Kleinian group

From Wikipedia, the free encyclopedia

In mathematics, a Kleinian group, named after Felix Klein, is a finitely generated discrete group Γ of orientation preserving conformal (i.e. angle-preserving) maps of the open unit ball B³ in to itself. Some mathematicians extend the definition Kleinian groups to allow orientation reversing conformal maps.

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, which acts by Möbius transformations on the Riemann sphere. Classically, a Kleinian group was required to act properly discontinuously on an open subset of the Riemann sphere, but modern usage allows any discrete subgroup.

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 complement is called the domain of discontinuity. Ahlfors' finiteness theorem implies that Ω(G) / G is a Riemann surface orbifold of finite type.

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 projective 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: The Vision of Felix Klein, by David Mumford, Caroline Series and David Wright)
• Adam Majewski, Fractals - Limit sets of kleinian groups, (undated) (links and additional references).
• Pablo Arés Gastesi, Kleinian and Fuchsian groups.
• Jos Leys, The Kleinian galleries (undated). (An art gallery of fractals based on Kleinian groups).

[edit] See also

• Klein four-group
• Schottky group