A Note on Geometric Upper Bounds for the Exponent of Convergence of Convex Cocompact Kleinian Groups

Bernd O. Stratmann

Abstract In this note we obtain by purely geometric means that for convex cocompact Kleinian groups the exponent of convergence is bounded from above by an expression which depends mainly on the diameter of the convex core of the associated infinite-volume hyperbolic manifold. This result is derived via refinements of Sullivan’s shadow lemma and of estimates for the growth of the orbital counting function and Poincaré series. We finally obtain spectral and fractal implications, such as lower bounds for the bottom of the spectrum of the Laplacian on these manifolds, and upper bounds for the decay of the area of neighbourhoods of the associated limit sets.

1 Introduction

This note presents a study of the lacunarity of limit sets of convex cocompact Kleinian groups acting on hyperbolic spaces. We derive some deeper insight into the geometric nature of the principle that if a geometrically finite complex dynamical system is deformed such that it approaches an element on the boundary of the relevant parameter-space which is not a cusp (i.e. not geometrically finite), then the complexity of the associated attractors (i.e. limit sets) increases dramatically, in the sense that their Hausdorff dimensions tend to the topological dimension of the smallest surrounding Euclidean space (see e.g. [2, 5, 12, 24, 29]). In order to state the results, recall that to each convex cocompact Kleinian group G acting on hyperbolic (N + 1) -space 𝔻^{N+1} we can associate a certain parameter δ = δ(G) , called the exponent of convergence of G. It is well-known that δ is equal to the Hausdorff dimension of the limit set L(G) . Another way of saying this is that δ represents the Hausdorff dimension of the set of all bounded geodesic movements emanating from some point on the associated infinite-volume hyperbolic (N + 1) -manifold M_{G} (in fact, most recently the latter equality has been verified in [2] for arbitrary Kleinian groups (see also [25])). The main result of the paper is that we obtain by purely geometric means an upper bound for δ in terms of certain intrinsic parameters. More precisely, if σ denotes the hyperbolic diameter of the convex core of M_{G} (i.e. the smallest convex submanifold of M_{G} such that the inclusion map is a homotopy equivalence) and ω the minimal inner ending depth of M_{G} (see Definition 2.1), then (Theorem 4.1)

δ ≤ N − K_{N} / e^{N(3σ+ω)},

where K_{N} > 0 denotes some universal constant which depends exclusively on N . We remark that for geometrically finite Kleinian groups for which δ > N/2 a similar result has been obtained by Burger and Canary in [4] (also, note that in [5] a lower bound for δ has been derived). Their proof relies heavily on certain deep results from spectral theory, which has the disadvantage that it does not give access to the range^{1} δ ≤ N/2 , and also that it gives rather limited insight into the fractal geometry behind. Since our approach is purely geometric, we do not have to struggle with the ‘spectral border’ at N/2 . The method we use can be summarised briefly as follows. We give a refinement of Sullivan’s shadow lemma, by specifying in terms of σ one of the constants occurring in this lemma (Lemma 3.1). This then leads to ‘good’ estimates from below for the growth of the orbital counting function (Lemma 3.2), which eventually gives, for each s > δ and each element x of the convex core of M_{G} (Lemma 3.3),

∑_{g∈G} e^{−sd(x,g(x))} ≥ (2e^{3σ})^{−N} s/(s − δ).

Finally, we show that our main result theorem 4.1 has two interesting immediate implications. The first of these is a lower bound for the smallest eigenvalue λ_{0}(M_{G}) of the Laplacian on M_{G}. For δ > N/2 we obtain the result (Proposition 5.1)

λ_{0}(M_{G}) > (NK_{N}/2) / e^{N(3σ+ω)}.

The second implication is about the fractal nature of L(G) . Namely, for the N - dimensional spherical area area_{N}(U_{ϵ}(L(G)) of ϵ -neighbourhoods of L(G) we obtain the result (Proposition 5.2)

lim_{ϵ→0}(log area_{N}(U_{ϵ}(L(G))) / log ϵ ≥ K_{N} / e^{N(3σ+ω)}.

Remarks:

(1) The geometric method presented suggests that σ can be used as a rough measure for the degree of lacunarity^{2} of L(G) , in the sense that if we deform G such that σ increases, then the lacunarity of the limit set decreases.

(2) It should certainly be possible to extend the geometric analysis in this paper also to geometrically finite groups with parabolic elements. This could be achieved by refining in [27] the ‘Dirichlet-type theorem’, the ‘global measure formula’ and the ‘horoball counting theorem’, specifying all the appearing constants there in terms of the diameter and the injective radii of the compact part of the convex core.

1Note, for N = 2 we have that δ ≤ 1 implies that G is either a function group or a finite covering of a group which is quasi-conformal equivalent to a Fuchsian group (cf.[7]).

2The concept of lacunarity for a fractal set (‘lacuna’ latin for ‘gap’) was introduced by Mandelbrot. His introduction of this concept in [15] (p. 312) starts with the following sentence.

”A second skeleton rattles in the closets of most models of the distribution of galaxies”.

Similar as for the concept ‘fractal’, for the concept ‘lacunarity’ there exists no rigorous definition in the literature. Nevertheless, roughly speaking, a fractal set F (given by some ‘construction rule (CR) ’) is to be called lacunar if its ‘complementary gaps’ (created by means of (CR) ) tend to be large, in the sense that if one alters the geometrical realisation of F (without effecting the combinatorics of (CR) ) such that relatively the gaps get smaller, then the lacunarity of F drops.

2 Preliminaries

In this section we recall briefly the background concerning convex cocompact groups and Patterson measures which will be relevant in the sequel. Convex cocompact Kleinian groups. For N = 1, 2, ⋯, consider the Poincaré ball-model (𝔻^{N+1}, d) of the (N + 1) - dimensional hyperbolic space with visible boundary S^{N} . We assume that the reader is familiar with the hyperbolic geometry in this model (see e.g. [1]). Let Con(N) denote the group of (orientation preserving) isometries of (𝔻^{N+1}, d) . Throughout we shall mainly be concerned with Kleinian groups G, i.e. discrete subgroups of Con(N) , which are non-elementary and convex cocompact. This means that G is finitely generated, has no abelian subgroup of finite index and no parabolic elements, and that a Dirichlet fundamental domain for the action of G on 𝔻^{N+1} is bounded by finitely many faces (i.e. hyperbolic hyper-planes of codimension 1 ). By Selberg’s lemma ([23]), i.e. by passing (if necessary) to a torsion-free normal subgroup of finite index (an operation which does not effect the limit set), we can assume without loss of generality that G has no torsion. Let Ω(G) ⊂ S^{N} denote the set of ordinary points of G, that is the set of points where G still acts discontinuously on S^{N} . Recall that the boundary of the associated convex cocompact Kleinian manifold N_{G} = (𝔻^{N+1}∪Ω(G))/G decomposes into a finite set Σ̂ of cocompact hyperbolic manifolds of codimension 1 (without punctures) . Equivalently, if F denotes the Dirichlet fundamental domain (based at the origin) for the action of G on 𝔻^{N+1} ∪ Ω(G) , then F̄ (the closure of F in ℝ^{N+1} ) intersects Ω(G) in a finite set Σ of pairwise disjoint polygons, which are fundamental domains representing the manifolds in Σ̂ . Furthermore, if M_{G} := 𝔻^{N+1}/G denotes the interior of N_{G}, then M_{G} is an infinite volume hyperbolic (N + 1) -manifold, and a fundamental geometric object associated to it, is its convex core C(M_{G}) , which is the smallest convex submanifold of M_{G} such that the inclusion map is a homotopy equivalence. More precisely, if H(L(G)) denotes the convex hull in 𝔻^{N+1} of the limit L(G) := S^{N} \ Ω(G) of G, then C(M_{G}) := H(L(G))/G. Throughout, we let σ denote the hyperbolic diameter of C(M_{G}) . Note that σ is finite if and only if G is convex cocompact. Finally, we recall the notion of the retraction map R : (Ω(G) ∪ 𝔻^{N+1}) \ H(L(G)) → ∂H(L(G)) , which is defined as follows (see e.g. [13]). For a point x ∈ 𝔻^{N+1} , let B(x) be the largest hyperbolic ball centred at x such that its interior is disjoint from H(L(G)) . Since d is strictly convex, B(x) intersects ∂H(L(G)) in a single point, and this point is R(x) . Now, the Chabauty-limits of hyperbolic balls in 𝔻^{N+1} are horoballs, hence for each ξ ∈ Ω(G) we take the maximal horoball H(ξ) centred at ξ whose interior is disjoint from H(L(G)) . Then R(ξ) is the point of intersection of ∂H(L(G)) and H(ξ) .

