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A [[Discrete subgroup|discrete subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555201.png" /> of the group of all fractional-linear mappings (cf. [[Fractional-linear mapping|Fractional-linear mapping]])
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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/k/k055/k055520/k0555202.png" /></td> </tr></table>
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of the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555203.png" /> that acts properly discontinuous. This means that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555204.png" /> of points of accumulation of orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555205.png" />, for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555206.png" />, called the limit set of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555207.png" />, is a proper subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555208.png" />. The complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k0555209.png" />, called the discontinuity set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552010.png" />, is open and has the property that each of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552011.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552012.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552014.png" />, where
+
A [[Discrete subgroup|discrete subgroup]]  $  \Gamma $
 +
of the group of all fractional-linear mappings (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/k/k055/k055520/k05552015.png" /></td> </tr></table>
+
$$
 +
\gamma ( z)  = \
  
is the [[Stabilizer|stabilizer]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552017.png" />. If a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552018.png" /> is not one of the fixed points of the elliptic elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552021.png" /> is the identity mapping, and for each elliptic fixed point, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552022.png" /> is a [[Cyclic group|cyclic group]] of finite order. The basic theory of Kleinian groups was laid down in the fundamental papers of H. Poincaré [[#References|[1]]] and F. Klein [[#References|[2]]] in the 19th century; the name "Kleinian group" goes back to Poincaré.
+
\frac{a z + b }{c z + d }
 +
,\ \
 +
a d - b c = 1 ,
 +
$$
  
The limit set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552023.png" /> is either empty, consists of one or two points, or is infinite. The first two cases correspond to the elementary groups (in particular, all cyclic groups). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552024.png" /> is infinite, then it is a nowhere-dense perfect subset (cf. [[Perfect set|Perfect set]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552025.png" /> of positive [[Logarithmic capacity|logarithmic capacity]]. Often the elementary groups are not included among the Kleinian groups.
+
of the extended complex plane  $  \overline{\mathbf C}\; $
 +
that acts properly discontinuous. This means that the set $  \Lambda ( \Gamma ) $
 +
of points of accumulation of orbits  $  \{ {\gamma ( z _ {0} ) } : {\gamma \in \Gamma } \} $,
 +
for all points  $  z _ {0} \in \mathbf C $,  
 +
called the limit set of the group  $  \Gamma $,  
 +
is a proper subset of  $  \overline{\mathbf C}\; $.  
 +
The complement  $  \Omega ( \Gamma ) = \overline{\mathbf C}\; \setminus  \Lambda ( \Gamma ) $,
 +
called the discontinuity set of  $  \Gamma $,  
 +
is open and has the property that each of its points  $  z $
 +
has a neighbourhood  $  U _ {z} $
 +
for which  $  \gamma ( U _ {z} ) \cap U _ {z} = \emptyset $
 +
for all  $  \gamma \in \Gamma \setminus  \Gamma _ {z} $,
 +
where
  
The quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552026.png" /> has a natural complex (conformal) structure in which the projection
+
$$
 +
\Gamma _ {z}  = \{ {\gamma \in \Gamma } : {\gamma ( z) = z } \}
 +
$$
  
<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/k/k055/k055520/k05552027.png" /></td> </tr></table>
+
is the [[Stabilizer|stabilizer]] of  $  z $
 +
in  $  \Gamma $.
 +
If a point  $  z \in \Omega ( \Gamma ) $
 +
is not one of the fixed points of the elliptic elements of  $  \Gamma $,
 +
then  $  \Gamma _ {z} = \{ J \} $,
 +
where  $  J $
 +
is the identity mapping, and for each elliptic fixed point,  $  \Gamma _ {z} $
 +
is a [[Cyclic group|cyclic group]] of finite order. The basic theory of Kleinian groups was laid down in the fundamental papers of H. Poincaré [[#References|[1]]] and F. Klein [[#References|[2]]] in the 19th century; the name  "Kleinian group" goes back to Poincaré.
  
is holomorphic, and can be expressed as a finite or countable union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552028.png" /> of Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552029.png" />; this covering is ramified over projections of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552030.png" /> with non-trivial stabilizers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552031.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552032.png" /> itself splits up into connected components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552033.png" /> whose number is 1, 2 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552034.png" />. If the subgroup
+
The limit set  $  \Lambda ( \Gamma ) $
 +
is either empty, consists of one or two points, or is infinite. The first two cases correspond to the elementary groups (in particular, all cyclic groups). If  $  \Lambda ( \Gamma ) $
 +
is infinite, then it is a nowhere-dense perfect subset (cf. [[Perfect set|Perfect set]]) of $  \overline{\mathbf C}\; $
 +
of positive [[Logarithmic capacity|logarithmic capacity]]. Often the elementary groups are not included among the Kleinian groups.
  
