determined up to an automorphism of $ K $. Its group of covering transformations is a Fuchsian group of signature $ ( g; k _ {1} \dots k _ {n} ) $.

The finitely-generated Fuchsian groups of the first kind of fixed signature $ ( g; k _ {1} \dots k _ {n} ) $ can be parametrized by the points of some $ ( 3g - 3 + n) $- dimensional complex manifold homeomorphic to a cell, that is, a Teichmüller space $ T ( g; k _ {1} \dots k _ {n} ) $( see ). Here two points of the Teichmüller space correspond to the same Fuchsian group (up to conjugacy in the automorphism group of the disc) if and only if these points are equivalent relative to a certain discrete group of holomorphic transformations of $ T ( g; k _ {1} \dots k _ {n} ) $— the so-called modular group $ \mathop{\rm Mod} ( g; k _ {1} \dots k _ {n} ) $. There is an isomorphism

$$ T ( g; k _ {1} \dots k _ {n} ) \rightarrow \ T ( g; \infty \dots \infty ) $$

( $ n $ times $ \infty $), under which $ \mathop{\rm Mod} ( g; k _ {1} \dots k _ {n} ) $ is mapped to a subgroup of finite index in $ \mathop{\rm Mod} ( g; \infty \dots \infty ) $.

If a Fuchsian group of signature $ ( g; k _ {1} \dots k _ {n} ) $ contains a subgroup of finite index of signature $ ( h; l _ {1} \dots l _ {m} ) $, then the space $ T ( g; k _ {1} \dots k _ {n} ) $ can be imbedded in a unique way as a closed subset of $ T ( h; l _ {1} \dots l _ {m} ) $. In certain exceptional cases these spaces coincide [10]. For example, $ T ( 2) = T ( 0; 2, 2, 2, 2, 2, 2) $; this means that every compact Riemann surface of genus 2 admits a hyper-elliptic involution and so is a hyper-elliptic curve.

For Fuchsian groups of signature $ ( 0; k _ {1} , k _ {2} , k _ {3} ) $, called triangular groups, and only for these, the Teichmüller space consists of a single point. Every triangular group is a subgroup of index 2 in the group generated by reflections relative to the sides of a triangle with angles $ \pi /k _ {1} $, $ \pi /k _ {2} $, $ \pi /k _ {3} $( see Reflection group). An example of a triangular group is the modular Kleinian group; its signature is equal to $ ( 0; 2, 3, \infty ) $.

Every finitely-generated Fuchsian group of the second kind is topologically isomorphic (as a group of the disc) to a finitely-generated Fuchsian group of the first kind and admits a similar geometric description, with the difference that some pairs of sides $ c _ {i} $, $ c _ {i} ^ \prime $ of the fundamental polygon have no common points, even improper ones, and that the corresponding generators $ \gamma _ {i} $ are hyperbolic transformations. The compactified quotient space is a Riemann surface with boundary.

Every infinitely-generated Fuchsian group is a free product of cyclic subgroups. Its fundamental domain can be constructed as the limit of the fundamental domains of finitely-generated groups (see [5]).

References

[1]	L. Fuchs, "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 (1880) pp. 151–169
[2]	H. Poincaré, "Théorie des groupes Fuchsiennes" Acta. Math. , 1 (1882) pp. 1–62
[3]	R. Fricke, F. Klein, "Vorlesungen über die Theorie der automorphen Funktionen" , 1–2 , Teubner (1897–1912)
[4a]	L.V. Ahlfors, "The complex analytic structure of the space of closed Riemann surfaces" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 45–66
[4b]	L. Bers, "Quasi-conformal mappings and Teichmüller's theorem" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 89–119
[5]	S.L. Krushkal', B.N. Apanasov, N.A. Gusevskii, "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc. (1986) (Translated from Russian)
[6]	S.M. Natanson, "Invariant lines of Fuchsian groups" Russian Math. Surveys , 27 (1972) pp. 161–177 Uspekhi Mat. Nauk , 27 : 4 (1972) pp. 145–160
[7]	E.B. Vinberg, O.V. Svartsman, "Riemann surfaces" J. Soviet Math. , 14 (1980) pp. 985–1020 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 16 (1978) pp. 191–245
[8]	J. Lehner, "Discontinuous groups and automorphic functions" , Amer. Math. Soc. (1964)
[9]	W. Magnus, "NonEuclidean tesselations and their groups" , Acad. Press (1974)
[10]	D. Singerman, "Finitely maximal Fuchsian groups" J. London Math. Soc. , 6 : 1 (1972) pp. 29–38

Comments

Relations with Riemann surfaces are discussed in [a3], [a2]. [a1] presents the theory of Fuchsian groups and automorphic functions.

Let $ \Gamma $ be a discrete subgroup of $ \mathop{\rm PSL} ( 2 , \mathbf R ) $, the group of fractional transformations $ z \mapsto ( a z + b) / ( c z + d) $; $ a , b , c , d \in \mathbf R $, $ a d - b c = 1 $. For any $ z $ in the extended complex plane and any sequence of distinct elements $ \gamma _ {i} $ of $ \Gamma $ a cluster point of $ \{ \gamma _ {i} z \} $ is called a limit point of $ \Gamma $. If there are 0, 1 or $ 2 $ limit points, $ \Gamma $ is conjugate to a group of motions of the plane. Otherwise there are infinitely many limit points and $ \Gamma $ is called a Fuchsoid group. A Fuchsoid group is a Fuchsian group if it is finitely generated. For a real point $ x \in \mathbf R \cup \{ \infty \} $ let $ \Gamma _ {x} $ be the stabilizer in $ \Gamma $ of $ x $. The point $ x $ is called a cusp or parabolic cusp if $ \Gamma _ {x} $ is a free cyclic group generated by a parabolic transformation $ \neq \{ 1 \} $( cf. Fractional-linear mapping). The cusps of $ \Gamma $ are represented by vertices of a fundamental polygon on the real axis.

References

[a1]	L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951)
[a2]	L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1974)
[a3]	H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6
[a4]	A.F. Beardon, "The geometry of discrete groups" , Springer (1983)
[a5]	B. Maskit, "Kleinian groups" , Springer (1988)