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A discrete group of holomorphic transformations (cf. [[Discrete group of transformations|Discrete group of transformations]]) of an (open) disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418901.png" /> onto the Riemann sphere, that is, of a disc or a half-plane onto the complex plane. Most often one takes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418902.png" /> the upper half-plane
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418903.png" /></td> </tr></table>
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 +
A discrete group of holomorphic transformations (cf. [[Discrete group of transformations|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
 +
 
 +
$$
 +
= \{ {z \in \mathbf C } : { \mathop{\rm Im}  z > 0 } \}
 +
$$
  
 
or the unit disc
 
or the unit disc
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418904.png" /></td> </tr></table>
+
$$
 +
= \{ {z \in \mathbf C } : {| z | < 1 } \}
 +
.
 +
$$
  
 
In the first case the elements of a Fuchsian group are Möbius transformations (cf. [[Fractional-linear mapping|Fractional-linear mapping]])
 
In the first case the elements of a Fuchsian group are Möbius transformations (cf. [[Fractional-linear mapping|Fractional-linear mapping]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418905.png" /></td> </tr></table>
+
$$
 +
z  \mapsto \
 +
 
 +
\frac{az + b }{cz + d }
 +
 
 +
$$
  
with real coefficients, and a Fuchsian group is nothing other than a [[Discrete subgroup|discrete subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418906.png" />. In the second case the elements of a Fuchsian group are Möbius transformations with pseudo-unitary matrices.
+
with real coefficients, and a Fuchsian group is nothing other than a [[Discrete subgroup|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418907.png" /> 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|Kleinian group]]).
+
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|Kleinian group]]).
  
 
Arbitrary Fuchsian groups were first studied by H. Poincaré (see [[#References|[2]]]) in 1882 in connection with the [[Uniformization|uniformization]] problem. He called the groups Fuchsian in honour of L. Fuchs, whose paper [[#References|[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.
 
Arbitrary Fuchsian groups were first studied by H. Poincaré (see [[#References|[2]]]) in 1882 in connection with the [[Uniformization|uniformization]] problem. He called the groups Fuchsian in honour of L. Fuchs, whose paper [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418908.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f0418909.png" />, or a straight line in the sense of Lobachevskii geometry, is called elementary. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189010.png" /> is a non-elementary Fuchsian group, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189011.png" /> of limit points of the orbit of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189012.png" /> lying on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189013.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189014.png" /> and is called the limit set of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189015.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189016.png" /> is called a Fuchsian group of the first kind if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189017.png" />, and of the second kind otherwise (then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189018.png" /> is a nowhere-dense perfect subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189019.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189020.png" /> a polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189021.png" /> of the Lobachevskii plane with sides
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189022.png" /></td> </tr></table>
+
$$
 +
a _ {1} , b _ {1}  ^  \prime  , a _ {1}  ^  \prime  , b _ {1} \dots a _ {g} ,\
 +
b _ {g}  ^  \prime  , a _ {g}  ^  \prime  , b _ {g} , c _ {1} ,\
 +
c _ {1}  ^  \prime  \dots c _ {n} , c _ {n}  ^  \prime
 +
$$
  
 
in such a way that
 
in such a way that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189023.png" /></td> </tr></table>
+
$$
 +
a _ {i}  = \
 +
\alpha _ {i} ( a _ {i}  ^  \prime  ),\ \
 +
b _ {i}  = \
 +
\beta _ {i} ( b _ {i}  ^  \prime  ),\ \
 +
c _ {i}  = \
 +
\gamma _ {i} ( c _ {i}  ^  \prime  )
 +
$$
  
 
for some elements
 
for some elements
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189024.png" /></td> </tr></table>
+
$$
 +
\alpha _ {1} \dots \alpha _ {g} ,\ \
 +
\beta _ {1} \dots \beta _ {g} ,\ \
 +
\gamma _ {1} \dots \gamma _ {n}  $$
  
which generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189025.png" />, with defining relations
+
which generate $  \Gamma $,  
 +
with defining relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189026.png" /></td> </tr></table>
+
$$
 +
\prod _ {i = 1 } ^ { g }
 +
( \alpha _ {i} \beta _ {i} \alpha _ {i}  ^ {-} 1 \beta _ {i}  ^ {-} 1 )
 +
\prod _ {i = 1 } ^ { n }
 +
\gamma _ {i}  = 1,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189027.png" /></td> </tr></table>
+
$$
 +
\gamma _ {i} ^ {k _ {i} }  = 1,\  i = 1 \dots n,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189028.png" /> is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189029.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189030.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189031.png" /> leaves the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189033.png" /> common to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189035.png" /> fixed. It is elliptic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189036.png" />, and parabolic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189037.png" />; in the latter case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189038.png" /> lies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189039.png" />, that is, it is an improper point of the Lobachevskii plane. Every elliptic or parabolic element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189040.png" /> is conjugate to a power of some unique generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189041.png" />. The angles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189042.png" /> at the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189044.png" />, are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189045.png" />; the sum of all remaining angles is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189046.png" />. The sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189048.png" />, and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189052.png" />, 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189053.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189054.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189055.png" />, where
+
The area of a fundamental domain of $  \Gamma $
 +
is equal to $  - 2 \pi \chi ( \Gamma ) $,  
 +
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189056.png" /></td> </tr></table>
+
$$
 +
\chi ( \Gamma )  = \
 +
\chi ( g; k _ {1} \dots k _ {n} )  = \
 +
2 - 2g - \sum _ {i = 1 } ^ { n }
 +
\left ( 1 - {
 +
\frac{1}{k _ {i} }
 +
} \right ) .
 +
$$
  
The collection of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189058.png" /> are taken to be unordered, is a topological invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189059.png" />, regarded as a group of homeomorphisms of the disc, and is called its signature. The only restriction on the signature is the condition
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\chi ( g; k _ {1} \dots k _ {n} )  < 0.
 +
$$
  
For a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189061.png" /> of finite index in a Fuchsian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189062.png" /> the Riemann–Hurwitz formula
+
For a subgroup $  \Delta $
 +
of finite index in a Fuchsian group $  \Gamma $
 +
the Riemann–Hurwitz formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189063.png" /></td> </tr></table>
+
$$
 +
\chi ( \Delta )  = \
 +
\chi ( \Gamma )
 +
[ \Gamma : \Delta ]
 +
$$
  
 
holds. In every Fuchsian group there is a subgroup of finite index that has no elements of finite order.
 
holds. In every Fuchsian group there is a subgroup of finite index that has no elements of finite order.
  
The quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189064.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189065.png" /> for which the quotient mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189066.png" /> is holomorphic. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189067.png" /> is a Riemann surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189068.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189069.png" /> is a regular branched covering with branching indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189070.png" />. Conversely, the uniformization theorem asserts that for any compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189071.png" /> with given points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189072.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189073.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189074.png" /> is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189075.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189076.png" />) satisfying the condition (*), there is a regular holomorphic branched covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189077.png" /> that is branched over precisely the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189078.png" /> with branching indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189079.png" />, respectively. The covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189080.png" /> is uniquely determined up to an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189081.png" />. Its group of covering transformations is a Fuchsian group of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189082.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189083.png" /> can be parametrized by the points of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189084.png" />-dimensional complex manifold homeomorphic to a cell, that is, a [[Teichmüller space|Teichmüller space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189085.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189086.png" /> — the so-called modular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189087.png" />. There is an isomorphism
+
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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189088.png" /></td> </tr></table>
+
$$
 +
T ( g; k _ {1} \dots k _ {n} )
 +
\rightarrow \
 +
T ( g; \infty \dots \infty )
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189089.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189090.png" />), under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189091.png" /> is mapped to a subgroup of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189092.png" />.
+
( $  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189093.png" /> contains a subgroup of finite index of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189094.png" />, then the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189095.png" /> can be imbedded in a unique way as a closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189096.png" />. In certain exceptional cases these spaces coincide [[#References|[10]]]. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189097.png" />; this means that every compact Riemann surface of genus 2 admits a hyper-elliptic involution and so is a [[Hyper-elliptic curve|hyper-elliptic curve]].
+
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 [[#References|[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|hyper-elliptic curve]].
  
For Fuchsian groups of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189098.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f04189099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890101.png" /> (see [[Reflection group|Reflection group]]). An example of a triangular group is the modular Kleinian group; its signature is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890102.png" />.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890104.png" /> of the fundamental polygon have no common points, even improper ones, and that the corresponding generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890105.png" /> are hyperbolic transformations. The compactified quotient space is a Riemann surface with boundary.
+
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 [[#References|[5]]]).
 
