Kleinian group

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 Gpof a point p will typically accumulate on the boundary of the
closed ball ar{B}^3.

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


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