GEOMETRIC EXPONENTS AND KLEINIAN GROUPS

CHRISTOPHER J. BISHOP

May 17, 1996

Abstract. We define a number of different geometric critical exponents associated to a compact set in the plane which generalize the index of Besicovitch and Taylor on the line. Although these exponents may differ for general sets, we show that they are all equal when Λ is the limit set of a non-elementary, analytically finite Kleinian group and they agree with the classical Poincaré exponent. We also prove that if area(Λ) = 0 then the exponents equal dim_{H}(Λ) and give an example to show this can fail when area(Λ) > 0. We show that for analytically finite groups the Hausdorff and Minkowski dimensions of the limit set must agree. Another consequence of our argument is that if {Ω_{j}} is an enumeration of the components of Ω = S^{2} \ Λ, then

∑_{j} diam(Ω_{j})^{2} < ∞.

This had been conjectured by Maskit in 1974.

1991 Mathematics Subject Classification. 30F40 ; Secondary 28A78, 30C35, 31A15 . Key words and phrases. Hausdorff dimension, Minkowski dimension, Kleinian groups, limit sets,

critical exponents, Whitney squares, diameter sum, analytically finite, geometrically finite, Poincaré exponent, Maskit conjecture.

The author is partially supported by NSF Grant DMS 92-04092 and an Alfred P. Sloan research fellowship.

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 1

1. Statement of results

The purpose of this paper is to simplify and extend results of Patterson and Sullivan (and more recent results of Jones and myself) concerning the dimension of Kleinian limit sets. We will see that a variety of measures of the ‘size’ of a compact set all agree for limit sets of a wide class of groups (although they can differ for general sets). The quantities we will consider include Hausdorff dimension, Minkowski (or box counting) dimension, and several geometric exponents: δ_{Whit}, δ_{diam}, δ_{area}, δ_{inrad}, δ_{cir}. An easy consequence of our arguments is that the Hausdorff and Minkowski dimensions agree; thus limit sets of analytically finite groups are ‘dimension regular’ in the sense of Tricot [34] and ‘fractal’ in the sense of Taylor [32].

We start by reviewing the definitions of the various dimensions and exponents. Suppose K is a compact set in ℝ^{2} and let N(K, ϵ) be the minimal number of ϵ

balls needed to cover K. We define the upper Minkowski dimension as

\overline{dim_{M}}(K) = lim sup_{ϵ→0} \frac{log N(K, ϵ)}{log 1/ϵ},

and the lower Minkowski dimension

\underline{dim_{M}}(K) = lim inf_{ϵ→0} \frac{log N(K, ϵ)}{log 1/ϵ}.

If the two values agree, the common value is simply called the Minkowski dimension of K and is denoted dim_{M}(K). We define the Hausdorff content of K as

H_{α}^{∞}(K) = inf{∑ r_{j}^{α} : K ⊂ ∪_{j}D(x_{j}, r_{j})},

(the infimum is over all coverings of K by disks ) and

dim_{H}(K) = inf{α : H_{α}^{∞}(K) = 0}.

2 CHRISTOPHER J. BISHOP

This is the Hausdorff dimension of K and it is easy to see that dim_{H}(K) ≤ \underline{dim_{M}}(K). For general sets, the Minkowski dimension need not even exist and if it does exist, it can be strictly larger than the Hausdorff dimension. Recall that a dyadic interval I ⊂ ℝ is an interval of the form [j2^{−n}, (j +1)2^{−n}] and a dyadic square in ℝ^{2} is simply any product of equal length dyadic intervals. It is easy to see that in the definition of either dimension we need only consider coverings of K by dyadic squares.

For K ⊂ [0, 1], Besicovitch and Taylor [6] considered the following critical exponent. Suppose {I_{j}} are the complementary components of K in [0, 1] and |I| denotes the length of I. They showed that if K has zero length then

dim_{H}(K) ≤ δ_{BT}(K) ≡ inf{s : ∑_{j} |I_{j}|^{s} < ∞}.

Later Tricot [33] sharpened this by observing that δ_{BT} = \overline{dim_{M}}(K) if |K| = 0.

We want to generalize the Besicovitch-Taylor index to the plane (similar definitions clearly make sense in higher dimensions as well). Suppose K ⊂ ℝ^{2} is compact and {Ω_{j}} is an enumeration of the bounded complementary components. We define

δ_{inrad}(K) = inf{s : ∑_{j} inrad(Ω_{j})^{s} < ∞},
δ_{area}(K) = inf{s : ∑_{j} area(Ω_{j})^{s/2} < ∞},
δ_{diam}(K) = inf{s : ∑_{j} diam(Ω_{j})^{s} < ∞},

(Here “inrad” denotes the radius of the largest disk in Ω_{j}.) If K is a compact set on the sphere S^{2}, then these quantities are measured with respect to the spherical metric. Each exponent generalizes δ_{BT}, but they disagree with each other for general sets (see Section 13). Moreover they may all disagree with Minkowski dimension.

Figure 1.1. A Whitney decomposition

Next we define an exponent generalizing δ_{BT} which maintains the relationship to Minkowski dimension. A Whitney decomposition of a open set Ω ⊂ ℝ^{n} is a collection of cubes {Q_{j}} (disjoint except for boundaries) so that Ω = ∪_{j}Q_{j} and such that

diam(Q_{j}) ≃ dist(Q_{j}, ∂Ω).

(In this paper, a ≃ b means that the ratio a/b is bounded and bounded away from zero.) The existence of a Whitney decomposition for any open set is a standard fact in real analysis (e.g., Theorem VI.1 of [27]). One can simply take a maximal collection of dyadic cubes in Ω such that diam(Q) ≤ dist(Q, ∂Ω). In ℝ^{2}, this gives a decomposition satisfying diam(Q) ≤ dist(Q_{j}, ∂Ω) ≤ 4diam(Q_{j}).

Given a compact set K let Ω = ℝ^{n} \ K and suppose {Q_{j}} is a Whitney decomposition of Ω. Define

δ_{Whit}(K) = inf{s : ∑_{Q_{j},diam(Q_{j})≤1} diam(Q_{j})^{s} < ∞}.

We need to omit the Whitney squares of large diameter, or the series would not converge. It is easy to check that the definition does not depend on the particular choice of Whitney decomposition. Moreover, for any compact set K ⊂ ℝ^{n} we will see δ_{Whit}(K) ≤ \overline{dim_{M}}(K) with equality if area(K) = 0. See Section 2.

4 CHRISTOPHER J. BISHOP

If I ⊂ ℝ is an open interval and {J_{j}} the corresponding Whitney decomposition, it is easy to see that

∑ |J_{j}|^{s} ≃ C_{s}|I|^{s},

for any s > 0. Thus for K ⊂ [0, 1], the sum in the definition of δ_{Whit} is comparable to the sum in the definition the Besicovitch-Taylor index. Thus δ_{Whit} = δ_{BT}.

We now turn our attention to a special class of sets in ℝ^{2} for which all these different exponents agree.

Consider a group G of Möbius transformations acting on the two sphere S^{2}. Such transformations are identified elements of PSL(2, ℂ) in a natural way and G is called Kleinian if it is discrete in this topology (i.e., the identity is isolated in G). Such a group G also acts as isometries on the hyperbolic 3-ball 𝔹^{3}. G is called elementary if it contains a finite index Abelian subgroup. In this paper we will consider only non-elementary groups. For a non-elementary group, the limit set, Λ(G), is the accumulation set (on S^{2}) of the orbit of any point z_{0} ∈ S^{2} (and is independent of the point). The complement Ω(G) = S^{2} \ Λ is called the ordinary set. In this paper we will assume Ω is non-empty and that the group is normalized so ∞ ∈ Ω.

For any Kleinian group, the quotient R = Ω/G is a union of (branched) Riemann surfaces. We say that G is analytically finite if R = R_{1} ∪ ⋯ ∪ R_{s} is a finite union of finite type surfaces (i.e., each R_{j} is compact or compact with a finite number of punctures and branch points). The Ahlfors finiteness theorem ([2], [5]) says that if G is finitely generated then G is analytically finite.

If z_{0} ∈ Ω(G) then the Poincaré exponent is defined as

(1.1) δ_{Poin}(G) = inf{s : ∑_{g∈G} dist(g(z_{0}), Λ)^{s} < ∞},

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 5

where distance is in the spherical metric. It is easy to show it does not depend on the choice of z_{0}. Usually this exponent is simply denoted δ(G), but in this paper we will always call it δ_{Poin} to avoid confusion with the many other exponents we consider.