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 [[#References|[5]]]).
Line 73: Line 222:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,  "Théorie des groupes Fuchsiennes"  ''Acta. Math.'' , '''1'''  (1882)  pp. 1–62</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Fricke,  F. Klein,  "Vorlesungen über die Theorie der automorphen Funktionen" , '''1–2''' , Teubner  (1897–1912)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  L. Bers,  "Quasi-conformal mappings and Teichmüller's theorem"  R. Nevanlinna (ed.)  et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press  (1960)  pp. 89–119</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.L. Krushkal',  B.N. Apanasov,  N.A. Gusevskii,  "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc.  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Lehner,  "Discontinuous groups and automorphic functions" , Amer. Math. Soc.  (1964)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  W. Magnus,  "NonEuclidean tesselations and their groups" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Singerman,  "Finitely maximal Fuchsian groups"  ''J. London Math. Soc.'' , '''6''' :  1  (1972)  pp. 29–38</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,  "Théorie des groupes Fuchsiennes"  ''Acta. Math.'' , '''1'''  (1882)  pp. 1–62</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Fricke,  F. Klein,  "Vorlesungen über die Theorie der automorphen Funktionen" , '''1–2''' , Teubner  (1897–1912)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  L. Bers,  "Quasi-conformal mappings and Teichmüller's theorem"  R. Nevanlinna (ed.)  et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press  (1960)  pp. 89–119</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.L. Krushkal',  B.N. Apanasov,  N.A. Gusevskii,  "Kleinian groups and uniformization in examples and problems" , Amer. Math. Soc.  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Lehner,  "Discontinuous groups and automorphic functions" , Amer. Math. Soc.  (1964)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  W. Magnus,  "NonEuclidean tesselations and their groups" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Singerman,  "Finitely maximal Fuchsian groups"  ''J. London Math. Soc.'' , '''6''' :  1  (1972)  pp. 29–38</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Relations with Riemann surfaces are discussed in [[#References|[a3]]], [[#References|[a2]]]. [[#References|[a1]]] presents the theory of Fuchsian groups and automorphic functions.
 
Relations with Riemann surfaces are discussed in [[#References|[a3]]], [[#References|[a2]]]. [[#References|[a1]]] presents the theory of Fuchsian groups and automorphic functions.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890106.png" /> be a discrete subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890107.png" />, the group of fractional transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890108.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890110.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890111.png" /> in the extended complex plane and any sequence of distinct elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890112.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890113.png" /> a cluster point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890114.png" /> is called a limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890115.png" />. If there are 0, 1 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890116.png" /> limit points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890117.png" /> is conjugate to a group of motions of the plane. Otherwise there are infinitely many limit points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890118.png" /> is called a Fuchsoid group. A Fuchsoid group is a Fuchsian group if it is finitely generated. For a real point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890119.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890120.png" /> be the stabilizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890121.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890122.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890123.png" /> is called a cusp or parabolic cusp if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890124.png" /> is a free cyclic group generated by a parabolic transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890125.png" /> (cf. [[Fractional-linear mapping|Fractional-linear mapping]]). The cusps of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041890/f041890126.png" /> are represented by vertices of a fundamental polygon on the real axis.
+
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|Fractional-linear mapping]]). The cusps of $  \Gamma $
 +
are represented by vertices of a fundamental polygon on the real axis.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.R. Ford,  "Automorphic functions" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Ahlfors,  L. Sario,  "Riemann surfaces" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.M. Farkas,  I. Kra,  "Riemann surfaces" , Springer  (1980)  pp. Sect. III.6</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.F. Beardon,  "The geometry of discrete groups" , Springer  (1983)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Maskit,  "Kleinian groups" , Springer  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.R. Ford,  "Automorphic functions" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Ahlfors,  L. Sario,  "Riemann surfaces" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.M. Farkas,  I. Kra,  "Riemann surfaces" , Springer  (1980)  pp. Sect. III.6</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.F. Beardon,  "The geometry of discrete groups" , Springer  (1983)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Maskit,  "Kleinian groups" , Springer  (1988)</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


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)
How to Cite This Entry:
Fuchsian group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fuchsian_group&oldid=19299
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article