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 $\backslash pi\_1$ of a hyperbolic 3-manifold, then the quotient space $H^3/\backslash Gamma$ 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 $\backslash bar\{B\}^3$.

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

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 $S^2\_\backslash infty$, which are Möbius transformations. There are isomorphisms

$$

\mbox{Mob}(S^2_\infty) \cong \mbox{Conf}(B^3) \cong \mbox{Isom}(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 $CP^1$.

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

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The canonical hyperbolic metric on the unit ball $B^3$ is given by

$ds^2=\; \backslash frac\{4\; \backslash left|\; dx\; \backslash right|^2\; \}\{\backslash left(\; 1-|x|^2\; \backslash right)^2\}$

for $x\backslash in\; B^3$.

References

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• 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).

See also

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• Klein_four-group
• Schottky group

Discrete groups | Lie groups | Automorphic forms

Grupa kleinowska | Группа Клейна