Definition 2.1 Let G be a convex cocompact Kleinian group, and let Δ_{i} denote the largest spherical disc contained in the i -th element S_{i} of Σ , and let η_{i} be its Euclidean centre. With Δ̂_{i} denoting the convex hull of Δ_{i} in 𝔻^{N+1} , the inner depth of the end corresponding to S_{i} is defined by ω_{i} := d(Δ̂_{i}, R(η_{i})) . The number ω := min_{i=1,···,n}{ω_{i}} will be referred to as the minimal inner ending depth of N_{G}.

Remarks:

(1) For N = 1 , i.e. if G is a non-elementary convex cocompact Fuchsian group, we remark that the boundary of N_{G} decomposes into a finite set of loops, and \overline{F} , the closure of the fundamental domain of G, intersects S^{1} in a finite set Σ of pairwise disjoint intervals. The images of Σ under R are the boundary geodesics γ_{i}, i.e. the loops in M_{G} where the ‘funnels’ are attached to the compact convex core. If l_{i} denotes the hyperbolic length of γ_{i}, then a routine calculation in hyperbolic geometry gives that sinh ω_{i} sinh(l_{i}/2) = 1.

(2) For N = 2 , i.e. if G is a non-elementary convex cocompact ‘classical’ Kleinian group, we remark that H(L(G)) is the union of ideal tetrahedra in 𝔻^{3} with endpoints in L(G) . Hence, the boundary ∂H(L(G)) is made of geodesics and totally geodesic ideal triangles. By a result of Thurston ([32], [10], [13]), we have that ∂C(M_{G}) is a set Σ^{∗} of cocompact Riemann surfaces (where one uses the natural path metric induced by the length of geodesic segments in ∂C(M_{G}) ). The elements in Σ^{∗} are ‘crumpled surfaces’ (i.e. images of pleated surfaces) which correspond to the elements in Σ . In order to interpret the inner ending depths ω_{i} in terms of ∂C(M_{G}) , one may now apply a theorem of Sullivan (see [10]), which states that on the components of Ω(G) the retraction map is a K -bilipschitz map onto ∂H(L(G) . Unfortunately, here the constant K depends strongly on the group, and it seems as if so far this dependence has not been explicitly specified in the literature (the interested reader might find [6] and [16] helpful).

The Patterson measure. Let μ denote the Patterson measure supported on the limit set L(G) . We shall assume that μ is constructed with respect to the base point 0 ∈ 𝔻^{N+1} . More precisely, recall that to G we can associate its exponent of convergence δ = δ(G) , which is defined as the exponent of convergence of the Poincaré series

∑_{g∈G} e^{−sd(y,g(y))},

for some arbitrary y ∈ 𝔻^{N+1} . (For arbitrary non-elementary Kleinian group it is easy to see that 0 ≤ δ ≤ N ; also, note that in the geometrically finite case we have δ < N (cf. [28],[33])). Now, for some sequence of positive numbers ϵ_{k} tending to zero (and given that the Poincaré series diverges for s = δ (otherwise, a slowly varying function has to be introduced which then forces this divergence ([18])), μ is the weak limit of the sequence of measures

