Kleinian group


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Kleinian group

[′klī·nē·ən ‚grüp]
(mathematics)
A group of conformal mappings of a Riemann surface onto itself which is discontinuous at one or more points and is not discontinuous at more than two points.
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In the Riemannian case (as opposed to the pseudo-Riemannian case) an important class of locally conformally flat Riemannian n-manifolds arises when the developing map (conformal immersion into [S.sup.n]) is injective in which case the manifold is the quotient of an open subset of [S.sup.n] by a Kleinian group [12].
Yau, "Conformally flat manifolds, Kleinian groups and scalar curvature," Inventiones Mathematicae, vol.
It also is proved that a Kleinian group [GAMMA] has a generating set consisting of elements whose traces are real ([14, Theorem 6.3].) We give a new and independent criterion for the result of [14] that also applies to Kleinian groups.
For the sake of completeness, we record in Section 2 some fundamentals on hyperbolic geometry, fundamental domains and on Fuchsian and Kleinian groups. In Section 3, we recall a result from [12] and develop a necessary proposition to prove our main result and its corollaries.
By Theorem I (and II) of [Su], there is no invariant measurable tangent line field on the conservative part of the action of any Kleinian group on [Mathematical Expression Omitted].
Here the Ahlfors-Bers quasiconformal deformation theory for Kleinian groups and rigidity theorems of Sullivan [Su] will be used to show that the Fixed Point Index Lemma of [HS] holds for conformal mappings of circle domains with a certain extension property.
The topics include a measure-theoretic result for approximation by Delone sets, self-similar tilings of fractal blow-ups, dimensions of limit sets of Kleinian groups, an overview of complex fractal dimensions: from fractal strings to fractal drums and back, and eigenvalues of the Laplacian on domains with fractal boundary.
Sharing his appreciation for the beauty of such mathematical objects as Kleinian groups and hyperbolic knots, Bonahon (University of Southern California) first introduces 2-dimensional geometry in chapters of incremental difficulty.
They discuss topics related to Kleinian groups, classical Riemann surface theory, mapping class groups, geometric group theory, and statistical mechanics.
Kleinian groups and hyperbolic 3-manifolds; proceedings.
The lecture topics include quasiconformal mappings, Teichmiller spaces, and Kleinian groups; calcul infinitesimal stochastique; rational approximations to algebraic numbers, and many others.
The topics include Bers and partial differential equations, measurable Riemann mappings, the Ahlfor-Bers creation of the modern theory of Kleinian groups, the Bers embeddings and (some of) its ramifications, the Weil-Petersson geometry of a family of Riemann surfaces, and the early history of moduli and TeichmEller spaces.