Kleinian group

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In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i.e. angle-preserving) maps of the open unit ball B³ in R³ to itself.

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 hyperbolic 3-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 Poincaré 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 Möbius 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.

Contents 1 Example 2 Example 3 Metric 4 References 5 See also

[edit] 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.

[edit] Example

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

[edit] Metric

The canonical hyperbolic metric on the unit ball B³ is given by

for .

[edit] References

• Michael Kapovich, Hyperbolic Manifolds and Discrete Groups, (2000) Birkhauser, Boston ISBN 0-8176-3904-7
• 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).

[edit] See also

• Klein four-group
• Schottky group