Hausdorff dimension and Kleinian groups

Abstract

Let G be a non-elementary, finitely generated Kleinian group, Λ(G) its limit set and Ω(G) = C\Λ(G) its set of discontinuity. Let δ(G) be the critical exponent for the Poincaré series and let Λ c be the conical limit set of G. We prove that 1.6. The Minkowski dimension of Λ equals the Hausdorff dimension. 7. If area(Λ) = 0 then δ(G) = dim(Λ(G)).

Critical exponent of graphed Teichmüller representations on ℍ 2 ×ℍ 2

“…We can fix H n and play with smaller groups, or fix the group and play with other spaces. The first question has been intensively studied, let us cite at least [3,23,25,26,28,36,37]. One of the main result of these papers is the relation between δ and the hausdorff dimension of the limit set.…”

“…Let us still give a very rough (and not exact) definition: let N (r) be the number of balls of radius r that are necessary to cover a subset A of a metric space, then when r is very small N (r) behaves more or less as 1/r α where α is the Hausdorff dimension of A. The most evolved result in this direction [3] says that for discrete non-elementary isometry group Γ of H n (ie. Card Λ Γ = +∞), the critical exponent δ(Γ H 3 ) is equal to the Hausdorff dimension of the conical limit set.…”

On the Patterson-Sullivan measure of some discrete group of isometries

“…Bishop and P.W. Jones [3] shows that this conjecture would imply that δ Γ = 2 and µ σ is infinite when Γ is a non geometrically finite group of finite type. In higher dimension we have the following result :…”

Trace theorem for quasi-Fuchsian groups

“…The definition below can be found in [27] It is known that a limit set of a finitely generated quasi-Fuchsian group (which is not Fuchsian) has Hausdorff dimension strictly greater than 1 (see Corollary 1.7 in [12]). …”

“…It is known that (C\Λ(G))/G is a Riemann surface for an arbitrary Kleinian group G. The following definition is taken from [12]. Definition 2.5.…”