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In mathematics, a Kleinian group is a finitely generated discrete group of conformal (i.e. angle-preserving)
self-maps of the open unit ball <math>B^3<math> in <math>R^3<math>.
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
Mobius transformations. There are isomorphisms
- <math>
Mob(S^2_\infty) \cong Conf(B^3) \cong 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.
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