<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/k/k055/k055520/k05552035.png" /></td> </tr></table>
+
The quotient space  $  \Omega ( \Gamma ) / \Gamma $
 +
has a natural complex (conformal) structure in which the projection
  
is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552037.png" /> is called an invariant component. There can be at most two invariant components. Kleinian groups with invariant components have acquired the name Kleinian function groups.
+
$$
 +
\pi :  \Omega ( \Gamma )  \rightarrow  \Omega ( \Gamma ) / \Gamma
 +
$$
 +
 
 +
is holomorphic, and can be expressed as a finite or countable union  $  \cup _ {j} S _ {j} $
 +
of Riemann surfaces  $  S _ {j} $;
 +
this covering is ramified over projections of points  $  z \in \Omega ( \Gamma ) $
 +
with non-trivial stabilizers  $  \Gamma _ {z} $.
 +
$  \Omega ( \Gamma ) $
 +
itself splits up into connected components  $  \Omega _ {j} $
 +
whose number is 1, 2 or  $  \infty $.
 +
If the subgroup
 +
 
 +
$$
 +
\Gamma _ {\Omega _ {j}  }  = \
 +
\{ {\gamma \in \Gamma } : {\gamma ( \Omega _ {j} ) = \Omega _ {j} } \}
 +
$$
 +
 
 +
is the same as $  \Gamma $,  
 +
then $  \Omega _ {j} $
 +
is called an invariant component. There can be at most two invariant components. Kleinian groups with invariant components have acquired the name Kleinian function groups.
  
 
===Examples.===
 
===Examples.===
  
 +
1) Fuchsian groups (cf. [[Fuchsian group|Fuchsian group]]). Each such group leaves invariant some circle (or line)  $  l $,
 +
preserves the direction of circulation and  $  \Lambda ( \Gamma ) \subset  l $.
 +
In order that a (non-elementary) Kleinian group  $  \Gamma $
 +
is Fuchsian, it is necessary and sufficient that it does not contain loxodromic elements. According to the Klein–Poincaré uniformization theorem, every [[Riemann surface|Riemann surface]], apart from a few simple cases, is uniformizable by a Fuchsian group acting, for example, in the upper half-plane  $  H = \{ {z \in \mathbf C } : { \mathop{\rm Im}  z > 0 } \} $,
 +
that is, it is representable in the form  $  H / \Gamma $
 +
up to conformal equivalence. If one introduces into  $  H $
 +
the hyperbolic Poincaré metric
  
1) Fuchsian groups (cf. [[Fuchsian group|Fuchsian group]]). Each such group leaves invariant some circle (or line) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552038.png" />, preserves the direction of circulation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552039.png" />. In order that a (non-elementary) Kleinian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552040.png" /> is Fuchsian, it is necessary and sufficient that it does not contain loxodromic elements. According to the Klein–Poincaré uniformization theorem, every [[Riemann surface|Riemann surface]], apart from a few simple cases, is uniformizable by a Fuchsian group acting, for example, in the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552041.png" />, that is, it is representable in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552042.png" /> up to conformal equivalence. If one introduces into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552043.png" /> the hyperbolic Poincaré metric
+
$$
 +
d s  = \
  
<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/k/k055/k055520/k05552044.png" /></td> </tr></table>
+
\frac{| d z | }{ \mathop{\rm Im}  z }
 +
,
 +
$$
  
then the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552045.png" /> become non-Euclidean (hyperbolic) motions. Poincaré has also put forward a similar interpretation for an arbitrary Kleinian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552046.png" />, based on extending the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552047.png" /> to the half-space
+
then the elements of $  \Gamma $
 +
become non-Euclidean (hyperbolic) motions. Poincaré has also put forward a similar interpretation for an arbitrary Kleinian group $  \Gamma $,  
 +
based on extending the action of $  \Gamma $
 +
to the half-space
  
<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/k/k055/k055520/k05552048.png" /></td> </tr></table>
+
$$
 +
\mathbf R _ {+}  ^ {3}  = \
 +
\{ {x = ( x _ {1} , x _ {2} , x _ {3} ) } : {
 +
x _ {1} + i x _ {2} \in \mathbf C , x _ {3} > 0 } \}
 +
.
 +
$$
  
Namely, since each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552049.png" /> is a superposition of a countable number of inversions with respect to circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552050.png" />, it is possible to consider inversions with respect to the corresponding hemispheres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552051.png" /> supported by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552052.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552053.png" /> extended in this way acts discontinuously in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552054.png" /> and its elements become hyperbolic motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552055.png" />.
+
Namely, since each element of $  \Gamma $
 +
is a superposition of a countable number of inversions with respect to circles $  L \subset  \overline{\mathbf C}\; $,  
 +
it is possible to consider inversions with respect to the corresponding hemispheres in $  \mathbf R _ {+}  ^ {3} $
 +
supported by the $  L $.  
 +
The group $  \Gamma $
 +
extended in this way acts discontinuously in $  \mathbf R _ {+}  ^ {3} $
 +
and its elements become hyperbolic motions of $  \mathbf R _ {+}  ^ {3} $.
  
