Kleinian group
Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

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 orientationpreserving isometries of 3dimensional hyperbolic space H^{3}, and as orientation preserving conformal maps of the open unit ball B^{3} in R^{3} 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 nonempty 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 years earlier, in 1877, 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 nonempty open subset of the Riemann sphere, but modern usage allows any discrete subgroup.
When Γ is isomorphic to the fundamental group of a hyperbolic 3manifold, then the quotient space H^{3}/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 B^{3} 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 .
The boundary of the closed ball is called the sphere at infinity, and is denoted . The set of accumulation points of Gp in is called the limit set of G, and usually denoted . The complement 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 is a Riemann surface orbifold of finite type.
The unit ball B^{3} with its conformal structure is the Poincaré model of hyperbolic 3space. When we think of it metrically, with metric
it is a model of 3dimensional hyperbolic space H^{3}. The set of conformal selfmaps of B^{3} becomes the set of isometries (i.e. distancepreserving maps) of H^{3} under this identification. Such maps restrict to conformal selfmaps of , which are Möbius transformations. There are isomorphisms
The subgroups of these groups consisting of orientationpreserving 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 P^{1}(C).
Finiteness conditions
 A Kleinian group is said to be of finite type if its region of discontinuity has a finite number of orbits of components under the group action, 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 H^{3}/Γ has finite volume. Any Kleinian group of finite covolume is finitely generated.
 A Kleinian group is called geometrically finite if it has a fundamental polyhedron (in hyperbolic 3space) 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 norm 1 elements 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 H^{3}/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, O_{d}), where is the ring of integers of the imaginary quadratic field for d a positive squarefree integer.
Elementary and reducible Kleinian groups
A Kleinian group is called elementary if its limit set is finite, in which case the limit set has 0, 1, or 2 points. Examples of elementary 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.
QuasiFuchsian groups
A Kleinian group that preserves a Jordan curve is called a quasiFuchsian group. When the Jordan curve is a circle or a straight line these are just conjugate to Fuchsian groups under conformal transformations. Finitely generated quasiFuchsian groups are conjugate to Fuchsian groups under quasiconformal 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 C_{i} 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 H^{3}/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 3space. The group of symmetries of the tessellation is a Kleinian group.
Fundamental groups of hyperbolic 3manifolds
The fundamental group of any oriented hyperbolic 3manifold 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 3manifold.
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 quasiFuchsian 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 spacefilling 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 spacefilling curves associated to pseudoAnosov maps.
See also
 Ahlfors measure conjecture
 Density theorem for Kleinian groups
 Ending lamination theorem
 Tameness theorem (Marden's conjecture)
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 0003486X, JSTOR 1970638, MR 0297992
 Bers, Lipman; Kra, Irwin, eds. (1974), A crash course on Kleinian groups, Lecture Notes in Mathematics, 400, Berlin, New York: SpringerVerlag, doi:10.1007/BFb0065671, MR 0346152
 Cannon, James W.; Thurston, William P. (2007) [1982], "Group invariant Peano curves", Geometry & Topology, 11: 1315–1355, doi:10.2140/gt.2007.11.1315, ISSN 14653060, MR 2326947
 Fricke, Robert; Klein, Felix (1897), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen (in German), Leipzig: B. G. Teubner, ISBN 9781429705516, JFM 28.0334.01
 Fricke, Robert; Klein, Felix (1912), Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen (in German), Leipzig: B. G. Teubner., ISBN 9781429705523, 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, MR 0521758
 Kapovich, Michael (2009) [2001], Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/9780817649135, ISBN 9780817649128, MR 1792613
 Klein, Felix (1883), "Neue Beiträge zur Riemann'schen Functionentheorie", Mathematische Annalen, Springer Berlin / Heidelberg, 21 (2): 141–218, doi:10.1007/BF01442920, ISSN 00255831, JFM 15.0351.01
 Kra, Irwin (1972), Automorphic forms and Kleinian groups, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., MR 0357775
 Krushkal, S.L. (2001), "K/k055520", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Maclachlan, Colin; Reid, Alan W. (2003), The arithmetic of hyperbolic 3manifolds, Graduate Texts in Mathematics, 219, Berlin, New York: SpringerVerlag, ISBN 9780387983868, MR 1937957
 Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 287, Berlin, New York: SpringerVerlag, ISBN 9783540177463, MR 959135
 Matsuzaki, Katsuhiko; Taniguchi, Masahiko (1998), Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 9780198500629, MR 1638795
 Mumford, David; Series, Caroline; Wright, David (2002), Indra's pearls, Cambridge University Press, ISBN 9780521352536, MR 1913879
 Poincaré, Henri (1883), "Mémoire sur Les groupes kleinéens", Acta Mathematica, Springer Netherlands, 3: 49–92, doi:10.1007/BF02422441, ISSN 00015962, 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 00494704, MR 2227047
 Thurston, William (1980), The geometry and topology of threemanifolds, Princeton lecture notes
 Thurston, William P. (1982), "Threedimensional manifolds, Kleinian groups and hyperbolic geometry", American Mathematical Society. Bulletin. New Series, 6 (3): 357–381, doi:10.1090/S027309791982150030, ISSN 00029904, MR 648524
External links
 A picture of the limit set of a quasiFuchsian 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
 Animations of Kleinian group limit sets
 Images related to Kleinian groups by McMullen
 Weisstein, Eric W. "Kleinian Group". MathWorld.