# Category: Kleinian groups

Tameness theorem
In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a
Double limit theorem
In hyperbolic geometry, Thurston's double limit theorem gives condition for a sequence of quasi-Fuchsian groups to have a convergent subsequence. It was introduced in , theorem 4.1) and is a major ste
Ahlfors measure conjecture
In mathematics, the Ahlfors 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 was introdu
The geometry and topology of three-manifolds
The geometry and topology of three-manifolds is a set of widely circulated but unpublished notes by William Thurston from 1978 to 1980 describing his work on 3-manifolds. The notes introduced several
Schottky group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by Friedrich Schottky.
Kleinian model
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where is a discrete subgroup of PSL(2,C). Here, the subgroup , a Kleinian group, is defin
Riley slice
In the mathematical theory of Kleinian groups, the Riley slice of Schottky space is a family of Kleinian groups generated by two parabolic elements. It was studied in detail by and named after Robert
Hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all
Bers slice
In the mathematical theory of Kleinian groups, Bers slices and Maskit slices, named after Lipman Bers and Bernard Maskit, are certain slices through the moduli space of Kleinian groups.
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of t
Density theorem for Kleinian groups
In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by and , states that every finitely generated Kl
Quasi-Fuchsian group
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 limit set is equal to the Jordan curve the q
Kleinian group
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 th
Arithmetic hyperbolic 3-manifold
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are part
Hilbert's twenty-second problem
Hilbert's twenty-second problem is the penultimate entry in the celebrated list of 23 Hilbert problems compiled in 1900 by David Hilbert. It entails the uniformization of analytic relations by means o
Mumford's compactness theorem
In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metri
Geometric finiteness
In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be des
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisf
Ahlfors finiteness theorem
In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by La
Simultaneous uniformization theorem
In mathematics, the simultaneous uniformization theorem, proved by , states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus using a quasi-Fuchsian gro
(2,3,7) triangle group
In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces
Ending lamination theorem
In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their
Indra's Pearls (book)
Indra's Pearls: The Vision of Felix Klein is a geometry book written by David Mumford, Caroline Series and David Wright, and published by Cambridge University Press in 2002 and 2015. The book explores
Jørgensen's inequality
In the mathematical theory of Kleinian groups, Jørgensen's inequality is an inequality involving the traces of elements of a Kleinian group, proved by Troels Jørgensen. The inequality states that if A