2) Quasi-Fuchsian groups. These are direct generalizations of Fuchsian groups. A quasi-Fuchsian group is a Kleinian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552056.png" /> that leaves invariant some oriented [[Jordan curve|Jordan curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552057.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552060.png" /> is called a group of genus one, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552061.png" />, it is said to have genus two. The Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552063.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552064.png" /> is the interior and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552065.png" /> is the exterior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552066.png" />, are homeomorphic. Moreover, for example, any two homeomorphic Riemann surfaces of finite type (that is, closed surfaces with a finite number of punctures) can be uniformized by the same quasi-Fuchsian group. Finitely-generated quasi-Fuchsian groups reduce to Fuchsian ones (are conjugate with them) by means of quasi-conformal automorphisms of the plane.
+
2) Quasi-Fuchsian groups. These are direct generalizations of Fuchsian groups. A quasi-Fuchsian group is a Kleinian group $  \Gamma $
 +
that leaves invariant some oriented [[Jordan curve|Jordan curve]] $  l \subset  \mathbf C $.  
 +
Then $  \Lambda ( \Gamma ) \subset  l $.  
 +
If $  \Lambda ( \Gamma ) = l $,  
 +
then $  \Gamma $
 +
is called a group of genus one, while if $  l \setminus  \Lambda ( \Gamma ) \neq \emptyset $,  
 +
it is said to have genus two. The Riemann surfaces $  D _ {1} / \Gamma $
 +
and $  D _ {2} / \Gamma $
 +
where $  D _ {1} $
 +
is the interior and $  D _ {2} $
 +
is the exterior of $  l $,  
 +
are homeomorphic. Moreover, for example, any two homeomorphic Riemann surfaces of finite type (that is, closed surfaces with a finite number of punctures) can be uniformized by the same quasi-Fuchsian group. Finitely-generated quasi-Fuchsian groups reduce to Fuchsian ones (are conjugate with them) by means of quasi-conformal automorphisms of the plane.
  
3) Schottky groups. These are Kleinian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552067.png" /> with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552069.png" />, for which there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552070.png" /> non-intersecting Jordan curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552071.png" /> bounding a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552072.png" />-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552073.png" /> such that
+
3) Schottky groups. These are Kleinian groups $  \Gamma $
 +
with generators $  \gamma _ {1} \dots \gamma _ {p} $,  
 +
$  p \geq  1 $,  
 +
for which there exist $  2 p $
 +
non-intersecting Jordan curves $  l _ {1} , l _ {1} ^ { \prime } \dots l _ {p} , l _ {p} ^ { \prime } $
 +
bounding a $  2 p $-
 +
connected domain $  D $
 +
such 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/k/k055/k055520/k05552074.png" /></td> </tr></table>
+
$$
 +
\gamma _ {j} ( D) \cap D  = \emptyset ,\ \
 +
\gamma _ {j} ( l _ {j} )  = l _ {j} ^ { \prime } ,\ \
 +
j = 1 \dots p .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552075.png" /> is free, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552076.png" /> is a closed surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552077.png" /> and all the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552078.png" /> are hyperbolic or loxodromic. All closed Riemann surfaces are uniformized by Schottky groups (this is Koebe uniformization).
+
Here $  \Gamma $
 +
is free, $  \Omega ( \Gamma ) $
 +
is a closed surface of genus $  p $
 +
and all the elements $  \gamma \in \Gamma \setminus  \{ J \} $
 +
are hyperbolic or loxodromic. All closed Riemann surfaces are uniformized by Schottky groups (this is Koebe uniformization).
  
4) Degenerate groups. These are non-elementary finitely-generated Kleinian groups whose discontinuity sets are simply-connected domains. There is an extremely-complicated proof of the existence of such groups; meanwhile no explicit examples have been constructed (1978). Degenerate groups are a special case of groups with one invariant simply-connected component, called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552080.png" />-groups.
+
4) Degenerate groups. These are non-elementary finitely-generated Kleinian groups whose discontinuity sets are simply-connected domains. There is an extremely-complicated proof of the existence of such groups; meanwhile no explicit examples have been constructed (1978). Degenerate groups are a special case of groups with one invariant simply-connected component, called $  b $-
 +
groups.
  
At the basis of the geometric approach to the study of Kleinian groups is the notion of a fundamental domain, that is, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552081.png" /> containing one point of each orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552083.png" />, and such that each non-empty component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552084.png" /> of it is connected. For example, for Schottky groups one can take for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552085.png" /> the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552086.png" /> indicated in its definition, and adjoining to it points of the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552087.png" />. Often only the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552088.png" /> is called the fundamental domain. For any Kleinian group one can choose a canonical fundamental domain bounded by circular arcs. The properties of the fundamental domain enable one to elucidate the structure of a Kleinian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552089.png" />. One of the methods for constructing Kleinian groups are the so-called combination theorems, which give conditions under which a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552090.png" /> generated by given Kleinian groups is again a Kleinian group. For example, if one takes Fuchsian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552091.png" /> acting, respectively, in discs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552092.png" /> that are sufficiently far apart, and if one takes the compact surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552093.png" /> representing them, of respective genera <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552094.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552095.png" /> is a function group representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552096.png" /> surfaces of genera <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552098.png" />. The methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k05552099.png" />-dimensional topology relating to the study of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520100.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520101.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520102.png" /> is the boundary, turn out to be very suitable.
+
At the basis of the geometric approach to the study of Kleinian groups is the notion of a fundamental domain, that is, a set $  \omega \subset  \Omega ( \Gamma ) $
 +
containing one point of each orbit $  \Gamma z _ {0} $,  
 +
$  z _ {0} \in \Omega ( \Gamma ) $,  
 +
and such that each non-empty component $  \omega \cap \Omega _ {j} $
 +
of it is connected. For example, for Schottky groups one can take for $  \omega $
 +
the domain $  D $
 +
indicated in its definition, and adjoining to it points of the curves $  l _ {1} \dots l _ {p} $.  
 +
Often only the interior of $  \omega $
 +
is called the fundamental domain. For any Kleinian group one can choose a canonical fundamental domain bounded by circular arcs. The properties of the fundamental domain enable one to elucidate the structure of a Kleinian group $  \Gamma $.  
 +
One of the methods for constructing Kleinian groups are the so-called combination theorems, which give conditions under which a group $  \Gamma $
 +
generated by given Kleinian groups is again a Kleinian group. For example, if one takes Fuchsian groups $  \Gamma _ {1} \dots \Gamma _ {n} $
 +
acting, respectively, in discs $  U _ {1} \dots U _ {n} $
 +
that are sufficiently far apart, and if one takes the compact surfaces $  U _ {j} / \Gamma _ {j} $
 +
representing them, of respective genera $  p _ {j} $,  
 +
then $  \Gamma = \langle  \Gamma _ {1} \dots \Gamma _ {n} \rangle $
 +
is a function group representing $  n + 1 $
 +
surfaces of genera $  p _ {1} \dots p _ {n} $
 +
and $  p _ {1} + \dots + p _ {n} $.  
 +
The methods of $  3 $-
 +
dimensional topology relating to the study of a $  3 $-
 +
dimensional manifold $  ( \mathbf R _ {+}  ^ {3} \cup \Omega ( \Gamma )) / \Gamma $,  
 +
for which $  \Omega ( \Gamma ) / \Gamma $
 +
is the boundary, turn out to be very suitable.
  
The analytic approach to the theory of Kleinian groups is connected with the study of automorphic forms (cf. [[Automorphic form|Automorphic form]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520103.png" /> is a non-elementary Kleinian group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520104.png" />, then for integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520105.png" /> the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520106.png" /> converges (at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520107.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520108.png" />); the corresponding Poincaré theta-series
+
The analytic approach to the theory of Kleinian groups is connected with the study of automorphic forms (cf. [[Automorphic form|Automorphic form]]). If $  \Gamma $
 +
is a non-elementary Kleinian group and $  \infty \in \Omega ( \Gamma ) $,  
 +
then for integers $  q \geq  2 $
 +
the series $  \sum _ {\gamma \in \Gamma }  | \gamma  ^  \prime  ( z) |  ^ {q} $
 +
converges (at the points $  z \in \Omega ( \Gamma ) $
 +
with $  \Gamma _ {z} = \{ J \} $);  
 +
the corresponding Poincaré theta-series
  
