What is a Kleinian group?

1974 ◽  
pp. 1-14 ◽  
Author(s):  
Lipman Bers
Keyword(s):  
Kleinian Group

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

THE DISCONTINUITY SET FOR A KLEINIAN GROUP AND TOPOLOGY OF ITS KLEINIAN MANIFOLD

1993 ◽  
Vol 04 (01) ◽  
pp. 167-177 ◽  
Author(s):  
ANDREW V. TETENOV
Keyword(s):  
Kleinian Group ◽  
Kleinian Groups ◽  
Infinite Volume ◽  
And Topology ◽  
Finiteness Theorems

The aim of the paper is to proof the finiteness theorems for the discontinuity set Ω (G) of Kleinian groups G on n-sphere, n ≥ 3 and to study the relation between the structure of Ω (G) and the topology of infinite volume hyperbolic (n + 1)-manifold M (G), π1 (M) = G. In particular, we describe the homotopy and homology of these manifolds.

THE TRACE FUNCTION AND COMPLEX KLEINIAN GROUPS IN $P_{\mathbb{C}}^{2}$

2008 ◽  
Vol 19 (07) ◽  
pp. 865-890 ◽  
Author(s):  
JUAN-PABLO NAVARRETE
Keyword(s):  
Fixed Points ◽  
Kleinian Group ◽  
Cyclic Subgroup ◽  
Kleinian Groups ◽  
Trace Function ◽  
Limit Set

It is well known that the elements of PSL(2, ℂ) are classified as elliptic, parabolic or loxodromic according to the dynamics and their fixed points; these three types are also distinguished by their trace. If we now look at the elements in PU(2,1), then there are the equivalent notions of elliptic, parabolic or loxodromic elements; Goldman classified these transformations by their trace. In this work we extend the classification of elements of PU(2,1) to all of PSL(3, ℂ); we also extend to this setting the theorem that classifies them according to their trace. We use the notion of limit set introduced by Kulkarni, and calculate the limit set of every cyclic subgroup of PSL(3, ℂ) acting on [Formula: see text]. Given a classical Kleinian group it is possible to "suspend" this group to a subgroup of PSL(3, ℂ); we also calculate the limit set of this suspended group.

A variation formula for the topological entropy of convex-cocompact manifolds

2010 ◽  
Vol 31 (6) ◽  
pp. 1849-1864 ◽  
Author(s):  
SAMUEL TAPIE
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Formula ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Kleinian Group ◽  
Limit Set ◽  
Analytic Family ◽  
Variation Formula ◽  
Sectional Curvatures ◽  
Convex Cocompact

AbstractLet (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

scholarly journals Jørgensen’s Inequality and Algebraic Convergence Theorem in Quaternionic Hyperbolic Isometry Groups

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Huani Qin ◽  
Yueping Jiang ◽  
Wensheng Cao
Keyword(s):  
Hyperbolic Space ◽  
Convergence Theorem ◽  
Kleinian Group ◽  
Isometry Groups ◽  
Quaternionic Hyperbolic Space ◽  
Algebraic Convergence ◽  
Real Hyperbolic Space ◽  
Jørgensen’S Inequality ◽  
Algebraic Limit

We obtain an analogue of Jørgensen's inequality in quaternionic hyperbolic space. As an application, we prove that if ther-generator quaternionic Kleinian group satisfies I-condition, then its algebraic limit is also a quaternionic Kleinian group. Our results are generalizations of the counterparts in then-dimensional real hyperbolic space.

A 4-Dimensional Kleinian Group

1994 ◽  
Vol 344 (1) ◽  
pp. 391 ◽  
Author(s):  
B. H. Bowditch ◽  
G. Mess
Keyword(s):  
Kleinian Group

Amalgamation and the invariant trace field of a Kleinian group

1991 ◽  
Vol 109 (3) ◽  
pp. 509-515 ◽  
Author(s):  
Walter D. Neumann ◽  
Alan W. Reid
Keyword(s):  
Kleinian Group ◽  
Commensurability Class

Let Γ be a Kleinian group of finite covolume and denote by Γ(2) the subgroup generated by {γ2:γ ∈ Γ}. In [9] the trace field of Γ(2) was shown to be an invariant of the commensurability class of Γ. In [8] this field was termed the invariant trace field of Γ and further properties of this field were studied. Following the notation of [8] we denote the invariant trace field of Γ by k(Γ).

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