Hausdorff dimension and Kleinian groups

Kleinian groups of small Hausdorff dimension are classical Schottky groups

Let H be the hyperbolic 3-space. A subgroup Γ of PSL(2,C) = Iso(H) is called a Kleinian group if it is discrete. Let x ∈ H. The orbit of x under action of Γ is denoted by Γx. The limit set ΛΓ of Γ is definited as ΛΓ = Γx ∩ ∂H. By definition, ΛG is smallest closed Γ-invariant subset of ∂H. The group Γ is called second kind if ΛΓ 6= ∂H, otherwise it is said to be first kind. The set ΩΓ = ∂H 3 − Λ...

Analyticity of Hausdorff dimension of limit sets of Kleinian groups

We show that the Hausdorff dimension of the limit set is a real analytic function on the deformation space of a certain class of convex co-compact Kleinian groups, which includes all convex co-compact function groups. This extends a result of Ruelle [21] for quasifuchsian groups.

Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups

This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H/Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3/Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the ...

Calculating Hausdorff Dimension of Julia Sets and Kleinian Limit Sets

Homological Dimension and Critical Exponent of Kleinian Groups

We prove an inequality between the relative homological dimension of a Kleinian group Γ ⊂ Isom(Hn) and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere.