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 B³ in R³. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... some unit spheres In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used. ...

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. The projective linear group of a vector space V over a field F is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. The projective special linear... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action. ...

When Γ is isomorphic to the fundamental group π₁ of a three-dimensional hyperbolic 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. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where Γ is a discrete subgroup of PSL(2,C). ...

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 . This article is about mathematical concept. ... The word discrete comes from the Latin word discretus which means separate. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

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). In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...

The unit ball B³ with its conformal structure is the Poincare 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 In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...

The subgroups of these groups consisting of orientation-preserving transformations are all isomorphic to the matrix group In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

PSL(2,C)

via the usual identification of the unit sphere with the complex projective line CP¹. In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...

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. The symmetry groups of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) are generated by reflections and rotations in space. ... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In the mathematical field of geometric topology, a handlebody is a particular kind of manifold. ...

Example

Crystallographic groups. Let T be a periodic tessellation of hyperbolic 3-space. The group of symmetries of the tessellation is a Kleinian group. A crystallographic group is a topologically discrete subgroup of the group of isometries of some geometric space (typically, not necessarily a Euclidean space) with a compact fundamental domain. ... Periodicity is the quality of occurring at regular intervals (e. ... A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ...

Metric

The canonical hyperbolic metric on the unit ball B³ is given by See: International System of Units, colloquially called the Metric System, and also metrication. ...

for .

References

• 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

Klein_four-group In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V), named after Felix Klein, is the group C2 × C2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). ...