Further Remarks on the Exponent of Convergence and the Hausdorff Dimension of the Limit Set of Kleinian Groups

2011 ◽  
pp. 229-236
Author(s):  
Martial R. Hille
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Groups ◽  
Limit Set ◽  
Exponent Of Convergence

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 Hausdorff dimension of the limit sets of infinitely generated Kleinian groups

2000 ◽  
Vol 128 (1) ◽  
pp. 123-139 ◽  
Author(s):  
KATSUHIKO MATSUZAKI
Keyword(s):  
Hausdorff Dimension ◽  
Discrete Subgroup ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Full Dimension

In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension.

Osculatory Packings by Spheres

1970 ◽  
Vol 13 (1) ◽  
pp. 59-64 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Pairwise Disjoint ◽  
Exponent Of Convergence ◽  
Residual Set ◽  
Open Set ◽  
Finite Lebesgue Measure

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

scholarly journals Surface groups acting on spaces

2017 ◽  
Vol 39 (7) ◽  
pp. 1843-1856
Author(s):  
GEORGIOS DASKALOPOULOS ◽  
CHIKAKO MESE ◽  
ANDREW SANDERS ◽  
ALINA VDOVINA
Keyword(s):  
Hausdorff Dimension ◽  
Group Action ◽  
Metric Space ◽  
Hyperbolic Plane ◽  
Harmonic Map ◽  
Surface Group ◽  
Surface Groups ◽  
Limit Set ◽  
Convex Cocompact

Harmonic map theory is used to show that a convex cocompact surface group action on a $\text{CAT}(-1)$ metric space fixes a convex copy of the hyperbolic plane (i.e. the action is Fuchsian) if and only if the Hausdorff dimension of the limit set of the action is equal to 1. This provides another proof of a result of Bonk and Kleiner. More generally, we show that the limit set of every convex cocompact surface group action on a $\text{CAT}(-1)$ space has Hausdorff dimension $\geq 1$, where the inequality is strict unless the action is Fuchsian.

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).

FOX-ARTIN ARC AND CONSTRUCTING KLEINIAN GROUPS ACTING ON S3 WITH LIMIT SET A WILD CANTOR SET

1995 ◽  
Vol 06 (01) ◽  
pp. 19-32 ◽  
Author(s):  
NIKOLAY GUSEVSKII ◽  
HELEN KLIMENKO
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Wild Cantor Set ◽  
Wild Cantor Sets

We construct purely loxodromic, geometrically finite, free Kleinian groups acting on S3 whose limit sets are wild Cantor sets. Our construction is closely related to the construction of the wild Fox–Artin arc.

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