Kleinian group

Encyclopedia : K : KL : KLE : 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 [B^3] in [R^3].

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 [\pi_1] of a hyperbolic 3-manifold, then the quotient space [H^3/\Gamma] 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 [G]. But the orbit [Gp] of a point [p] will typically accumulate on the boundary of the closed ball [\bar^3].

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

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

[\mbox(S^2_\infty) \cong \mbox(B^3) \cong \mbox(H^3)]

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 [CP^1].

Contents

0.1 Example
0.2 Example

1 Metric
2 References
3 See also

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 [H^3/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 [B^3] is given by

[ds^2= \frac]

for [x\in B^3].

References

Michael Kapovich, Hyperbolic Manifolds and Discrete Groups, (2000) Birkhauser, Boston ISBN 0-817-63904-7
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
David Wright, [Welcome to the Indra's Pearls Web Site], (2003) (A website devoted to the book Indra's Pearls, by David Mumford, Caroline Series and David Wright)
Adam Majewski, [Fractals - Limit sets of kleinian groups], (undated) (links and additional references).
Jos Leys, [The Kleinian galleries] (undated). (An art gallery of fractals based on Kleinian groups).

Kleinian group

Example

Example

Metric

References

See also