The exponent δ_{Poin}(G) is can also be defined by extending G’s action on S^{2} to a group of isometries of the 3-ball 𝔹^{3} ⊂ ℝ^{3} (with its hyperbolic metric ρ), and then considering any of the series

∑_{G}(1 − |g(0)|)^{s}, ∑_{G} |g′(0)|^{s}, ∑_{G} e^{−sρ(0,g(0))}.

It is easy to see that these series all give the same critical exponent, δ_{Poin}, (e.g. Lemma 3.2 of [7]). Jones and I proved that δ_{Poin}(G) ≤ dim_{H}(Λ) for any non-elementary Kleinian group (Theorem 1.1 of [7]). Theorem 1.1. Suppose G is an analytically finite Kleinian group such that Λ ≠ S^{2}. Then the Minkowski dimension of Λ exists and δ_{Poin} = δ_{Whit} ≤ dim_{M}(Λ) = dim_{H}(Λ). If area(Λ) = 0 then δ_{Poin} = dim_{H}(Λ). If Ω(G) = S^{2} \ Λ(G) has infinitely many components, then (1.2) δ_{Poin} = δ_{inrad} = δ_{area} = δ_{diam}. If, in addition, area(Λ) = 0 then they are all equal to dim_{H}(Λ).

For Fuchsian and geometrically finite groups the equality δ_{Poin} = dim_{H}(Λ) is due to Patterson and Sullivan respectively (see [24] and its references, particularly [26] and [30]). The equality dim_{M} = dim_{H} was proven for geometrically finite groups by Stratmann and Urbanski in [28]. The equalities δ_{Poin} = dim_{H}(Λ) = dim_{M}(Λ) were established for finite generated groups with zero area limit set of by Jones and myself in [7]. Besides giving new results, the proof of Theorem 1.1 will also give simpler, more direct, proofs of these previously known cases.

6 CHRISTOPHER J. BISHOP

It is obvious that δ_{area} ≤ 2 for any Kleinian group and hence by the theorem δ_{diam} ≤ 2 for any analytically finite group. Thus if {Ω_{j}} is an enumeration of the components of Ω(G), we have

∑_{j} diam(Ω_{j})^{s} < ∞,

for any s > 2 (the diameter is in the spherical metric, or in the Euclidean metric if ∞ ∈ Ω(G) and we sum over the bounded components). Maskit proved this with s = 4 in [22] and conjectured it should be true with s = 2 (he also proved it with s = 3; personal communication).

Theorem 1.2. If G is an analytically finite Kleinian group then

∑_{j} diam(Ω_{j})^{2} < ∞.

If every component of Ω was merely a Euclidean rescaling of one of a finite number of “large” components this would be trivial since the sum could be compared with the sum of areas of the components. This is what happens for geometrically finite groups and groups without parabolics. The idea of the proof of the general case is to associate to each component a measure of how much it is distorted from a simple rescaling and then estimate the number of components with large distortion.

An interesting special case occurs when G is a circle packing group, that is, every component of Ω(G) is a Euclidean disk. (Note: some authors use the term “circle packing” only if the components are disks and Λ has zero area, but here we only mean that the components are disks.) See Figure 1.2. In this case we automatically get δ_{inrad} = δ_{area} = δ_{diam}. If K = S^{2} \ ∪_{j}D(x_{j}, r_{j}) is the complement of a collection of

Figure 1.2. A circle packing limit set

open disks we define the circle packing exponent as

δ_{cir} = inf{s : ∑_{j} r_{j}^{s} < ∞}.

If we sum the Whitney decomposition of a single disk D_{j} we get

∑_{Q_{k}⊂D_{j}} diam(Q_{j})^{s} ≃ r_{j}^{s},

if s > 1 and equals ∞ if s ≤ 1. Thus δ_{Whit}(K) = max(1, δ_{cir}). However, since circle packing groups have connected Λ it is known that δ_{Poin} > 1 ([7]) and thus

(1.3) δ_{Whit} = δ_{inrad} = δ_{area} = δ_{diam} = δ_{cir},

for any analytically finite circle packing group. Furthermore, they all equal dim_{H}(Λ) if area(Λ) = 0. For finitely generated circle packing groups this had been proven by Parker [25], using the results of [7] to reduce to the geometrically finite case. Circle packing groups are also considered in [19] and [8].

Notation: In this paper A ≃ B means that A/B is bounded and bounded away from 0. Given a square S in the plane and λ > 0, λS denotes the concentric square

8 CHRISTOPHER J. BISHOP

with diam(λS) = λdiam(S). D(x, r) denotes a disk in ℝ^{2} with center x and radius r. B(x, r) denotes a ball on S^{2} or 𝔹^{3} in the spherical metric. B_{ρ}(x, r) denotes a ball in 𝔹^{3} (or M = 𝔹^{3}/G) with respect to the hyperbolic metric.

The remaining sections of this paper are organized as follows:

Section 2: δ_{Whit}(K) ≤ \overline{dim_{M}}(K) for any compact K, and equality holds if area(K) = 0. Section 3: δ_{Poin} ≤ δ_{Whit} for analytically finite groups with equality if Ω(G)/G is compact. We also show δ_{Poin} ≤ δ_{inrad} ≤ δ_{diam}. Section 4: We define the isometric circle of a Möbius transformation and give a lemma of Maskit. Section 5: Proof of Theorem 1.1 for geometrically finite groups and groups without parabolics. Section 6: Proof of Maskit’s conjecture. Section 7: We state an estimate for the Poincaré sum which is used in the next two sections and proven in Section 12. Section 8: δ_{diam} ≤ δ_{Poin}. Section 9: δ_{Whit} ≤ δ_{Poin}. Section 10: We show all the exponents equal dim_{H}(Λ) when area(Λ) = 0. Section 11: We give an example to show δ_{Poin} < 2 and area(Λ) > 0 is possible for analytically finite groups. Section 12: Review some facts about the heat kernel on M = 𝔹^{3}/G and prove the estimate on the Poincaré sum used earlier. Section 13: Describe the restrictions between δ_{inrad}, δ_{area}, δ_{diam} and δ_{Whit} which hold for general compact sets.

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 9

I thank Bernie Maskit for several very helpful conversations. This paper was partial motivated by his question from [22] as well as the results of [25]

2. Whitney squares and Minkowski dimension

Here we recall the proof of the following lemma. This is a known fact, but the proof is very short and important for what follows, so we include it for the convenience of the reader. Lemma 2.1. If K is compact then δ_{Whit}(K) ≤ \overline{dim_{M}}(K). If, in addition, area(K) = 0 then δ_{Whit}(K) = \overline{dim_{M}}(K) . Proof. Suppose {Q_{j}} is a Whitney decomposition of Ω = ℝ^{n} \ K. For each Q_{j} with diam(Q_{j}) ≤ diam(K), there is a dyadic cube Q′_{j} of the same size which hits K and satisfies dist(Q_{j}, Q′_{j}) ≤ Cdiam(Q_{j}). Clearly each Q′_{j} is associated to only a bounded number of Whitney cubes. Therefore the number of dyadic cubes of size 2^{−n} which hit K is at least C2^{n(δ_{Whit}−ϵ)} (for n large enough, depending on ϵ). Thus δ_{Whit}(K) ≤ \overline{dim_{M}}(K).

Conversely, if K has zero area, Q is a dyadic square hitting K and {Q_{k}} is the collection of Whitney squares for Ω contained in Q, then

∑_{k} diam(Q_{k})^{2} = diam(Q)^{2}.

Hence for any s ≤ 2, (since diam(Q) ≤ 1), ∑_{k} diam(Q_{k})^{s} ≥ diam(Q)^{s}.

Since there are more than C2^{n(\overline{dim_{M}} (K)−ϵ)} such squares Q, the sum over the whole Whitney collection is greater than

C2^{−ns}2^{n(\overline{dim_{M}} (K)−ϵ)},

10 CHRISTOPHER J. BISHOP

which diverges if s < \overline{dim_{M}}(K) − ϵ. Thus δ_{Whit}(K) ≥ \overline{dim_{M}}(K), as desired.

We can have strict inequality if K has positive area. For example, one can choose a set of disjoint disks D(x_{j}, r_{j}) ⊂ D(0, 1), so that K = \overline{D(0, 1)} \ ∪_{j}D_{j}, is nowhere dense, has positive area and r_{j} → 0 as fast as we wish. If we sum the Whitney decomposition of a single disk we get

∑_{Q_{k}⊂D_{j}} diam(Q_{j})^{s} ≃ r_{j}^{s},

if s > 1 and equals ∞ if s ≤ 1. Thus δ_{Whit}(K) = max(1, δ_{cir}). By taking r_{j} → 0 very fast, we can get

δ_{Whit}(K) = 1 < 2 = \overline{dim_{M}}(K).

