Not to be confused with Klein four-group, the direct product of two copies of the cyclic group of order 2.
Algebraic structure → Group theory Group theory
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In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B3 in R3. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H3, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.
In mathematics, a Kleiniangroup is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable...
is allowed to be a Kleiniangroup (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian...
In the mathematical theory of Kleiniangroups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently...
group PSL(2,Z) is thought of as a discrete subgroup of PSL(2,R). The modular group is a lattice in PSL(2,R), but it is not cocompact. Kleiniangroups...
states that every finitely generated Kleiniangroup is an algebraic limit of geometrically finite Kleiniangroups, and was independently proven by Ohshika...
subgroup Γ {\displaystyle \Gamma } , a Kleiniangroup, is defined so that it is isomorphic to the fundamental group π 1 ( N ) {\displaystyle \pi _{1}(N)}...
In mathematics, a Schottky group is a special sort of Kleiniangroup, first studied by Friedrich Schottky (1877). Fix some point p on the Riemann sphere...
states that every finitely generated Kleiniangroup is an algebraic limit of geometrically finite Kleiniangroups, and was independently proven by Ohshika...
{R} )} are obtained in this way (up to commensurability). Arithmetic Kleiniangroups are constructed similarly except that F {\displaystyle F} is required...
conjecture, now a theorem, states that the limit set of a finitely-generated Kleiniangroup is either the whole Riemann sphere, or has measure 0. The conjecture...
discrete group of isometries (Ratcliffe 1994, 12.7). Density theorem for Kleiniangroups Greenberg, L. (1966), "Fundamental polyhedra for kleiniangroups", Annals...
discrete groups, Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467, ISBN 978-0-8176-4912-8, Zbl 1180.57001 Maskit, Bernard (1988), Kleiniangroups, Grundlehren...
Modular group Congruence subgroup Kleiniangroup Discrete Heisenberg group Clifford–Klein form Borel subgroup Arithmetic group Dunkl operator Modular form Langlands...
field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects...
Henrik Abel in 1827. Bianchi group Classical modular curve Fuchsian group J-invariant Kleiniangroup Mapping class group Minkowski's question-mark function...
theory of Kleiniangroups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleiniangroup. The...
Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleiniangroups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations...
corresponding group is called isometry group of X. If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleiniangroups, for...
hyperbolic groups. The rank problem is decidable for torsion-free Kleiniangroups. The rank problem is open for finitely generated virtually abelian groups (that...
corresponding group is an example of a hyperbolic triangle group. Poincaré also gave a 3-dimensional version of this result for Kleiniangroups: in this case...
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