μ_{k} := \frac{∑_{g∈G} e^{−(δ+ϵ_{k})d(0,g(0))}δ_{g(0)}}{∑_{g∈G} e^{−(δ+ϵ_{k})d(0,g(0))}} ,

where δ_{x} denotes the Dirac measure at x ∈ 𝔻^{N+1} . One of the most important properties of the Patterson measure is its δ -conformality, that is for arbitrary Borel sets E ⊂ S^{N} and for every g ∈ G we have that

μ(g^{−1}E) = ∫_{E} (\frac{1 − |g(0)|^{2}}{|ξ − g(0)|^{2}} )^{δ} dμ(ξ).

For a non-elementary convex cocompact group G the Patterson measure has been analysed extremely well (see e.g. [17],[19],[28],[30]). For instance, then μ is equivalent to the δ -dimensional Hausdorff measure on L(G) . This result is an immediate consequence of the fact that C(M_{G}) is a compact subset of 𝔻^{N+1} , combined with Sullivan’s shadow lemma (cf. [28]), which in its original form states that there exist constants c, κ > 0 such that for each g ∈ G with d(0, g(0)) sufficiently large, we have that

c^{−1} e^{−δd(0,g(0))} ≤ μ(Π(B(g(0), κ))) ≤ c e^{−δd(0,g(0))}.

Here, B(z, ρ) denotes the hyperbolic ball of radius ρ centred at z, and Π the ‘shadow projection from the origin’, which is given for arbitrary E ⊂ 𝔻^{N+1} by Π(E) := {ξ ∈ S^{N} : s_{ξ} ∩E ≠ ∅}, where s_{ξ} denotes the hyperbolic ray in 𝔻^{N+1} from 0 to ξ ∈ S^{N} .

3 Refined shadowing and counting, and a lower estimate for the Poincaré series

In this paragraph we derive three lemmata. The first lemma gives a refinement of Sullivan’s shadow lemma for arbitrary Kleinian groups. Using this refinement, we then obtain for convex cocompact Kleinian groups a lower bound for the orbital counting function, which consequently leads to an estimate from below for the growth of the Poincaré series. We remark that ‘rough versions’ of these lemmata are wellknown (cf. [28],[19], [17]). New in our statements here is that we have specified the constants occurring in these lemmata, in particular their dependence on the diameter σ of the convex core C(M_{G}) .

Lemma 3.1 Let G < Con(N) be an arbitrary non-elementary Kleinian group and let ρ > 0 . For each g ∈ G such that d(0, g(0)) > ρ, we have that

μ(Π(B(g(0), ρ))) ≤ (4 cosh(ρ/2))^{N} e^{−δd(0,g(0))}.

Proof: Let z ∈ 𝔻^{N+1} such that d(0, z) > ρ. An elementary Euclidean argument gives that if ξ ∈ Π(B(z, ρ) , then

|z − ξ| · |z + ξ| < (1 − |z|^{2}) cosh ρ.

Now, consider the hyperbolic half-space which contains z and which is bounded by the hyper-plane through 0 orthogonal to the ‘vector’ z. Let H denote the intersection of S^{N} with the boundary of this half-space. Clearly, we have that ξ ∈ H , which implies that |z + ξ| ≥ √(1 + tanh^{2}(ρ/2)) . Hence, it follows that

|z − ξ| < \frac{cosh ρ}{√(1 + tanh^{2}(ρ/2))}(1 − |z|^{2}) = cosh(ρ/2) (1 − |z|^{2}).

Using this observation, we now have, for g ∈ G with d(0, g(0)) > ρ, that

1 ≥ μ(g^{−1}(Π(B(g(0), ρ)))) = ∫_{Π(B(g(0),ρ))} (\frac{1 − |g(0)|^{2}}{|ξ − g(0)|^{2}} )^{δ} dμ(ξ)
≥ (cosh(ρ/2))^{−δ} ∫_{Π(B(g(0),ρ))}(1 − |g(0)|^{2})^{−δ}dμ(ξ)
≥ cosh(ρ/2)^{−N} (1 − |g(0)|^{2})^{−δ} μ(Π(B(g(0), ρ))).

