Kleinian group

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)