<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/k/k055/k055520/k055520109.png" /></td> </tr></table>
+
$$
 +
\sum _ {\gamma \in \Gamma }
 +
f ( \gamma ( z) )
 +
\gamma ^ {\prime q } ( z) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520110.png" /> is a meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520111.png" />, give automorphic forms of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520112.png" />. For finitely-generated Kleinian groups the dimension of the space of such forms can be calculated by means of the [[Riemann–Roch theorem|Riemann–Roch theorem]]. The geometric structure of such groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520113.png" /> is described by the Ahlfors theorem, according to which the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520114.png" /> for these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520115.png" /> consists of a finite number of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520116.png" /> of finite type, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520117.png" /> can be ramified over each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520118.png" /> only at a finite number of points. This result admits quantitative refinements. Homological methods are also used, based on the study of the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520119.png" /> in vector spaces of polynomials (see [[#References|[5]]]). Methods of the theory of [[Quasi-conformal mapping|quasi-conformal mapping]] [[#References|[6]]], [[#References|[7]]] play an essential role in the theory of Kleinian groups on the plane. In particular, the theory of deformations of Kleinian groups, closely related to the theory of moduli of Riemann surfaces (see [[Moduli of a Riemann surface|Moduli of a Riemann surface]] and [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]]) relies on these methods. Along these lines certain new classes of Kleinian groups have emerged. Meanwhile, however, no complete classification has been obtained, not even for finitely-generated Kleinian groups.
+
where $  f $
 +
is a meromorphic function in $  \Omega ( \Gamma ) $,  
 +
give automorphic forms of weight $  ( - 2 q ) $.  
 +
For finitely-generated Kleinian groups the dimension of the space of such forms can be calculated by means of the [[Riemann–Roch theorem|Riemann–Roch theorem]]. The geometric structure of such groups $  \Gamma $
 +
is described by the Ahlfors theorem, according to which the space $  \Omega ( \Gamma ) / \Gamma $
 +
for these $  \Gamma $
 +
consists of a finite number of surfaces $  S _ {1} \dots S _ {n} $
 +
of finite type, and $  \pi  ^ {-} 1 $
 +
can be ramified over each $  S _ {j} $
 +
only at a finite number of points. This result admits quantitative refinements. Homological methods are also used, based on the study of the action of $  \Gamma $
 +
in vector spaces of polynomials (see [[#References|[5]]]). Methods of the theory of [[Quasi-conformal mapping|quasi-conformal mapping]] [[#References|[6]]], [[#References|[7]]] play an essential role in the theory of Kleinian groups on the plane. In particular, the theory of deformations of Kleinian groups, closely related to the theory of moduli of Riemann surfaces (see [[Moduli of a Riemann surface|Moduli of a Riemann surface]] and [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]]) relies on these methods. Along these lines certain new classes of Kleinian groups have emerged. Meanwhile, however, no complete classification has been obtained, not even for finitely-generated Kleinian groups.
  
By comparison with planar ones, Kleinian groups in a multi-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520121.png" />, defined as properly-discontinuous subgroups of the group of conformal automorphisms of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520122.png" />, have been much less extensively studied; here completely new phenomena occur.
+
By comparison with planar ones, Kleinian groups in a multi-dimensional Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n > 2 $,  
 +
defined as properly-discontinuous subgroups of the group of conformal automorphisms of the space $  \overline{ {\mathbf R  ^ {n} }}\; = \mathbf R  ^ {n} \cup \{ \infty \} $,  
 +
have been much less extensively studied; here completely new phenomena occur.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les groupes kleinéens"  ''Acta Math.'' , '''3'''  (1883)  pp. 49–92</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Klein,  "Neue Beiträge zur Riemannschen Funktionentheorie"  ''Math. Ann.'' , '''21'''  (1883)  pp. 141–218</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.R. Ford,  "Automorphic functions" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Lehner,  "Discontinuous groups and automorphic functions" , Amer. Math. Soc.  (1964)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I. Kra,  "Automorphic forms and Kleinian groups" , Benjamin  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.L. Krushkal',  "Quasi-conformal mappings and Riemann surfaces" , Winston &amp; Wiley  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Bers (ed.)  I. Kra (ed.) , ''A crash course on Kleinian groups'' , ''Lect. notes in math.'' , '''400''' , Springer  (1974)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B. Maskit,  "Kleinian groups" , Springer  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Mémoire sur les groupes kleinéens"  ''Acta Math.'' , '''3'''  (1883)  pp. 49–92</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Klein,  "Neue Beiträge zur Riemannschen Funktionentheorie"  ''Math. Ann.'' , '''21'''  (1883)  pp. 141–218</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.R. Ford,  "Automorphic functions" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Lehner,  "Discontinuous groups and automorphic functions" , Amer. Math. Soc.  (1964)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I. Kra,  "Automorphic forms and Kleinian groups" , Benjamin  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.L. Krushkal',  "Quasi-conformal mappings and Riemann surfaces" , Winston &amp; Wiley  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Bers (ed.)  I. Kra (ed.) , ''A crash course on Kleinian groups'' , ''Lect. notes in math.'' , '''400''' , Springer  (1974)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B. Maskit,  "Kleinian groups" , Springer  (1988)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For the definitions of loxodromic, elliptic, hyperbolic<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520123.png" /> fractional-linear transformations, cf. [[Fractional-linear mapping|Fractional-linear mapping]].
+
For the definitions of loxodromic, elliptic, hyperbolic $  \dots $
 +
fractional-linear transformations, cf. [[Fractional-linear mapping|Fractional-linear mapping]].
  