3. Whitney squares and the Poincaré series

In this section we explain the elementary relations between δ_{Poin} and δ_{Whit}.

Suppose Ω is a domain in S^{2} with more than two boundary points. Then Ω has a hyperbolic metric ρ defined by the covering map from the disk to Ω. Let d(z) = dist(z, ∂Ω). For a general domain (e.g., [11], Theorem 4.3),

\frac{(1 + o(1))|dz|}{d(z) log 1/d(z)} ≤ |dρ(z)| ≤ 2\frac{|dz|}{d(z)}. (3.1)

A set K ⊂ ℝ^{2} is called uniformly perfect if there is a constant C < ∞ so that

\frac{1}{C} \frac{|dz|}{d(z)} ≤ |dρ(z)| ≤ 2\frac{|dz|}{d(z)},

on each component Ω of S^{2} \ K. (This is one of many equivalent definitions; see [15] and [17].)

Canary [9] has shown that the limit set of any finitely generated group is uniformly perfect. In fact, his proof shows this is true under the weaker assumption that there is an ϵ_{0} > 0 so that any closed geodesic on Ω/G has length ≥ ϵ_{0}. This is certainly true Ω/G is a finite union of finite type surfaces, so his result is still true for analytically finite groups. (Also see Lemma 3.1 of [7].)

Lemma 3.1. If G is any non-elementary Kleinian group with Λ ≠ S^{2} then δ_{Poin} ≤ δ_{Whit}. If Ω(G)/G is compact then δ_{Poin} = δ_{Whit}.

Proof. Fix a point z_{0} ∈ Ω(G) (not an elliptic fixed point). There is a small hyperbolic disk around z_{0} (with radius r_{0} depending on z_{0}) which projects injectively to R = Ω/G under the quotient map. Thus points in G(z_{0}), the orbit of z_{0} under G, are separated by at least r_{0} in the hyperbolic metric. By (3.1) each Whitney square has a uniformly bounded hyperbolic diameter and area. Thus each Whitney square for Ω = S^{2} \ Λ contains at most a bounded number M (depending on z_{0} and G) of points in G(z_{0}). Therefore,

∑_{g∈G} dist(g(z_{0}), Λ)^{s} ≤ M ∑_{j} diam(Q_{j})^{s}

and hence δ_{Poin}(G) ≤ δ_{Whit}(Λ(G)).

Now suppose R = Ω(G)/G = R_{1} ∪ ⋯ ∪ R_{s} is a finite union of compact Riemann surfaces. We can choose points E = {z_{1}, …, z_{s}} ⊂ Ω, so that z_{j} projects into R_{j}, j = 1, …, s under the quotient map. By compactness, any point z ∈ Ω is a bounded hyperbolic distance from G(E), the orbit of E under G. For each square Q with dist(Q, Λ) ≤ 1, choose a closest point z_{Q} ∈ G(E). Then z_{Q} is only a bounded hyperbolic distance from Q so the uniform perfectness of Λ implies

diam(Q) ≤ Cdist(z_{Q}, Λ).

Furthermore, only a bounded number (say M) of the Q_{j}’s can associated to any given

12 CHRISTOPHER J. BISHOP

point of G(E). Thus

∑_{j} diam(Q_{j})^{s} ≤ MC^{s} ∑_{z_{j}∈E} ∑_{g∈G} dist(g(z_{j}), Λ)^{s},

and therefore δ_{Whit}(Λ(G)) ≤ δ_{Poin}(G).

Suppose that Ω(G) = S^{2} \ Λ(G) has infinitely many components. For each component Ω_{j} choose the point z_{j} ∈ G(E) ∩ Ω_{j} which maximizes dist(z, ∂Ω). Define

δ_{orbit}(G) = inf{s : ∑_{j} dist(z_{j}, Λ)^{s} < ∞}.

It’s easy to see that this is independent of the choice of E. δ_{orbit} is the group analog of the index δ_{inrad}, since it is the radius of the largest disk in Ω which is centered at a point of G(E). In particular, δ_{orbit} ≤ δ_{inrad}.

Lemma 3.2. If G is an analytically finite Kleinian group such that Ω(G) has infinitely many components, then δ_{Poin} = δ_{orbit}.

Proof. The direction δ_{orbit} ≤ δ_{Poin} is obvious since δ_{orbit} is defined by summing over a strict subseries of the Poincaré series defining δ_{Poin}. To prove the other direction we simply note that if Ω(G) has infinitely many components then we can find two distinct components Ω_{1}, Ω_{2} which are at positive (spherical) distance from each other. This is because if we fix a component Ω_{0} of Ω(G) then there are infinitely many disjoint quasidisks in S^{2} \ Ω_{0} each of which contain components of Ω(G) (see Maskit’s book [23]). Moreover the quasidisks have diameters tending to zero and accumulate at every point of ∂Ω_{0}. Thus is possible to choose two at positive distance from each other, and so the corresponding components of Ω(G) will also be at a positive distance.

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 13

Fix a point z_{0} ∈ Ω(G) in the definition of the Poincaré series. Fix orbit points z_{1} ∈ G(z_{0}) ∩ Ω_{1} and z_{2} ∈ G(z_{0}) ∩ Ω_{2} and balls B_{i} ⊂ Ω_{i} for i = 1, 2. Consider the cone C_{i} of hyperbolic geodesics in 𝔹^{3} which connect the origin 0 ∈ 𝔹^{3} to B_{i}. Since the components are at positive distance from each other, these cones are separated by a definite positive angle at 0. For any g ∈ G consider the image cones g(C_{1}) and g(C_{2}) which connect g(0) to g(B_{1}) and g(B_{2}). Since these cones are separated by a positive angle (independent of g), at least one of them makes a positive angle with the radial segment [0, g(0)]. Call this index i_{g} ∈ {1, 2}. Simple hyperbolic geometry now implies that

dist(g(z_{i_{g}}), ∂g(Ω_{i_{g}})) ≃ diam(g(Ω_{i_{g}})) ≃ 1 − |g(0)|.

This implies that each component of Ω(G) can contain only a bounded number of point of the form g(z_{i_{g}}). Thus

∑_{g∈G}(1 − |g(0)|)^{s} ≤ C ∑_{g∈G} dist(g(z_{i_{g}}))^{s},

and only a bounded number of distinct g(z_{i_{g}})’s lie in any one component of G. Thus the right hand side is bounded by a constant times the series defining δ_{orbit}. Thus δ_{Poin} = δ_{orbit}.

Since we have already noted that δ_{orbit} ≤ inrad, we get

Corollary 3.3. If G is an analytically finite Kleinian group so that Ω(G) has infinitely many components, then δ_{Poin} ≤ δ_{inrad} ≤ δ_{diam}.

14 CHRISTOPHER J. BISHOP

4. Isometric circles

Consider the Möbius transformation

g(z) = \frac{az + b}{cz + d},

and its derivative

g′(z) = \frac{ad − bc}{(cz + d)^{2}} = \frac{1}{(cz + d)^{2}} ,

(since we can identify Möbius transformations with elements of PSL(2, ℂ) we may normalize them so ad − bc = 1; assume that we have done this). The circle {z : |cz + d| = 1\}, with center c_{τ} = −\frac{d}{c} = τ^{−1}(∞) and radius r_{τ} = |c|^{−1/2} is called the isometric circle of τ. The mapping τ can be written as a composition τ(z) = f(h(z)), where h is the inversion through the isometric circle and f is a (Euclidean) isometry of the plane. (see [16], Sections 17 and 47).

Suppose G is a Kleinian group and Ω_{0} is some component of the ordinary set Ω(G) = S^{2} \ Λ(G). Let G_{0} = {g ∈ G : g(Ω_{0}) = Ω_{0}} be stabilzer ofΩ_{0}. Normalize the group G so that ∞ ∈ Ω_{0} and diam(Λ(G)) = 1. Suppose that Ω_{0} is not invariant under all of G and let Ω_{j} = g(Ω_{0}) be a distinct component of Ω(G) which is equivalent to Ω_{0}. Let S_{j} = {g ∈ G : g(Ω_{0}) = Ω_{j}} = gG_{0} be the corresponding coset. Let C_{j} be the set of centers of isometric circles of elements in S_{j}. Note that each point of C_{j} is of the form w = g^{−1}(∞) for g ∈ S_{j} and hence w ∉ Ω_{0} since ∞ ∈ Ω_{0} and g^{−1}(Ω_{0}) ≠ Ω_{0} by assumption.

Let h_{1} ∈ G_{0} and w_{1} = (gh_{1})^{−1}(∞) ∈ C_{j}. If h_{2} ∈ G_{0}, then

h_{2}(w) = h_{2}(h_{1}^{−1}(g^{−1}(∞))) = (gh_{1}h_{2}^{−1})^{−1}(∞),

is in C_{j} as well. In other words, the set C_{j} is invariant under G_{0}.

Figure 4.1. Bounding the diameter of Ω_{j}

Now let F be a fundamental region in Ω(G_{0}) for the action of G_{0}. Let F_{0} = F \Ω_{0}. By our remarks above, given any component Ω_{j} ≠ Ω_{0} of Ω(G) which is equivalent to Ω_{0} in G, there is an element g_{j} ∈ G so that g(Ω_{0}) = Ω_{j} and such that the center c_{j} of g_{j}’s isometric circle lies in F_{0}. Define

η_{j} = η(Ω_{j}) = dist(c_{j}, Ω_{0}).

This will serve as a measure of how distorted Ω_{j} is compared to Ω_{0}. In particular, if G has no parabolics, then \overline{F_{0}} ∩\overline{Ω_{0}} = ∅, so η is bounded away from zero independently of j and we shall see in the next section that this implies Ω_{j} is (up to a Euclidean similarity) a bounded distortion of Ω_{0}.

Let P be the finite set of cusp points on the boundary of F_{0}. The following is due to Maskit [22].

Lemma 4.1. Normalize G so that ∞ ∈ Ω_{0} and diam(Λ) = 1. Suppose Ω_{j} = g(Ω_{0}), c_{j} ∈ F_{0} is the center of the isometric circle of g, η_{j} = dist(c_{j}, P) and r_{j} is the radius of the isometric circle of g. Then there is a C = C(G) so that

C^{−1} \frac{r_{j}^{2}}{η_{j}} ≤ diam(Ω_{j}) ≤ C \frac{r_{j}^{2}}{η_{j}}

16 CHRISTOPHER J. BISHOP

Proof. Let p ∈ P be a parabolic cusp closest to c_{j}. See Figure 4.1. Let D be the largest disk in Ω(G_{0}) which is centered at c_{j}. By our remarks above, its radius is uniformly comparable to η_{j}. Thus when we invert it through the circle centered at c_{j} of radius r_{j} it is mapped to the exterior of the disk (also centered at c_{j}) of radius comparable to r_{j}^{2}/η_{j}. Thus Ω_{0}, which lies outside of D, is mapped by the inversion into a disk of this radius. Since g_{j} can be written as this inversion followed by a Euclidean isometry we deduce that Ω_{j} = g_{j}(Ω_{0}) has diameter at most Cr_{j}^{2}/η_{j}. On the other hand, note that the inversion maps ∞ to c_{j} and maps the point p to a point at distance ≥ Cr_{j}^{2}/η_{j} from c_{j}. Since ∞ ∈ Ω_{0} and p ∈ ∂Ω_{0}, this gives the lower bound for diam(Ω_{j}).

Later we will want to apply the lemma in a slightly different setting when Ω is replaced by a Euclidean ball B ⊂ Ω_{0} which has p on its boundary. The argument above implies that if diam(B) ≃ 1 then

diam(g(B)) ≃ \frac{r^{2}}{η} , (4.1)

where r is the radius of the isometric circle of g and η is the distance of its center from p.

5. The geometrically finite case

Recall that if we fix a component Ω_{0} of Ω(G) and consider a distinct component Ω_{j} = g(Ω_{0}), then we can choose g = g_{j} so that the isometric circle for g_{j} has radius r_{j} and center c_{j} ∈ F_{0}. The set F_{0} may be taken to be a finite union of components each of which is bounded by a piecewise smooth curve which hits Λ at only a finite number of points, each a rank 1 parabolic fixed point of G_{0}. At each such point p there is a component of Ω(G_{0}) intersecting F_{0} and which contains a disk which is

Figure 5.1. The geometry of F

invariant under h and so that there is a neighborhood of p in which F_{0} looks like a cusp bounded by two circles tangent at p. In particular, there is a uniform bound C (depending only on F_{0}, hence only on the choice of Ω_{0}) so that for each z ∈ F_{0}, the component of Ω(G_{0}) contains a disk D_{z} centered at z whose radius is comparable to dist(z, P). (This is clear near the parabolic cusps of F_{0} and true by compactness outside the cusp regions.) See Figure 5.1. Lemma 5.1. Suppose p ∈ ∂F_{0} is a rank one parabolic fixed point of G_{0} (the stabilizer of Ω_{0}) and h is a parabolic element fixing p. If Ω(G) \ Ω_{0} contains a disk D invariant under h, then dist(c_{g}, p) > η(G) is bounded away from zero for all elements g ∈ G with c_{g} ∈ F_{0}. Proof. If Ω(G) has such a disk, it corresponds to a neighborhood of a puncture on the surface Ω(G)/G. By taking the disk smaller we may assume that ∞ is not contained in any G-image of the disk D. If D ∩ F_{0} contains the center c_{j} of the isometric circle corresponding to some g_{j} then g_{j} maps the disk D to the exterior of a disk of radius ≃ r_{j}^{2}/η_{j} and hence g_{j}(D) contains ∞, a contradiction. Thus the centers of isometric

Figure 5.2. Horoballs and cusps

circles are bounded away from p.

A horoball in Ω(G) is a Euclidean ball B ⊂ Ω ⊂ S^{2} which is invariant under a rank 1 parabolic subgroup of G. The fixed point p of the parabolic element is on the boundary of the horoball and corresponds to a cusp on the surface Ω(G)/G. Conversely, to each cusp on the quotient surface there is a equivalence class of horoballs in Ω(G).

We say that p is doubly cusped if there are two disjoint horoballs in Ω(G), both invariant under the parabolic subgroup fixing p. The following is due to Beardon and Maskit.

Theorem 5.2. ( [3]) If G is a Kleinian group then G is geometrically finite iff every point of Λ(G) is either a conical limit point, a rank 2 parabolic fixed point or a doubly cusped rank 1 parabolic fixed point.

In particular, their result implies that in a geometrically finite group, every rank 1 parabolic fixed point must be doubly cusped (this particular consequence is also proven in [7], Corollary 4.6). By Lemma 5.1 this means that in a geometrically finite group there is a η_{0} > 0 (depending only on G) so that for every component η(Ω_{j}) ≥ η_{0}.

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 19

Lemma 5.3. Suppose G is an analytically finite Kleinian group such that Ω(G) has infinitely many components {Ω_{j}} and that η_{j} is bounded away from zero independently of j. Then

δ_{inrad}(Λ) = δ_{area}(Λ) = δ_{diam}(Λ) = δ_{Whit}(Λ).

Proof. The proof of the theorem is easy because under this hypothesis, each component Ω_{j} = g_{j}(Ω_{0}) is only a rescaling of Ω_{0} with a bounded amount of distortion. More precisely suppose Ω_{j} = g_{j}(Ω_{0}) and let c_{j}, r_{j} be the center and radius of the isometric circle of g_{j}. Recall that we can write g_{j} = f_{j} ∘ h_{j} where

h_{j}(z) = \frac{r_{j}^{2}}{(z − c_{j})^{2}} + c_{j},

is inversion though the isometric circle of g_{j} and f_{j} is a Euclidean isometry. Clearly f_{j} does not effect any of the quantities in question. We can further decompose h_{j} as the composition of four maps: translation by −c_{j}, inversion through the unit circle, a dilation (multiplication by r_{j}^{2}) and translation by c_{j}. The inversion is a smooth map except at the origin, so will have derivative bounded and bounded away from zero on any annulus {η_{0} ≤ |z| ≤ η_{0}^{−1}}.

Thus g_{j} : Ω_{0} → Ω_{j} can be written as a composition of three maps: the first has uniformly bounded distortion on a uniform neighborhood of ∂Ω_{0}, the second is a Euclidean dilation and the third is a Euclidean isometry. From this we deduce that

inrad(Ω_{j}) ≃ diam(Ω_{j}) ≃ \sqrt{area(Ω_{j})},

with constants that depend only on Ω_{0} and η_{0}, but not on j. Thus

δ_{inrad} = δ_{diam} = δ_{area}.

Let {Q_{k}} be the Whitney squares for Ω_{0} with dist(Q_{k}, ∂Ω_{0}) ≤ 1. Each is mapped via g_{j} is to a region which satisfies

diam(g_{j}(Q_{k})) ≃ inrad(g_{j}(Q_{k})) ≃ dist(g_{j}(Q_{k}), ∂Ω_{j}) ≃ diam(Ω_{j})diam(Q_{j}).

Thus each g_{j}(Q_{k}) intersects a bounded number of Whitney squares Q̃_{k} for Ω_{j} and only a bounded number of Q̃_{k}’s are not hit. Thus the Whitney sum for Ω_{j} is bounded by

∑_{k} diam(Q̃_{k})^{s} ≤ Cdiam(Ω_{j})^{s} + diam(Ω_{j})^{s} ∑_{k} diam(Q_{k})^{s}
≤ Cdiam(Ω_{j})^{s}.

Thus δ_{Whit} ≤ δ_{diam}. Equality holds since there is a Whitney square in Ω_{j} with diameter comparable to diam(Ω_{j}).

Lemma 5.4. If G is geometrically finite then δ_{Poin} = δ_{Whit}.

Proof. We already know this result if Ω(G)/G is compact (Lemma 3.1) so we may as well assume there are cusps, i.e., Ω(G) contains horoballs. Since G is analytically finite, the quotient surface has at most finitely many cusps, so there are at most finitely many equivalence classes on horoballs in Ω(G). Let {B_{1}, …, B_{s}} be a choice of representative from each class. The collection of all G images of these balls will be denoted B. It will be convenient to assume that G is normalized so that ∞ ∈ Ω(G) and ∞ is not contained in any horoball (which we may assume by conjugating the group and taking smaller horoballs if necessary). We also assume that all the horoballs are pairwise disjoint. By making the horoballs even smaller we may assume that each horoball in our finite collection {B_{j}} is contained in a horoball of twice the Euclidean

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 21

diameter and so that the larger balls do not intersect the orbit G(∞). The larger balls will be denoted {B̃_{j}}.

Choose a finite number of points E = {z_{1}, …, z_{t}}, one from each equivalence class of components of Ω(G). There is M < ∞ so that Q is a Whitney square which is not contained in any horoball of B, then Q is within hyperbolic distance M of some G-image of a point of E. The argument of Section 3 then implies that

∑_{Q_{j}∩B=∅} diam(Q_{j})^{s} ≤ C ∑_{z∈G(E)} dist(z, Λ)^{s}.

Thus to prove δ_{Whit} ≤ δ_{Poin} we only have to consider Whitney squares contained in horoballs.

Given a ball B ⊂ Ω(G), it is easy to see that B contains at most C2^{n} Whitney squares for Ω(G) of size ≥ 2^{−n}diam(B). Therefore we get

∑_{Q_{j}⊂B} diam(Q_{j})^{s} ≤ Cdiam(B)^{s}.

Therefore it is enough to show that if s > δ_{Poin} then

∑_{B∈B} diam(B)^{s} < ∞.

By breaking the sum into a finite number of sums, we need only consider the sum over all horoballs are G-equivalent to a fixed horoball, say B_{1}.

Given a horoball B = g(B_{1}), let H_{1} = {h ∈ G : h(B_{1}) = B_{1}} be the subgroup of G fixing B_{1} and let S = {h ∈ G : h(B) = B} = gH_{1}. Let T = g^{−1}(∞) : g ∈ S be the collection of centers of the isometric circles of these elements. We want to see that T is a H_{1}-invariant set. To see this suppose gh_{1} ∈ S and h_{2} ∈ H_{1}. Then

h_{2}((gh_{1})^{−1}(∞)) = (gh_{1}h_{2}^{−1})^{−1}(∞) ∈ T,

Figure 5.3. Fundamental domain for a parabolic group

as desired. Thus for any horoball B ≠ B_{1} which is equivalent to B_{1}, we may choose g ∈ G with B = g(B_{1}) so that g^{−1}(∞) lies in a fundamental domain F for H_{1}, but outside B_{1}. See Figure 5.3.

Since G is geometrically finite, the parabolic fixed point p is doubly cusped, so there is a second horoball (disjoint from B_{1}) at p. Since B_{2} does not contain any orbits of ∞ the centers of any isometric circles can be chosen in F outside of both B_{1} and B_{2}. Thus if B = g(B_{1}) we see g can be chosen so that the center of the corresponding isometric circle is bounded away from p (independently of B). This means that B is a bounded distortion image of B_{1} under g. Therefore there is point z ∈ G(E) with

dist(z, B) ≃ dist(z, Λ) ≃ diam(B),

with constants that are independent of B. Thus

∑_{B∈B} diam(B)^{s} ≤ C ∑_{z∈G(E)} dist(z, Λ)^{s} < ∞,

if s > δ_{Poin}. This proves δ_{Whit} ≤ δ_{Poin}, as desired.

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 23

If G has no parabolics then every component of Ω(G) has η bounded away from zero. Combining this observation with the previous two lemmas and Lemma 3.1 gives

Corollary 5.5. Suppose G is an analytically finite Kleinian group such that Ω(G) has infinitely many components {Ω_{j}} and that either G has no parabolics or G is geometrically finite. Then

δ_{inrad}(Λ) = δ_{area}(Λ) = δ_{diam}(Λ) = δ_{Poin}(G) = δ_{Whit}(Λ).

6. The proof of Maskit’s conjecture

Before proceeding with the proof of Theorem 1.1 in the remaining cases (i.e., geometrically infinite with parabolics) we will prove Maskit’s diameter sum conjecture (Theorem 1.2). The estimates we use here are a model for the argument used to finish the proof of Theorem 1.1.

Suppose {Ω_{j}} is an enumeration of the components of Ω(G). We wish to show

∑_{j} diam(Ω_{j})^{2} < ∞.

The components of Ω(G) belong to a finite number of equivalence classes, so it is enough to prove that the sum over each class is finite. So fix a component Ω_{0} and normalize G so that ∞ ∈ Ω_{0}, 0 ∈ Λ and diam(Λ) = 1. Suppose {Ω_{j}} is an enumeration of the components which are equivalent to Ω_{0}. If {Ω_{j}} is a finite list, there is nothing to do, so we assume that there are infinitely many such components.

Let D_{0} = {|z| > R} where R is chosen so large that all G-images of D_{0} are disjoint (we may assume without loss of generality that ∞ is not an elliptic fixed point). Consider a component Ω_{j} = g_{j}(Ω_{0}) and decompose g_{j} = f_{j} ∘ h_{j} as a composition of inversion through the isometric circle followed by a Euclidean isometry. As in

24 CHRISTOPHER J. BISHOP

Section 4, c_{j} = g^{−1}(∞) be the center of the isometric circle, r_{j} its radius and η_{j} ≃ dist(c_{j}, P) (where P is the set of parabolic fixed points on ∂F_{0}, the fundamental domain of G(Ω_{0})).

By Lemma 4.1 we know

diam(Ω_{j}) ≃ \frac{r_{j}^{2}}{η_{j}} .

On the other hand, D_{j} = g^{−1}(D_{0}) is a disk containing c_{j} of diameter ≃ r_{j}^{2}. These disks are disjoint from their images under the parabolics fixing various points of P, so

diam(D_{j}) ≤ Cdist(c_{j}, P)^{2}.

Thus the union of the disks {D_{j}} is contained in a neighborhood F′ of F which also has a cusp at p. In particular, near each parabolic point p ∈ ∂F,

area(F′ ∩ D(p, r)) ≤ Cr^{3}.

Thus

∑_{j} diam(Ω_{j})^{2} ≤ C ∑_{j} \frac{r_{j}^{4}}{η_{j}^{2}}
≤ C ∑_{j} area(D_{j})dist(D_{j}, P)^{−2}
≤ C ∑_{n} 2^{2n} ∑_{j:2^{−n−1}<dist(c_{j},P)≤2^{−n}} area(D_{j})
≤ C ∑_{n} 2^{−3n}2^{2n}
< ∞.

This proves Theorem 1.2

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 25

7. An estimate for the Poincaré series

To prove δ_{diam} ≤ δ_{Poin} and δ_{Whit} ≤ δ_{Poin} we will use the following estimate for the Poincaré sum which will be proven in Section 12. In what follows,

V (x) = vol(B_{ρ}(x, 1)/G),

i.e., V (x) is the volume of a unit hyperbolic ball around the projection of x in M = 𝔹^{3}/G.

Lemma 7.1. Suppose x ∈ S^{2} and let y = (1 − r)x ∈ 𝔹^{3}. Given any s > δ_{Poin} there is a C_{s} < ∞ so that for any (spherical) ball B = B(x, r),

∑_{g(0)∈B}(1 − |g(0)|)^{s} ≤ \frac{C_{s}r^{s}e^{−sρ(y,G(0))}}{\sqrt{V (0)V (y)}} .

We will only need to apply this is a very special case which we now describe in detail. According to the Margulis lemma (e.g., page 134 of [4]) there is an ϵ > 0 (independent of G) so that for any y ∈ 𝔹, then the group generated by

G_{y} = {g ∈ G : ρ(y, g(y)) ≤ ϵ},

is almost-nilpotent and hence elementary.

Suppose p ∈ S^{2} is a parabolic fixed point and let H ⊂ G be the parabolic subgroup which fixes p. Fix the H-invariant Euclidean ball T ⊂ 𝔹^{3} so that each element of H moves points in T (the “thin part”) less than this ϵ. Then for any y ∈ T the subgroup G_{y} contains H and is elementary. By the classification of elementary groups given on page 1085 of [20], G_{y} is either H or conjugate to one of a finite number of possibilities which contain H as a finite index subgroup. In each case the elliptics have order at most 6 and the elliptic fixed points in ∂T are uniformly separated in the hyperbolic

26 CHRISTOPHER J. BISHOP

metric. This implies that for y ∈ ∂T, V (y) is bounded away from zero by a constant which depends only on the choice of T, i.e., only on ϵ and the conjugacy class of p.

Thus Lemma 7.1 implies that if y = (1 − r)x is on the boundary of the thin part T of some parabolic we have

∑_{g(0)∈B}(1 − |g(0)|)^{s} ≤ C_{s}r^{s}. (7.1)

with a constant that depends on s and G, but not on y. More accurate estimates are available for geometrically finite groups. See e.g., [29].

Another result which we will need in the next two sections is Lemma 7.2. If G is analytically finite and δ_{Poin} < 1 then G is geometrically finite.

This is proven in [7] under the weaker hypothesis δ_{Poin} ≤ 1, but we will sketch a different proof of the lemma for completeness. Also see [10].

If Λ is totally disconnected then Ω(G) is connected and so G is finitely generated. In this case, a theorem of Abikoff and Maskit [1] implies G is geometrically finite. If Λ is not totally disconnected then the same is true of the boundary of any component of Ω. Choose a component Ω_{0} and let G_{0} ⊂ G be the subgroup fixing Ω_{0}. Then Ω_{0} is an invariant component for G_{0} and G_{0} is finitely generated, so the decomposition theorem for function groups [1] says there is a finitely generated subgroup H ⊂ G_{0} with connected limit set. Let Ω_{1} be a (simply connected) component of Ω(H) and Φ : 𝔻 → Ω_{1} be a Riemann mapping from the unit disk. If z_{0} ∈ Ω_{1} and {z_{j}} = H(z_{0}), and δ_{Poin}(H) < 1 then

∑_{z∈H(z_{0})} dist(z, ∂Ω_{1})^{s} < ∞,

for some s < 1. This means that the set F = ∩_{j}D(z_{j}, 2dist(z_{j}, ∂Ω_{j})),

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 27

has dimension ≤ s < 1. On the other hand, Φ^{−1}({z_{j}}) is the orbit in 𝔻 of a finitely generated Fuchsian group of the first kind, which implies it is non-tangentially dense almost everywhere on the unit circle, which in turn implies Φ^{−1}(F) has full measure. This contradicts Makarov’s theorem [21] and proves δ_{Poin}(G) ≥ δ_{Poin}(H) ≥ 1, as desired.

8. The diameter sum and Poincaré series

In this section we will show that δ_{diam} ≤ δ_{Poin} (except for the proof of Lemma 7.1). The argument is almost the same as in Section 6, except that we use the estimate of the previous section instead of the fact that ∑_{k} area(D_{j}) < ∞.

As in Section 6, let {Ω_{j}} be an enumeration of the components equivalent to a fixed component Ω_{0} which has been normalized so that ∞ ∈ Ω_{0}, 0 ∈ Λ and diam(Λ) = 1. Also, since we have equality for geometrically finite groups we may assume that G is geometrically infinite. This implies that δ_{Poin} ≥ 1 by Lemma 7.2.

Note,

∑_{j} diam(Ω_{j})^{s} ≤ C ∑_{j} \frac{r_{j}^{2s}}{η_{j}^{s}}
≤ C ∑_{j} dist(c_{j}, Λ)^{s}dist(c_{j}, P)^{−s}
≤ C ∑_{n} ∑_{j:2^{−n−1}<dist(c_{j},P)≤2^{−n}} dist(c_{j}, Λ)^{s}2^{sn}.

For each n we only have to consider the points {c_{j}} which lie in the intersection of the annulus 2^{−n−1} ≤ |z − p| < 2^{−n} with F. These points lie in a narrow rectangle of width 2^{−2n} and length 2^{−n}. This strip can be covered by approximately 2^{n} disks of diameter approximately 2^{−2n}. The tops of the hemispheres with these disks as bases lie may be chosen to be on the boundary of the thin part associated to p, if n is large

28 CHRISTOPHER J. BISHOP

enough. Thus equation (7.1) implies that the Poincaré sum for each disk is bounded by C2^{−2sn}. Since there are 2^{n} disks, the Poincaré sum for the whole strip is bounded by C2^{(1−2s)n}. Thus since we are assuming s > 1,

∑_{j} diam(Ω_{j})^{s} ≤ C ∑_{n} 2^{sn}2^{n(1−2s)} ≤ C ∑_{n} 2^{n(1−s)} < ∞.

This proves δ_{diam} ≤ δ_{Poin}.

9. The Whitney and Poincaré sums

In this section we will prove δ_{Whit} ≤ δ_{Poin}. The proof is almost identical to the proof of δ_{diam} ≤ δ_{Poin}, except that we apply Maskit’s lemma (Lemma 4.1) to horoballs instead of components.

We already know the result if G is geometrically finite (see Section 5) so we may assume G is geometrically infinite. Thus we may assume δ_{Poin} ≥ 1.

We use the same notation as in the proof of Lemma 5.4. From the proof of Lemma 5.4 we know that it is enough to show that

∑_{B} diam(B)^{s} < ∞,

if s > δ_{Poin} and the sum is over all horoballs which are equivalent to a fixed horoball B_{1}. Estimate (4.1) gives us

diam(B_{j}) ≤ C \frac{r_{j}^{2}}{η_{j}} ,

if B_{j} = g_{j}(B_{1}).

The proof from this point is identical to the argument in Section 8. Let {B_{j}} = {g_{j}(B_{1})} be an enumeration of the horoballs which are equivalent to B_{1} and assume g_{j} has been chosen so that g^{−1}(∞) lies in a fixed fundamental domain for H_{1} (the parabolic subgroup fixing the point p ∈ ∂B_{1}). The part of the fundamental domain

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 29

near the fixed point p is bounded by two circles tangent at p and so can be covered by a union of rectangles of size approximately 2^{−n} × 2^{−2n}. Thus exactly as in Section 8 we get

∑_{j} diam(B_{j})^{s} ≤ C ∑_{j} \frac{r_{j}^{2}}{η_{j}}
≤ C ∑_{j} dist(c_{j}, Λ)^{s}dist(c_{j}, p)^{s}
≤ ∑_{n=1}^{∞} 2^{sn} ∑_{j:2^{−n−1}<dist(x_{j},p)≤2^{n}} dist(c_{j}, Λ)^{s}.

As in Section 8 we use (7.1) to show that the inner sum is bounded by C2^{n(1−2s)} if s > δ_{Poin}. Thus since s > δ_{Poin} ≥ 1, we get

∑_{j} diam(B)^{s} ≤ C_{s} ∑_{n} 2^{sn}2^{n(1−2s)} < ∞,

as desired.

10. Limit sets with zero area

We have now proven that if G is analytically finite then all the critical exponents we have defined are the same, i.e.,

δ_{Poin} = δ_{inrad} = δ_{area} = δ_{Whit} = δ_{diam}.

Moreover, by Lemma 2.1 they all equal \overline{dim_{M}}(Λ) if Λ has zero area. By Theorem 1.1 of [7], δ_{Poin} ≤ dim_{H}(Λ), for any non-elementary Kleinian group, so with the trivial inequality dim_{H}(Λ) ≤ \overline{dim_{M}}(Λ), we deduce

δ_{Poin} = δ_{inrad} = δ_{area} = δ_{Whit} = δ_{diam} = dim_{H}(Λ) = dim_{M}(Λ),

whenever G is analytically finite with infinitely many components and area(Λ) = 0. If area(Λ) > 0 then dim_{H}(Λ) = dim_{M}(Λ) = 2, so the two dimensions agree in all

Figure 11.1. Constructing a limit set with positive area

cases.

11. Limit sets with positive area

There exist analytically finite groups with area(Λ) > 0, but δ_{Poin} < 2. The construction is given in Maskit’s book [23], so we will only sketch it here. Let C_{1}, C_{2}, C_{3}, C_{4} be circles of radius \sqrt{2}/2 centered at the points 1, i, −1, −i, and let G_{0} be the Fuchsian group generated by mapping the exterior of C_{1} to the interior of C_{3} and the exterior of C_{2} to the interior of C_{4} and preserving the circle C_{0} of radius \sqrt{2}/2 around the origin. Take a fundamental region for the group (say the exterior of all four circles) and let F be the component which lies inside C_{0}. Now choose a sequence of circles with disjoint interiors {D_{j}} inside F and assume diam(K_{2j}) = diam(K_{2j+1}) ≃ dist(K_{2j}, K_{2j+1}). Choose the circle so that E = F \∪_{j}D_{j} has no interior. Let G be the group generated by G_{0} and the transformations which map D_{2j} to the exterior of D_{2j+1}. In [23], Maskit proves that G is an analytically finite group where every component of Ω(G) is equivalent to the exterior of C_{0}.

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 31

However, G is a union of classical Schottky groups, so δ_{Poin} < 2 by Doyle’s theorem [14] and the fact that δ(∪_{n}G_{n}) = sup δ(G_{n}) if G = ∪_{n}G_{n} is a Kleinian group (see [7]). On the other hand, the accumulation set of the disks {D_{j}} is clearly in the limit set and it is easy to choose the disks so that this set has positive area. Thus area(Λ) > 0 but δ_{Poin} < 2.

In fact, this construction leads to sets of positive area, no matter how the disjoint disks are chosen. This is because the group is geometrically infinite, and so δ_{Poin} < 2 implies area(Λ) > 0 by the results of [7].

12. The heat kernel and Poincaré sum

In this section we will use an estimate on the heat kernel to deduce a bound for the number of orbit points in a certain region. This is turn will give the estimate of the Poincaré series which was stated in Section 7.

The heat kernel is the fundamental solution for the heat equation on M, i.e.,

u(x, t) = ∫_{M} u_{0}(y)K(x, y, t)dy,

is the solution of the heat equation u_{t} = Δu with initial value u_{0}. It can also be interpreted in terms of Brownian motion: K(x, y, t)dy is the distribution at time t of a Brownian motion started at the point x. For x ∈ M = 𝔹/G, let V (x) = vol(B_{M}(x, 1)) denote the hyperbolic volume of the unit ball centered at x. In [12] E.B. Davies proved:

Theorem 12.1. If ϵ > 0 then there is a constant C = C_{ϵ} such that

K(x, y, t) ≤ C_{ϵ}V (x)^{−1/2}V (y)^{−1/2}e^{(ϵ−λ_{0})t}e^{−ρ(x,y)^{2}(4+ϵ)^{−1}t^{−1}},

32 CHRISTOPHER J. BISHOP

for 1 ≤ t < ∞, where λ_{0} is the base eigenvalue for the Laplacian on M and ρ is the hyperbolic metric on M.

A different proof is given by Grigor’yan in [18].

We recall that if k(w, z, t) is the heat kernel on the hyperbolic ball 𝔹, then ([13], page 178)

k(w, z, t) = (4πt)^{−3/2} \frac{ρ(w, z)}{sinh(ρ(w, z))}e^{−t− \frac{ρ(w,z)^{2}}{4t}} .

Using the observation that

\frac{r}{sinh(r)} ≃ (1 + r)e^{−r},

for r > 0 gives,

(12.1)  k(w, z, t) ≃ t^{−3/2}(1 + ρ(w, z)) exp(−t − ρ(w, z) − \frac{ρ(w, z)^{2}}{4t}).

The exponent δ_{Poin} is also related to the base eigenvalue λ_{0} for the manifold M = 𝔹^{3}/G by

λ_{0} = \begin{cases}1, & δ_{Poin} ≤ 1 \\ δ_{Poin}(2 − δ_{Poin}), & 1 < δ_{Poin} ≤ 2 \end{cases} .

See [31]. This allows us to prove the following estimate (which was stated in [7]), Lemma 12.2. Let δ = δ_{Poin} = 1 + \sqrt{1 − λ_{0}}. Given x, y ∈ M = 𝔹/G, assume x corresponds to an orbit containing 0 via the covering map and y to an orbit {y_{j}}. Let ρ = ρ(x, y) and

N_{k} = #({y_{j} : ρ + k ≤ ρ(0, y_{j}) < ρ + k + 1}).

Then for any ϵ > 0,

N_{k} ≤ C_{ϵ} \frac{exp((1 + ϵ)δ(ρ + k)[1 − \frac{(1−λ_{0})^{1/2}}{δ} (1 − \sqrt{\frac{2kρ+k^{2}}{(ρ+k)^{2}}} )])}{V (x)^{1/2}V (y)^{1/2}},
≤ C_{ϵ} \frac{e^{(1+ϵ)δ(ρ+k)}}{V (x)^{1/2}V (y)^{1/2}}

GEOMETRIC EXPONENTS AND KLEINIAN GROUPS 33

where λ_{0} = δ(2 − δ).

Proof. Fix x, y ∈ M and let w ∈ 𝔹 correspond to x via the covering map and let {z_{j}} ⊂ 𝔹 be all the points corresponding to y. Then

K(x, y, t) = ∑_{j} k(w, z_{j}, t).

Using Theorem 12.1 and the formula for the heat kernel on the ball gives

N_{k}t^{−3/2}(1 + ρ + k)e^{−t−ρ−k− \frac{(ρ+k)^{2}}{4t}} ≤ \frac{C_{ϵ}e^{(ϵ−λ_{0})t}e^{− \frac{1}{4+ϵ} t^{−1}ρ^{2}}}{V (x)^{1/2}V (y)^{1/2}}

or

N_{k} ≤ \frac{C_{ϵ}t^{3/2}e^{(ϵ−λ_{0})t}e^{t+ρ+k+ \frac{(ρ+k)^{2}}{4t}} e^{− \frac{ρ^{2}}{(4+ϵ)t}}}{(1 + ρ + k)V (x)^{1/2}V (y)^{1/2}}
≤ \frac{C_{ϵ}t^{3/2}e^{(ϵ−λ_{0})t}e^{t+ρ+k+ \frac{(ρ+k)^{2}}{4t}} e^{− \frac{ρ^{2}}{(4+ϵ)t}}}{(1 + ρ + k)V (x)^{1/2}V (y)^{1/2}}
≤ \frac{C_{ϵ}t^{3/2}e^{(1−λ_{0})t}e^{ρ+k}e^{\frac{2ρk+k^{2}}{4t}} e^{ϵt}e^{\frac{ϵρ^{2}}{4(4+ϵ)t}}}{(1 + ρ + k)V (x)^{1/2}V (y)^{1/2}} .

To find the optimal value of t to put into the exponential we minimize

at + b + ct^{−1} = \frac{1}{a}(t − \sqrt{\frac{c}{a}})^{2} + b + 2\sqrt{ac},

where

a = 1 − λ_{0}, b = ρ + k, c = \frac{2ρk + k^{2}}{4t} .

This leads to the choice

t = \sqrt{\frac{c}{a}} = \frac{1}{2} \begin{matrix}√ & \overline{2ρk + k^{2}} \\ √ & \overline{1 − λ_{0}} \end{matrix} .

Thus

(12.2)  N_{k} ≤ \frac{C_{ϵ}[ρ(ρ + k)]^{3/4}e^{ρ+k+\sqrt{2ρk+k^{2}}\sqrt{1−λ_{0}}}e^{Cϵ(ρ+k)}}{(1 + ρ + k)V (x)^{1/2}V (y)^{1/2}} .

Since 1 + \sqrt{1 − λ_{0}} = δ, this gives (making C larger if necessary),

N_{k} ≤ C_{ϵ} \frac{e^{(1+Cϵ)δ(ρ+k)[1−^{\frac{\sqrt{1−λ_{0}}}{δ}} (1−\sqrt{\frac{2ρk+k^{2}}{(ρ+k)^{2}}} )]}}{V (x)^{1/2}V (y)^{1/2}} .