Since δ ≤ N , we have that (1−|g(0)|^{2})^{δ} ≤ (1+|g(0)|)^{2δ} (\frac{1−|g(0)|}{1+|g(0)|})^{δ} ≤ 4^{N} e^{−δd(0,g(0))} . Combining this with the estimate above, the lemma follows.

□

Lemma 3.2 Let G < Con(N) be a non-elementary convex cocompact Kleinian group. For each x ∈ C(M_{G}) and t ≥ 0 , we have for the orbital counting function N_{t}^{x}(G) := card {g ∈ G : d(x, g(x)) < t}, that

N_{t}^{x}(G) ≥ C_{N,σ} e^{δt},

where C_{N,σ} := min{e^{−3Nσ}; (4e^{2σ} cosh(σ/2))^{−N}}.

Proof: Without loss of generality we can assume that x = 0 . For ease of notation, let N_{t}(G) := N_{t}^{0}(G) . For t > 3σ, define A_{t} := {g ∈ G : t − 2σ ≤ d(0, g(0)) < t}. Since C(M_{G}) ⊂ 𝔻^{N+1} is compact, it follows that s_{ξ} ⊂ ⋃_{g∈G} B(g(0), σ) for each ξ ∈ L(G) . This implies that {Π(B(g(0), σ)) : g ∈ A_{t}} is a covering of L(G) . Hence, using lemma 3.1, we have that

1 = μ( ⋃_{g∈A_{t}} Π(B(g(0), σ))) ≤ ∑_{g∈A_{t}} μ(Π(B(g(0), σ)))
≤ (4 cosh(σ/2))^{N} ∑_{g∈A_{t}} e^{−δd(0,g(0))} ≤ (4 cosh(σ/2))^{N} e^{−(t−2σ)δ} card A_{t}
≤ (4e^{2σ} cosh(σ/2))^{N} e^{−δt} N_{t}(G).

For 0 ≤ t ≤ 3σ, we note that

N_{t}(G) ≥ 1 = e^{−3σδ} e^{3σδ} > e^{−3Nσ} e^{δt}.

Combining these two observations, the lemma follows. □

Lemma 3.3 Let G < Con(N) be a non-elementary convex cocompact Kleinian group. For each ϵ > 0 and x ∈ C(M_{G}) we have that

∑_{g∈G} e^{−(δ+ϵ)d(x,g(x))} ≥ C_{N,σ} \frac{δ + ϵ}{ϵ}.

Proof: As in the proof of lemma 3.2, assume that x = 0 . Using lemma 3.2, it follows that

∑_{g∈G} e^{−(δ+ϵ)d(0,g(0))} = lim_{T →∞} (N_{T} (G)e^{−(δ+ϵ)T} + (δ + ϵ) ∫_{0}^{T} e^{−(δ+ϵ)t}N_{t}(G)dt)

≥ C_{N,σ} lim_{T →∞} (e^{−ϵT} + (δ + ϵ) ∫_{0}^{T} e^{−ϵt}dt)
= C_{N,σ} (δ + ϵ)/ϵ.
□

4 An upper bound for the exponent of convergence

In this section we obtain the main result of this paper, namely for a convex cocompact Kleinian group we derive an upper bound for its exponent of convergence.

Theorem 4.1 Let G < Con(N) be a non-elementary convex cocompact Kleinian group with associated minimal inner ending depth ω. For the exponent of convergence δ of G we have that

δ ≤ N − K_{N} e^{−N(3σ+ω)},

where K_{N} := N 2^{−N} (4π)^{1/2−N} Γ((N + 2)/2)/Γ((N + 1)/2) (here, Γ denotes the Gamma-function).