 
One of the  "quantitative refinements" , or, more precisely, a quantitative extension, of the Ahlfors finiteness theorem is the Bers area inequality:
 
One of the  "quantitative refinements" , or, more precisely, a quantitative extension, of the Ahlfors finiteness theorem is the Bers area inequality:
  
<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/k/k055/k055520/k055520124.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{1}{2 \pi }
 +
\{ \textrm{
 +
hyperbolic  area  of  } \Omega \setminus  \Gamma \}  \leq  2 ( N- 1) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520125.png" /> is the (minimum) number of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055520/k055520126.png" />.
+
where $  N $
 +
is the (minimum) number of generators of $  \Gamma $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kra (ed.)  B. Maskit (ed.) , ''Riemann Surfaces and Related Topics (Proc. 1978 Stony Brook Conf.)'' , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.M. Farkas,  I. Kra,  "Riemann surfaces" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kra (ed.)  B. Maskit (ed.) , ''Riemann Surfaces and Related Topics (Proc. 1978 Stony Brook Conf.)'' , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.M. Farkas,  I. Kra,  "Riemann surfaces" , Springer  (1980)</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


A discrete subgroup $ \Gamma $ of the group of all fractional-linear mappings (cf. Fractional-linear mapping)

$$ \gamma ( z) = \ \frac{a z + b }{c z + d } ,\ \ a d - b c = 1 , $$

of the extended complex plane $ \overline{\mathbf C}\; $ that acts properly discontinuous. This means that the set $ \Lambda ( \Gamma ) $ of points of accumulation of orbits $ \{ {\gamma ( z _ {0} ) } : {\gamma \in \Gamma } \} $, for all points $ z _ {0} \in \mathbf C $, called the limit set of the group $ \Gamma $, is a proper subset of $ \overline{\mathbf C}\; $. The complement $ \Omega ( \Gamma ) = \overline{\mathbf C}\; \setminus \Lambda ( \Gamma ) $, called the discontinuity set of $ \Gamma $, is open and has the property that each of its points $ z $ has a neighbourhood $ U _ {z} $ for which $ \gamma ( U _ {z} ) \cap U _ {z} = \emptyset $ for all $ \gamma \in \Gamma \setminus \Gamma _ {z} $, where

$$ \Gamma _ {z} = \{ {\gamma \in \Gamma } : {\gamma ( z) = z } \} $$

is the stabilizer of $ z $ in $ \Gamma $. If a point $ z \in \Omega ( \Gamma ) $ is not one of the fixed points of the elliptic elements of $ \Gamma $, then $ \Gamma _ {z} = \{ J \} $, where $ J $ is the identity mapping, and for each elliptic fixed point, $ \Gamma _ {z} $ is a cyclic group of finite order. The basic theory of Kleinian groups was laid down in the fundamental papers of H. Poincaré [1] and F. Klein [2] in the 19th century; the name "Kleinian group" goes back to Poincaré.

The limit set $ \Lambda ( \Gamma ) $ is either empty, consists of one or two points, or is infinite. The first two cases correspond to the elementary groups (in particular, all cyclic groups). If $ \Lambda ( \Gamma ) $ is infinite, then it is a nowhere-dense perfect subset (cf. Perfect set) of $ \overline{\mathbf C}\; $ of positive logarithmic capacity. Often the elementary groups are not included among the Kleinian groups.

The quotient space $ \Omega ( \Gamma ) / \Gamma $ has a natural complex (conformal) structure in which the projection

$$ \pi : \Omega ( \Gamma ) \rightarrow \Omega ( \Gamma ) / \Gamma $$

is holomorphic, and can be expressed as a finite or countable union $ \cup _ {j} S _ {j} $ of Riemann surfaces $ S _ {j} $; this covering is ramified over projections of points $ z \in \Omega ( \Gamma ) $ with non-trivial stabilizers $ \Gamma _ {z} $. $ \Omega ( \Gamma ) $ itself splits up into connected components $ \Omega _ {j} $ whose number is 1, 2 or $ \infty $. If the subgroup

$$ \Gamma _ {\Omega _ {j} } = \ \{ {\gamma \in \Gamma } : {\gamma ( \Omega _ {j} ) = \Omega _ {j} } \} $$

is the same as $ \Gamma $, then $ \Omega _ {j} $ is called an invariant component. There can be at most two invariant components. Kleinian groups with invariant components have acquired the name Kleinian function groups.

Examples.

