In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable...

is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian...

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. Kleinian groups...

In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently...

states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups, and was independently proven by Ohshika...

mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders...

subgroup Γ {\displaystyle \Gamma } , a Kleinian group, 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 Kleinian group, first studied by Friedrich Schottky (1877). Fix some point p on the Riemann sphere...

quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). Hyperbolic 3-manifolds of finite volume have a particular...

or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups. For example, Cannon & Thurston (2007) showed that the circle at infinity...

In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If...

The term "fantasy" became a central issue with the development of the Kleinian group as a distinctive strand within the British Psycho-Analytical Society...

states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups, and was independently proven by Ohshika...

The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group. For Euclidean symmetry transformations rather...

{R} )} are obtained in this way (up to commensurability). Arithmetic Kleinian groups are constructed similarly except that F {\displaystyle F} is required...

surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important...

conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture...

discrete group of isometries (Ratcliffe 1994, 12.7). Density theorem for Kleinian groups Greenberg, L. (1966), "Fundamental polyhedra for kleinian groups", Annals...

discrete groups, Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467, ISBN 978-0-8176-4912-8, Zbl 1180.57001 Maskit, Bernard (1988), Kleinian groups, Grundlehren...

Henrik Abel in 1827. Bianchi group Classical modular curve Fuchsian group J-invariant Kleinian group Mapping class group Minkowski's question-mark function...

of Cannon and Thurston can be thought of in terms of Kleinian representations of the surface group H = π 1 ( S ) {\displaystyle H=\pi _{1}(S)} . As a subgroup...

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...

theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The...

Modular group Congruence subgroup Kleinian group Discrete Heisenberg group Clifford–Klein form Borel subgroup Arithmetic group Dunkl operator Modular form Langlands...

Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. 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 Kleinian groups, for...

hyperbolic groups. The rank problem is decidable for torsion-free Kleinian groups. The rank problem is open for finitely generated virtually abelian groups (that...

eventually Donald Winnicott and Paula Heimann, who moved away from the Kleinian group'. Subsequently, 'some new refugees, notably Michael Balint and Michael...

mathematical theory of Kleinian groups, Jørgensen's inequality is an inequality involving the traces of elements of a Kleinian group, proved by Troels Jørgensen (1976)...

attributed to Felix Klein who used it to study subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalently...