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 <math>B^3</math> in <math>R^3</math> 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 <math>\pi_1</math> of a hyperbolic 3-manifold, then the quotient space <math>H^3/\Gamma</math> 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 <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 Poincaré 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 Möbius transformations. There are isomorphisms
- <math>
\mbox{Mob}(S^2_\infty) \cong \mbox{Conf}(B^3) \cong \mbox{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>.
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[edit] 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.
[edit] 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.
[edit] Metric
The canonical hyperbolic metric on the unit ball <math>B^3</math> is given by
- <math>ds^2= \frac{4 \left| dx \right|^2 }{\left( 1-|x|^2 \right)^2}</math>
for <math>x\in B^3</math>.
[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).