1) Fuchsian groups (cf. Fuchsian group). Each such group leaves invariant some circle (or line) $ l $, preserves the direction of circulation and $ \Lambda ( \Gamma ) \subset l $. In order that a (non-elementary) Kleinian group $ \Gamma $ is Fuchsian, it is necessary and sufficient that it does not contain loxodromic elements. According to the Klein–Poincaré uniformization theorem, every Riemann surface, apart from a few simple cases, is uniformizable by a Fuchsian group acting, for example, in the upper half-plane $ H = \{ {z \in \mathbf C } : { \mathop{\rm Im} z > 0 } \} $, that is, it is representable in the form $ H / \Gamma $ up to conformal equivalence. If one introduces into $ H $ the hyperbolic Poincaré metric

$$ d s = \ \frac{| d z | }{ \mathop{\rm Im} z } , $$

then the elements of $ \Gamma $ become non-Euclidean (hyperbolic) motions. Poincaré has also put forward a similar interpretation for an arbitrary Kleinian group $ \Gamma $, based on extending the action of $ \Gamma $ to the half-space

$$ \mathbf R _ {+} ^ {3} = \ \{ {x = ( x _ {1} , x _ {2} , x _ {3} ) } : { x _ {1} + i x _ {2} \in \mathbf C , x _ {3} > 0 } \} . $$

Namely, since each element of $ \Gamma $ is a superposition of a countable number of inversions with respect to circles $ L \subset \overline{\mathbf C}\; $, it is possible to consider inversions with respect to the corresponding hemispheres in $ \mathbf R _ {+} ^ {3} $ supported by the $ L $. The group $ \Gamma $ extended in this way acts discontinuously in $ \mathbf R _ {+} ^ {3} $ and its elements become hyperbolic motions of $ \mathbf R _ {+} ^ {3} $.

2) Quasi-Fuchsian groups. These are direct generalizations of Fuchsian groups. A quasi-Fuchsian group is a Kleinian group $ \Gamma $ that leaves invariant some oriented Jordan curve $ l \subset \mathbf C $. Then $ \Lambda ( \Gamma ) \subset l $. If $ \Lambda ( \Gamma ) = l $, then $ \Gamma $ is called a group of genus one, while if $ l \setminus \Lambda ( \Gamma ) \neq \emptyset $, it is said to have genus two. The Riemann surfaces $ D _ {1} / \Gamma $ and $ D _ {2} / \Gamma $ where $ D _ {1} $ is the interior and $ D _ {2} $ is the exterior of $ l $, are homeomorphic. Moreover, for example, any two homeomorphic Riemann surfaces of finite type (that is, closed surfaces with a finite number of punctures) can be uniformized by the same quasi-Fuchsian group. Finitely-generated quasi-Fuchsian groups reduce to Fuchsian ones (are conjugate with them) by means of quasi-conformal automorphisms of the plane.

3) Schottky groups. These are Kleinian groups $ \Gamma $ with generators $ \gamma _ {1} \dots \gamma _ {p} $, $ p \geq 1 $, for which there exist $ 2 p $ non-intersecting Jordan curves $ l _ {1} , l _ {1} ^ { \prime } \dots l _ {p} , l _ {p} ^ { \prime } $ bounding a $ 2 p $- connected domain $ D $ such that

$$ \gamma _ {j} ( D) \cap D = \emptyset ,\ \ \gamma _ {j} ( l _ {j} ) = l _ {j} ^ { \prime } ,\ \ j = 1 \dots p . $$

Here $ \Gamma $ is free, $ \Omega ( \Gamma ) $ is a closed surface of genus $ p $ and all the elements $ \gamma \in \Gamma \setminus \{ J \} $ are hyperbolic or loxodromic. All closed Riemann surfaces are uniformized by Schottky groups (this is Koebe uniformization).

4) Degenerate groups. These are non-elementary finitely-generated Kleinian groups whose discontinuity sets are simply-connected domains. There is an extremely-complicated proof of the existence of such groups; meanwhile no explicit examples have been constructed (1978). Degenerate groups are a special case of groups with one invariant simply-connected component, called $ b $- groups.

At the basis of the geometric approach to the study of Kleinian groups is the notion of a fundamental domain, that is, a set $ \omega \subset \Omega ( \Gamma ) $ containing one point of each orbit $ \Gamma z _ {0} $, $ z _ {0} \in \Omega ( \Gamma ) $, and such that each non-empty component $ \omega \cap \Omega _ {j} $ of it is connected. For example, for Schottky groups one can take for $ \omega $ the domain $ D $ indicated in its definition, and adjoining to it points of the curves $ l _ {1} \dots l _ {p} $. Often only the interior of $ \omega $ is called the fundamental domain. For any Kleinian group one can choose a canonical fundamental domain bounded by circular arcs. The properties of the fundamental domain enable one to elucidate the structure of a Kleinian group $ \Gamma $. One of the methods for constructing Kleinian groups are the so-called combination theorems, which give conditions under which a group $ \Gamma $ generated by given Kleinian groups is again a Kleinian group. For example, if one takes Fuchsian groups $ \Gamma _ {1} \dots \Gamma _ {n} $ acting, respectively, in discs $ U _ {1} \dots U _ {n} $ that are sufficiently far apart, and if one takes the compact surfaces $ U _ {j} / \Gamma _ {j} $ representing them, of respective genera $ p _ {j} $, then $ \Gamma = \langle \Gamma _ {1} \dots \Gamma _ {n} \rangle $ is a function group representing $ n + 1 $ surfaces of genera $ p _ {1} \dots p _ {n} $ and $ p _ {1} + \dots + p _ {n} $. The methods of $ 3 $- dimensional topology relating to the study of a $ 3 $- dimensional manifold $ ( \mathbf R _ {+} ^ {3} \cup \Omega ( \Gamma )) / \Gamma $, for which $ \Omega ( \Gamma ) / \Gamma $ is the boundary, turn out to be very suitable.

