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 B³ in R³.

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 π₁ of a three-dimensional hyperbolic 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 B³ 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 Λ(G).

The unit ball B³ with its conformal structure is the Poincare model of hyperbolic 3-space. When we think of it metrically, it is denoted H³. The set of conformal self-maps of B³ becomes the set of isometries (i.e. distance-preserving maps) of H³ under this identification. Such maps restrict to conformal self-maps of , which are Mobius transformations. There are isomorphisms

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

PSL(2,C)

via the usual identification of the unit sphere with the complex projective line CP¹.

Example

Reflection groups . Let C_i 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 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 Schottky group .

Example

Crystallographic groups. Let T 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 B³ is given by

for .

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