Kleinian group

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 $B^3$ in $mathbb\{R\}^3$ to itself. Some mathematiciansextend 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 "PGL"₂("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 $pi\_1$ 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, anddiscrete 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 $S^2\_infty$.The set of accumulation points of "Gp" in $S^2\_infty$ is called the"limit set" of $G$, and usually denoted $Lambda(G)$. The complement$Omega(G)=S^2\_infty\; -\; Lambda(G)$ is called the "domain of discontinuity". Ahlfors'finiteness theorem implies that $Omega(G)/G$ is a Riemann surface orbifold of finite type.

The unit ball $B^3$ with its conformal structure is the Poincaré modelof 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

:$mbox\{Mob\}(S^2\_infty)\; cong\; mbox\{Conf\}(B^3)\; cong\; mbox\{Isom\}(H^3)$

The subgroups of these groups consisting of orientation-preserving transformations areall isomorphic to the projective 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 collectionof disjoint closed disks. The group generated by inversion in each circle is a Kleinian group. Thelimit set is a Cantor set, and the quotient $H^3/G$ is a mirror orbifold with underlyingspace 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 hyperbolic3-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:$ds^2=\; frac\{4\; left|\; dx\; ight|^2\; \}\{left(\; 1-|x|^2\; ight)^2\}$for $xin\; B^3$.

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, " [http://klein.math.okstate.edu/IndrasPearls/ Welcome to the Indra's Pearls Web Site] ", (2003) "(A website devoted to the book "", by David Mumford, Caroline Series and David Wright)"
* Adam Majewski, " [http://republika.pl/fraktal/kleinian.html Fractals - Limit sets of kleinian groups] ", (undated) "(links and additional references)".
* Pablo Arés Gastesi, " [http://www.math.tifr.res.in/~pablo/teichmuller/node5.html Kleinian and Fuchsian groups] ".
* Jos Leys, " [http://www.josleys.com/kleinianindex.htm The Kleinian galleries] " (undated). "(An art gallery of fractals based on Kleinian groups)".

See also

*Klein four-group
*Schottky group