Proof: Consider the i -th element of Σ , which by definition corresponds to an end of inner depth ω_{i}. Without loss of generality, assume that 0 = R(η_{i}) . For g ∈ G, we shall now first give a lower bound for the N -dimensional spherical Lebesque measure ( λ_{N}^{∗} ) of Δ(g) := g(Δ_{i}) . Let w_{g} denote the point in ∂(g(Δ̂_{i})) which is uniquely determined by ω_{i} = d(w_{g}, g(0)) . Note that for the angle θ at g(0) (such that θ ≤ π) spanned by the geodesic segments connecting 0 with g(0) and g(0) with w_{g} , we have that θ ≥ π/2 . This follows immediately by combining the fact that the geodesic segment between 0 and g(0) is contained in H(L(G)) , and the fact that the interior of H_{g} is a subset of 𝔻^{N+1} \ H(L(G)) , where H_{g} denotes the horoball which is tangent to H(L(G)) at w_{g} and which is based at the centre g(η_{i}) of Δ(g) . Thus, the geodesic segment joining 0 and w_{g} is the hypotenuse of the triangle whose vertices are the points 0, g(0) and w_{g} . Hence,

e^{−d(0,w_{g})} ≥ e^{−ω_{i}} e^{−d(0,g(0))}.

Now, if d_{g} denotes the hyperbolic distance from 0 to the convex hull of Δ(g) in 𝔻^{N+1} , then we immediately have that e^{−d_{g}} ≥ e^{−d(0,w_{g})} . Thus, an elementary calculation gives that λ_{N}^{∗}(Δ(g)) ≥ (4π)^{−N} V_{N} e^{−Nd_{g}} , where V_{N} denotes the volume of the unit ball in ℝ^{N} . Summarising these estimates, it follows that

λ_{N}^{∗}(Δ(g)) ≥ (4π)^{−N} V_{N} e^{−Nω_{i}} e^{−Nd(0,g(0))}.

Since {Δ(g) : g ∈ G} is a family of pairwise disjoint discs in S^{N} , this latter estimate and lemma 3.3 (applied for ϵ = N − δ) imply that, with O_{N+1} denoting the area of the unit sphere in ℝ^{N+1} ,

O_{N+1} = λ_{N}^{∗}(Ω(G)) ≥ ∑_{g∈G} λ_{N}^{∗}(Δ(g))
≥ (4π)^{−N} V_{N} e^{−Nω_{i}} ∑_{g∈G} e^{−Nd(0,g(0))}
≥ C_{N,σ} (4π)^{−N} V_{N} e^{−Nω_{i}} N/(N − δ).

Using the facts V_{N} = π^{N/2}/Γ(N/2 + 1) and O_{N+1} = 2π^{(N+1)/2}/Γ((N + 1)/2) , and noting that for the constant C_{N,σ} of lemma 3.2 we have that C_{N,σ} ≥ 2^{−N} e^{−3Nσ} , it then follows that

δ ≤ N − K_{N} e^{−3Nσ} e^{−Nω_{i}},

where we have set K_{N} := N 2^{−N} (4π)^{1/2−N} \frac{Γ((N+2)/2)}{Γ((N+1)/2)} . Clearly, this estimate is optimal for ω_{i} = ω. Hence, choosing i such that the corresponding end has inner depth ω, the theorem follows.

□

By ‘Remark 1’ in paragraph 2, theorem 4.1 has the following immediate implication.

Corollary 4.2 If G < Con(1) is a convex cocompact Fuchsian group, then we have for its exponent of convergence δ that

δ ≤ 1 − \frac{1}{8e^{3σ}} max_{i}\{\frac{sinh(l_{i}/2)}{1 + cosh(l_{i}/2)}\}.

