In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C)|, is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by . has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball in . The group of Möbius transformations is also related as the non-orientation-preserving isometry group of , . So, 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 Felix Klein and Henri Poincaré, who named them after Felix Klein. The special case of Schottky groups had been studied a few years earlier, in 1877, by Friedrich Schottky.

Definitions

One modern definition of Kleinian group is as a group which acts on the 3-ball <math>B^3</math> as a discrete group of hyperbolic isometries. Hyperbolic 3-space has a natural boundary; in the ball model, this can be identified with the 2-sphere. We call it the sphere at infinity, and denote it by <math>S^2_\infty</math>. A hyperbolic isometry extends to a conformal homeomorphism of the sphere at infinity (and conversely, every conformal homeomorphism on the sphere at infinity extends uniquely to a hyperbolic isometry on the ball by Poincaré extension). It is a standard result from complex analysis that conformal homeomorphisms on the Riemann sphere are exactly the Möbius transformations, which can further be identified as elements of the projective linear group PGL(2,C).

  • A Kleinian group is said to be of type 1 if the limit set is the whole Riemann sphere, and of type 2 otherwise.

Examples

  • The Maskit slice through the moduli space of Kleinian groups

Bianchi groups

A Bianchi group is a Kleinian group of the form PSL(2, O<sub>d</sub>), where <math>\mathcal{O}_d</math> is the ring of integers of the imaginary quadratic field <math>\mathbb{Q}(\sqrt{-d})</math> 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.

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

thumb|right|Limit set of a quasi-Fuchsian group

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 the first kind, and otherwise it is said to be of the second kind.

Schottky groups

Let C<sub>i</sub> 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<sup>3</sup>/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, and the first explicit example was found by Jørgensen. gave examples of doubly degenerate groups and space-filling curves associated to pseudo-Anosov maps.

See also

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

References

  • A picture of the limit set of a quasi-Fuchsian group from .
  • A picture of the limit set of a Kleinian group from . 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