The analytic approach to the theory of Kleinian groups is connected with the study of automorphic forms (cf. Automorphic form). If $ \Gamma $ is a non-elementary Kleinian group and $ \infty \in \Omega ( \Gamma ) $, then for integers $ q \geq 2 $ the series $ \sum _ {\gamma \in \Gamma } | \gamma ^ \prime ( z) | ^ {q} $ converges (at the points $ z \in \Omega ( \Gamma ) $ with $ \Gamma _ {z} = \{ J \} $); the corresponding Poincaré theta-series

$$ \sum _ {\gamma \in \Gamma } f ( \gamma ( z) ) \gamma ^ {\prime q } ( z) , $$

where $ f $ is a meromorphic function in $ \Omega ( \Gamma ) $, give automorphic forms of weight $ ( - 2 q ) $. For finitely-generated Kleinian groups the dimension of the space of such forms can be calculated by means of the Riemann–Roch theorem. The geometric structure of such groups $ \Gamma $ is described by the Ahlfors theorem, according to which the space $ \Omega ( \Gamma ) / \Gamma $ for these $ \Gamma $ consists of a finite number of surfaces $ S _ {1} \dots S _ {n} $ of finite type, and $ \pi ^ {-} 1 $ can be ramified over each $ S _ {j} $ only at a finite number of points. This result admits quantitative refinements. Homological methods are also used, based on the study of the action of $ \Gamma $ in vector spaces of polynomials (see [5]). Methods of the theory of quasi-conformal mapping [6], [7] play an essential role in the theory of Kleinian groups on the plane. In particular, the theory of deformations of Kleinian groups, closely related to the theory of moduli of Riemann surfaces (see Moduli of a Riemann surface and Riemann surfaces, conformal classes of) relies on these methods. Along these lines certain new classes of Kleinian groups have emerged. Meanwhile, however, no complete classification has been obtained, not even for finitely-generated Kleinian groups.

By comparison with planar ones, Kleinian groups in a multi-dimensional Euclidean space $ \mathbf R ^ {n} $, $ n > 2 $, defined as properly-discontinuous subgroups of the group of conformal automorphisms of the space $ \overline{ {\mathbf R ^ {n} }}\; = \mathbf R ^ {n} \cup \{ \infty \} $, have been much less extensively studied; here completely new phenomena occur.

References

[1] H. Poincaré, "Mémoire sur les groupes kleinéens" Acta Math. , 3 (1883) pp. 49–92
[2] F. Klein, "Neue Beiträge zur Riemannschen Funktionentheorie" Math. Ann. , 21 (1883) pp. 141–218
[3] L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951)
[4] J. Lehner, "Discontinuous groups and automorphic functions" , Amer. Math. Soc. (1964)
[5] I. Kra, "Automorphic forms and Kleinian groups" , Benjamin (1972)
[6] S.L. Krushkal', "Quasi-conformal mappings and Riemann surfaces" , Winston & Wiley (1979) (Translated from Russian)
[7] L. Bers (ed.) I. Kra (ed.) , A crash course on Kleinian groups , Lect. notes in math. , 400 , Springer (1974)
[8] B. Maskit, "Kleinian groups" , Springer (1988)

Comments

For the definitions of loxodromic, elliptic, hyperbolic $ \dots $ fractional-linear transformations, cf. Fractional-linear mapping.

One of the "quantitative refinements" , or, more precisely, a quantitative extension, of the Ahlfors finiteness theorem is the Bers area inequality:

$$ \frac{1}{2 \pi } \{ \textrm{ hyperbolic area of } \Omega \setminus \Gamma \} \leq 2 ( N- 1) , $$

where $ N $ is the (minimum) number of generators of $ \Gamma $.

References

[a1] I. Kra (ed.) B. Maskit (ed.) , Riemann Surfaces and Related Topics (Proc. 1978 Stony Brook Conf.) , Princeton Univ. Press (1981)
[a2] H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980)
How to Cite This Entry:
Kleinian group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kleinian_group&oldid=18284
This article was adapted from an original article by S.L. Krushkal' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article