Kleinian 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 orientation-preserving isometries of 3-dimensional hyperbolic space H3, and as orientation-preserving conformal maps of the open unit ball "B"3 in R3 to itself. Therefore, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.
History.
The theory of general Kleinian groups was founded by and , 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 non-empty open subset of the Riemann sphere, but modern usage allows any discrete subgroup.
When Γ is isomorphic to the fundamental group formula_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 "B"3 have finite stabilizers, and discrete orbits under the group Γ. But the orbit Γ"p" of a point "p" will typically accumulate on the boundary of the closed ball formula_2.
 The boundary of the closed ball is called the sphere at infinity, and is denoted formula_3. The set of accumulation points of Γ"p" in formula_3 is called the limit set of Γ, and usually denoted formula_5. The complement formula_6 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 formula_7 is a Riemann surface orbifold of finite type.
The unit ball "B"3 with its conformal structure is the Poincaré model of hyperbolic 3-space. When we think of it metrically, with metric
it is a model of 3-dimensional hyperbolic space H3. The set of conformal self-maps of "B"3 becomes the set of isometries (i.e. distance-preserving maps) of H3 under this identification. Such maps restrict to conformal self-maps of formula_3, which are Möbius transformations. There are isomorphisms
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).
Variations.
There are some variations of the definition of a Kleinian group: sometimes 
Kleinian groups are allowed to be subgroups of PSL(2, C).2 (that is, of 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.
Examples.
Bianchi groups.
A Bianchi group is a Kleinian group of the form PSL(2, "O""d"), where formula_11 is the ring of integers of the imaginary quadratic field formula_12 for d a positive square-free 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 PSL(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, every Kleinian group preserving a circle or straight line in the Riemann sphere is conjugate to a Fuchsian group.
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 if 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 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 , and the first explicit example was found by Jørgensen. gave examples of doubly degenerate groups and space-filling curves associated to pseudo-Anosov maps.