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In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C). The group PSL(2, C) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space H3, and as orientation preserving conformal maps of the open unit ball B3 in R3 to itself. Therefore a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

There are some variations of the definition of a Kleinian group: sometimes Kleinian groups are allowed to be subgroups of PSL(2, C).2 (PSL(2, C) extended by complex conjugations), in other words to have orientation reversing elements, and sometimes they are assumed to be finitely generated, and sometimes they are required to act properly discontinuously on a non-empty open subset of the Riemann sphere. A Kleinian group is said to be of type 1 if the limit set is the whole Riemann sphere, and of type 2 otherwise.

The theory of general Kleinian groups was founded by Felix Klein (1883) and Henri Poincaré (1883), who named them after Felix Klein. The special case of Schottky groups had been studied a few year before by Schottky.

Definitions

By considering the ball's boundary, a Kleinian group can also be defined as a subgroup Γ of PGL(2,C), the complex projective linear group, which acts by Möbius transformations on the Riemann sphere. Classically, a Kleinian group was required to act properly discontinuously on a non-empty open subset of the Riemann sphere, but modern usage allows any discrete subgroup.

When Γ is isomorphic to the fundamental group \pi_1 of a hyperbolic 3-manifold, then the quotient space H3/Γ becomes a Kleinian model of the manifold. Many authors use the terms Kleinian model and Kleinian group interchangeably, letting the one stand for the other.

Discreteness implies points in B3 have finite stabilizers, and discrete orbits under the group G. But the orbit Gp of a point p will typically accumulate on the boundary of the closed ball $$\bar{B}^3.$$
An Apollonian gasket is an example of a limit set of a Kleinian group

The boundary of the closed ball is called the sphere at infinity, and is denoted S^2_\infty. The set of accumulation points of Gp in $$S^2_\infty$$ is called the limit set of G, and usually denoted $$\Lambda(G)$$. The complement $$\Omega(G)=S^2_\infty - \Lambda(G)$$ is called the domain of discontinuity or the ordinary set or the regular set. Ahlfors' finiteness theorem implies that if the group is finitely generated then $$\Omega(G)/G$$ is a Riemann surface orbifold of finite type.

The unit ball B3 with its conformal structure is the Poincaré model of hyperbolic 3-space. When we think of it metrically, with metric

$$ds^2= \frac{4 \left| dx \right|^2 }{\left( 1-|x|^2 \right)^2}$$

it is a model of 3-dimensional hyperbolic space H3. The set of conformal self-maps of B3 becomes the set of isometries (i.e. distance-preserving maps) of H3 under this identification. Such maps restrict to conformal self-maps of S^2_\infty, which are Möbius transformations. There are isomorphisms

$$\mbox{Mob}(S^2_\infty) \cong \mbox{Conf}(B^3) \cong \mbox{Isom}(\mathbf{H}^3).$$

The subgroups of these groups consisting of orientation-preserving transformations are all isomorphic to the projective matrix group: PSL(2,C) via the usual identification of the unit sphere with the complex projective line P1(C).
Finiteness conditions

A Kleinian group is said to be of finite type if its region of discontinuity has a finite number of components, and the quotient of each component by its stabilizer is a compact Riemann surface with finitely many points removed, and the covering is ramified at finitely many points.
A Kleinian group is called finitely generated if it has a finite number of generators. The Ahlfors finiteness theorem says that such a group is of finite type.
A Kleinian group Γ has finite covolume if H3/Γ has finite volume. Any Kleinian group of finite covolume is finitely generated.
A Kleinian group is called geometrically finite is it has a fundamental polyhedron (in hyperbolic 3-space) with finitely many sides. Ahlfors showed that if the limit set is not the whole Riemann sphere then it has measure 0.
A Kleinian group Γ is called arithmetic if it is commensurable with the group of units of an order of quaternion algebra A ramified at all real places over a number field k with exactly one complex place. Arithmetic Kleinian groups have finite covolume.
A Kleinian group Γ is called cocompact if H3/Γ is compact, or equivalently SL(2, C)/Γ is compact. Cocompact Kleinian groups have finite covolume.

A Kleinian group is called topologically tame if it is finitely generated and its hyperbolic manifold is homeomorphic to the interior of a compact manifold with boundary.
A Kleinian group is called geometrically tame if its ends are either geometrically finite or simply degenerate (Thurston 1980).

Examples
Bianchi groups

A Bianchi group is a Kleinian group of the form PSL(2, Od), where d is a positive square-free integer.
Elementary and reducible Kleinian groups

A Kleinian group is called degenerate if its limit set is finite, in which case the limit set has 0, 1, or 2 points. Examples of degenerate Kleinian groups include finite Kleinian groups (with empty limit set) and infinite cyclic Kleinian groups.

A Kleinian group is called reducible if all elements have a common fixed point on the Riemann sphere. Reducible Kleinian groups are elementary, but some elementary finite Kleinian groups are not reducible.
Fuchsian groups

Any Fuchsian group (a discrete subgroup of SL(2, R)) is a Kleinian group, and conversely any Kleinian group preserving the real line (in its action on the Riemann sphere) is a Fuchsian group. More generally, any Kleinian group preserving a circle or straight line in the Riemann sphere is conjugate to a Fuchsian group.
Koebe groups

A factor of a Kleinian group G is a subgroup H maximal subject to the following properties:
H has a simply connected invariant component D
A conjugate of an element h of H by a conformal bijection is parabolic or elliptic if and only if h is.
Any parabolic element of G fixing a boundary point of D is in H.
A Kleinian group is called a Koebe group if all its factors are elementary or Fuchsian.

