Kleinian group

In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i.e. angle-preserving) self-maps of the open unit ball <math>B^3<math> in <math>R^3<math>.

By considering the ball's boundary, a Kleinian group can also be defined as a subgroup Γ of PGL₂(C), the complex projective linear group, whose action by Möbius transformations at some point of the Riemann sphere is freely discontinuous.

When Γ is isomorphic to the fundamental group <math>\pi_1<math> of a three-dimensional hyperbolic manifold, then the quotient space <math>H^3/\Gamma<math> 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 <math>B^3<math> have finite stabilizers, and discrete orbits under the group <math>G<math>. But the orbit <math>Gp<math> of a point <math>p<math> will typically accumulate on the boundary of the closed ball <math>\bar{B}^3<math>.

The boundary of the closed ball is called the sphere at infinity, and is denoted <math>S^2_\infty<math>. The set of accumulation points of Gp in <math>S^2_\infty<math> is called the limit set of <math>G<math>, and usually denoted <math>\Lambda(G)<math>.

The unit ball <math>B^3<math> with its conformal structure is the Poincare model of hyperbolic 3-space. When we think of it metrically, it is denoted <math>H^3<math>. The set of conformal self-maps of <math>B^3<math> becomes the set of isometries (i.e. distance-preserving maps) of <math>H^3<math> under this identification. Such maps restrict to conformal self-maps of <math>S^2_\infty<math>, which are Möbius transformations. There are isomorphisms

<math>

\mbox{Mob}(S^2_\infty) \cong \mbox{Conf}(B^3) \cong \mbox{Isom}(H^3) <math>

The subgroups of these groups consisting of orientation-preserving transformations are all isomorphic to the matrix group

<math>PSL(2,C)<math>

via the usual identification of the unit sphere with the complex projective line <math>CP^1<math>.

Example

Reflection groups. Let <math>C_i<math> be the boundary circles of a finite collection of disjoint closed disks. The group generated by inversion in each circle is a Kleinian group. The limit set is a Cantor set, and the quotient <math>H^3/G<math> is a mirror orbifold with underlying space a ball. It is double covered by a handlebody; the corresponding index 2 subgroup is a Schottky group.

Example

Crystallographic groups. Let <math>T<math> be a periodic tessellation of hyperbolic 3-space. The group of symmetries of the tessellation is a Kleinian group.

Metric

The canonical hyperbolic metric on the unit ball <math>B^3<math> is given by

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

for <math>x\in B^3<math>.

References

• Bernard Maskit, Kleinian Groups, (1988) Springer-Verlag New York ISBN 0–387–17746–9
• Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyberbolic Manifolds and Kleinian Groups, (1998) Clarendon Press, Oxford ISBN 0–19–850062–9