5 Some spectral and fractal consequences

5.1 The bottom of the spectrum of the Laplacian

The result in theorem 4.1 immediately gives rise to a lower bound for the bottom λ_{0} of the spectrum of the Laplace operator (−Δ) on the associated infinite volume, convex cocompact, hyperbolic (N + 1) -manifold M_{G}. We remark that in the literature there are numerous publications dealing with various aspects concerning λ_{0} (see e.g. [3, 4, 5, 8, 9, 14, 20, 21, 22]). Recall the following well-known results from spectral theory for Kleinian groups. Lax and Phillips ([14]) have shown for a non-elementary geometrically finite Kleinian group in Con(2) that the spectrum of (−Δ) is continuous in [1, ∞) , and that on [0, 1) it consists exclusively of finitely many eigenvalues, which are positive and of finite multiplicity. The smallest of these latter eigenvalues is referred to as the bottom of the spectrum of the Laplacian and it is denoted by λ_{0}(M_{G}) . More precisely, for a geometrically finite Kleinian group in Con(N) define

λ_{0}(M_{G}) := sup{−λ : Δφ = λφ for some positive φ ∈ C^{∞}(M_{G})}.

By work of Patterson [18] and Sullivan [28, 31], we have that

λ_{0}(M_{G}) = \{ \begin{matrix}N^{2}/4  if & δ ≤ N/2 \\ δ(N − δ)  if & δ > N/2. \end{matrix}

Hence, combining this result and theorem 4.1, we derive the following statement.

Proposition 5.1 For the hyperbolic manifold M_{G} associated to a non-elementary convex cocompact Kleinian group G < Con(N) for which δ > N/2 , we have that

λ_{0}(M_{G}) > \frac{NK_{N}}{2} e^{−N(3σ+ω)}.

5.2 The limit set in the spirit of Minkowski

Also, as will be seen in this paragraph, theorem 4.1 gives rise immediately to an upper bound for the ϵ -asymptotic of the volume of the ϵ -neighbourhood of the limit set of a convex cocompact Kleinian group. Recall that for a subset A ⊂ ℝ^{n} the ϵ -parallel body U_{ϵ}(A) of A is given by

U_{ϵ}(A) := {x ∈ ℝ^{n} : |x − a| ≤ ϵ for some a ∈ A}.

Furthermore, recall that if N_{ϵ}(A) denotes the smallest number of sets of diameter at most ϵ that cover A, then the box-counting dimension dim_{B}(A) of A is given by (if the limit exists)

dim_{B}(A) = lim_{ϵ→0} \frac{log N_{ϵ}(A)}{− log ϵ}.

It turns out that the n -dimensional Euclidean volume ( vol_{n}) of the ϵ -parallel body and the box-counting dimension are two closely related concepts. Namely, we have the following well-known result (see e.g. [11]).

dim_{B}(A) = n − lim_{ϵ→0} \frac{log vol_{n}(U_{ϵ}(A))}{log ϵ}.

Hence, combining this general fact from fractal geometry and theorem 4.1, we obtain the following result.

Proposition 5.2 Let G < Con(N) be a non-elementary convex cocompact Kleinian group. For the spherical volume λ_{N}^{∗}(U_{ϵ}(L(G)) of the ϵ -parallel body of the limit set L(G) we have that

lim_{ϵ→0} \frac{log λ_{N}^{∗}(U_{ϵ}(L(G))}{log ϵ} ≥ K_{N} e^{−N(3σ+ω)}.

Proof: Using Frostman’s lemma (see e.g. [11]) and lemma 2 of [26], Sullivan’s shadow lemma immediately implies that the Hausdorff dimension and the boxcounting dimension of L(G) coincide and are equal to the exponent of convergence δ (note that equality of these three parameters has been verified also in the more general case of a geometrically finite Kleinian group ([26],[2])). Hence, combining theorem 4.1 and the afore-mentioned general result from fractal geometry, the statement follows.

□

Acknowledgment: I should like to thank the Mathematische Institut and the Institut für mathematische Stochastik at the University of Göttingen for their warm hospitality. In particular, I should like to thank Jan Thirase for enjoyable conversations on the subject matter of this paper.

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B.O. Stratmann, Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland. e-mail: bos@maths.st-and.ac.uk