Quasi-Fuchsian groups

A Kleinian group that preserves a Jordan curve is called a quasi-Fuchsian group. When the Jordan curve is a circle or a straight line these are just conjugate to Fuchsian groups under conformal transformations. Finitely generated quasi-Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal transformations. The limit set is contained in the invariant Jordan curve, and it is equal to the Jordan curve the group is said to be of type one, and otherwise it is said to be of type 2.
Schottky groups

Let Ci be the boundary circles of a finite collection of disjoint closed disks. The group generated by inversion in each circle has limit set a Cantor set, and the quotient H3/G is a mirror orbifold with underlying space a ball. It is double covered by a handlebody; the corresponding index 2 subgroup is a Kleinian group called a Schottky group.
Crystallographic groups

Let T be a periodic tessellation of hyperbolic 3-space. The group of symmetries of the tessellation is a Kleinian group.
Fundamental groups of hyperbolic 3-manifolds

The fundamental group of any oriented hyperbolic 3-manifold is a Kleinian group. There are many examples of these, such as the complement of a figure 8 knot or the Seifert–Weber space. Conversely if a Kleinian group has no nontrivial torsion elements then it is the fundamental group of a hyperbolic 3-manifold.
Degenerate Kleinian groups

A Kleinian group is called degenerate if it is not elementary and its limit set is simply connected. Such groups can be constructed by taking a suitable limit of quasi-Fuchsian groups such that one of the two components of the regular points contracts down to the empty set; these groups are called singly degenerate. If both components of the regular set contract down to the empty set, then the limit set becomes a space-filling curve and the group is called doubly degenerate. The existence of degenerate Kleinian groups was first shown indirectly by Bers (1970), and the first explicit example was found by Jørgensen. Cannon & Thurston (2007) gave examples of doubly degenerate groups and space-filling curves associated to pseudo-Anosov maps.

Tameness theorem (Marden's conjecture)
Ahlfors measure conjecture
density theorem for Kleinian groups
Ending lamination theorem

References

Bers, Lipman (1970), "On boundaries of Teichmüller spaces and on Kleinian groups. I", Annals of Mathematics. Second Series 91 (3): 570–600, doi:10.2307/1970638, ISSN 0003-486X, JSTOR 1970638, MR0297992
Bers, Lipman; Kra, Irwin, eds. (1974), A crash course on Kleinian groups, Lecture Notes in Mathematics, 400, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0065671, MR0346152
Cannon, James W.; Thurston, William P. (2007) , "Group invariant Peano curves", Geometry & Topology 11: 1315–1355, doi:10.2140/gt.2007.11.1315, ISSN 1465-3060, MR2326947
Fricke, Robert; Klein, Felix (1897) (in German), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen, Leipzig: B. G. Teubner, ISBN 978-1-4297-0551-6, JFM 28.0334.01
Fricke, Robert; Klein, Felix (1912) (in German), Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen, Leipzig: B. G. Teubner., ISBN 978-1-4297-0552-3, JFM 32.0430.01
Harvey, William James (1978), "Kleinian groups (a survey).", Séminaire Bourbaki, 29e année (1976/77), Exp. No. 491, Lecture Notes in Math., 677, Springer, Berlin, pp. 30–45,, doi:10.1007/BFb0070752, MR0521758
Kapovich, Michael (2009) , Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8, MR1792613
Klein, Felix (1883), "Neue Beiträge zur Riemann'schen Functionentheorie", Mathematische Annalen (Springer Berlin / Heidelberg) 21 (2): 141–218, doi:10.1007/BF01442920, ISSN 0025-5831, JFM 15.0351.01
Kra, Irwin (1972), Automorphic forms and Kleinian groups, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., MR0357775
Krushkal, S.L. (2001), "Kleinian group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
Maclachlan, Colin; Reid, Alan W. (2003), The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, 219, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98386-8, MR1937957
Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 287, Berlin, New York: Springer-Verlag, ISBN 978-3-540-17746-3, MR959135
Matsuzaki, Katsuhiko; Taniguchi, Masahiko (1998), Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-850062-9, MR1638795
Mumford, David; Series, Caroline; Wright, David (2002), Indra's pearls, Cambridge University Press, ISBN 978-0-521-35253-6, MR1913879
Poincaré, Henri (1883), "Mémoire sur Les groupes kleinéens", Acta Mathematica (Springer Netherlands) 3: 49–92, doi:10.1007/BF02422441, ISSN 0001-5962, JFM 15.0348.02
Series, Caroline (2005), "A crash course on Kleinian groups", Rendiconti dell'Istituto di Matematica dell'Università di Trieste 37 (1): 1–38, ISSN 0049-4704, MR2227047
Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes
Thurston, William P. (1982), "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", American Mathematical Society. Bulletin. New Series 6 (3): 357–381, doi:10.1090/S0273-0979-1982-15003-0, ISSN 0002-9904, MR648524

A picture of the limit set of a quasi-Fuchsian group from (Fricke & Klein 1897, p. 418).
A picture of the limit set of a Kleinian group from (Fricke & Klein 1897, p. 440). This was one of the first pictures of a limit set. A computer drawing of the same limit set
Images of limit sets of Kleinian groups on google
Animations of Kleinian group limit sets
Images related to Kleinian groups by McMullen
A program for drawing pictures related to Kleinian groups, by David Wright.