Now use (12.2) and the facts δ = 1 + \sqrt{1 − λ_{0}} and 2ρk + k^{2} ≤ (ρ + k)^{2}, to get

(12.3)  N_{k} ≤ \frac{C_{ϵ}e^{ρ+k+(ρ+k)\sqrt{1−λ_{0}}}e^{ϵ(ρ+k)}}{V (x)^{1/2}V (y)^{1/2}}
≤ \frac{C_{ϵ}e^{(1+ϵ)δ(ρ+k)}}{V (x)^{1/2}V (y)^{1/2}}

We will now use this to prove Lemma 7.1. Proof. This is clear from the estimate on the counting function. Let R = ρ(y, G(0)). More precisely, if we let A_{n} = {z ∈ B : e^{−n−1}r ≤ 1 − |z| < e^{−n}r}, s > δ_{Poin}, and ϵ = (s − δ_{Poin})/2, then

∑_{g(0)∈B}(1 − |g(0)|)^{s} ≤ ∑_{k} ∑_{g(0)∈B∩A_{k}}(1 − |g(0)|)^{s}
≤ ∑_{k} N_{k}(re^{−R−k})^{s}
≤ C_{s}r^{s}e^{−Rs}[V (x)V (y)]^{−1/2} ∑_{k} e^{(s−ϵ)k}e^{−sk}
≤ C_{s}r^{s}e^{−sρ(y,G(0))}[V (x)V (y)]^{−1/2}.

13. The geometric exponents for general sets

Theorem 13.1. If K ⊂ ℝ^{2} is a compact set with infinitely many complementary components then

δ_{inrad} ≤ δ_{area} ≤ δ_{diam},
max(1, 2 − \frac{δ_{inrad}}{δ_{area}} (2 − δ_{area})) ≤ δ_{Whit} ≤ 2.

Moreover, any four values which satisfy these constraints occur for some compact K.

In particular, the second constraint implies δ_{inrad} ≤ δ_{area} ≤ δ_{Whit} ≤ 2, but δ_{diam} may be larger than 2 (even infinite).

Proof. The inequalities δ_{inrad} ≤ δ_{area} ≤ δ_{diam} are obvious, so only the second constraint must be proven.

First, suppose K has a bounded complementary component Ω_{0}. Let {Q_{j}} be a Whitney decomposition of Ω_{0}, and let Q_{0} be a Whitney square of maximum size. For each n = 1, 2, …, note that the union of Whitney squares with

2^{n−10}diam(Q_{0}) ≤ diam(Q_{j}) ≤ 2^{−n}diam(Q_{0}),

topologically separate Q_{0} from K. Thus the sum of their diameters must be larger that diam(Q_{0}), i.e., the number of such squares is diam(Q_{0}) ≥ 2^{n}. Thus δ_{Whit}(K) ≥ 1.

If {Q_{k}} is a Whitney decomposition of a bounded component Ω_{j} then

∑_{k} diam(Q_{k})^{2} = 2 area(Ω_{0}).

Thus δ_{Whit} ≤ 2. Furthermore, for t ≤ 1,

2 ≤ ∑_{k}(\frac{diam(Q_{k})^{2}}{area(Ω_{j})} )
≤ ∑_{k}(\frac{diam(Q_{k})^{2}}{inrad(Ω_{j})^{2}} )(\frac{inrad(Q_{k})^{2}}{area(Ω_{j})} )
≤ (\frac{inrad(Q_{k})^{2}}{area(Ω_{j})} ) ∑_{k}(\frac{diam(Q_{k})^{2}}{inrad(Ω_{j})^{2}} )^{t}.

because inrad(Ω_{j}) ≥ diam(Q_{k}) for every Q_{k}. Thus for any s ≤ 2,

∑_{k} diam(Q_{k})^{s} ≥ 2area(Ω_{j})inrad(Ω_{j})^{s−2}.

36 CHRISTOPHER J. BISHOP

Now let 0 < t < 2 and use Hölder’s inequality with the conjugate indices

p = \frac{2}{t}, q = \frac{2}{2 − t},

to get

(13.1) ∑_{j} area(Ω_{j})^{t/2} ≤ 2(∑_{j} area(Ω_{j})inrad(Ω_{j})^{s−2})^{\frac{t}{2}} (∑_{j} inrad(Ω_{j})^{t \frac{2−s}{2−t}})^{\frac{2−t}{2}}
≤ 2(∑_{j} ∑_{Q_{k}⊂Ω_{j}} diam(Q_{k})^{s})^{\frac{t}{2}} (∑_{j} inrad(Ω_{j})^{t \frac{2−s}{2−t}})^{\frac{2−t}{2}} .

The two terms on the right hand side are finite if

s > δ_{Whit}(K), t\frac{2 − s}{2 − t} > δ_{inrad}(K),

respectively. Rewriting the second inequality as

t > \frac{2δ_{inrad}(K)}{2 − s + δ_{inrad}(K)},

and letting s = δ_{Whit}(K), we see that the left hand side of (13.1) is finite if

t > \frac{2δ_{inrad}(K)}{2 − δ_{Whit}(K) + δ_{inrad}(K)}.

Since the left hand side of (13.1) must be infinite if t < δ_{area}(K) we deduce

δ_{area} ≤ \frac{δ_{inrad}}{2 − δ_{Whit} + δ_{inrad}} .

Solving in terms of the other exponents gives δ_{Whit} ≥ 2 − \frac{δ_{inrad}}{δ_{area}} (2 − δ_{area}),

δ_{inrad} ≥ δ_{area}(\frac{2 − δ_{Whit}}{2 − δ_{area}} ),

as desired.

Next we give an example of a compact set K which shows that these are the only restrictions which hold for general sets. Consider the set K_{1} in Figure 13.1.

Figure 13.1.

The bounded complementary components are rectangles of dimension a_{n} × b_{n} where a_{n} ≤ b_{n}, ∑_{n} a_{n} < ∞ and b_{n} → 0. The various exponents become

δ_{inrad} = inf{s : ∑ b_{n}^{s} < ∞},
δ_{area} = inf{s : ∑(a_{n}b_{n})^{s/2} < ∞},
δ_{diam} = inf{s : ∑ a_{n}^{s} < ∞}.

If we set a_{n} = 2^{−n} and and b_{n} = n^{−β}, β > 0, then it easy to check

δ_{inrad} = δ_{area} = 0, δ_{diam} = \frac{1}{β} .

To compute the exponent δ_{Whit}, think of dividing each rectangle into b_{n}/a_{n} squares of size a_{n}. For s > 1, its easy to see that the Whitney sum corresponding to each subsquare is approximately a_{n}^{s}. Thus the sum over all Whitney squares in all the rectangles is approximately

∑_{n} a_{n}^{s−1}b_{n} = ∑_{n} 2^{−n(s−1)}n^{−β}.

This converges for all s > 1. Finally, the sum over the Whitney squares in the

Figure 13.2.

unbounded component sum for any s > 1 since the boundary of this component is just a rectifiable curve. Thus δ_{Whit}(K_{1}) = 1.

If we consider the set K_{2} in Figure 13.2, the complementary components are squares, with k_{n} squares of size c_{n}. If c_{n} = ⌊n^{−γ} log^{−2}(n)⌋ with γ > 1 and k_{n} = ⌊n^{α}⌋ with α ≤ γ then

δ_{inrad} = δ_{area} = δ_{diam} = \frac{1 + α}{γ},

and all values ≤ 2 can be obtained.

Next modify K_{2} by removing a finite number of points from each bounded component as in Figure 13.3 so that the largest remaining disk has diameter d_{n} ≤ c_{n}. This does not alter the values of δ_{area} or δ_{diam}, but if we choose d_{n} = n^{−δ}c_{n}, then

δ_{inrad}(K_{3}) = \frac{1 + α}{γ + δ},

can take any value between 0 and δ_{area}.

To compute δ_{Whit} divide each c_{n} × c_{n} square into (c_{n}/d_{n})^{2} equal sized subsquares. Note that the Whitney sum corresponding to each to these subsquares has size approximately d_{n}^{s}, if s > 1. Therefore the sum over all Whitney squares in the bounded

Figure 13.3.

components of K is approximately

∑_{n} k_{n}d_{n}^{s−2}c_{n}^{2} ≃ ∑_{n} n^{α}n^{−(δ+γ)(s−2)}n^{−2γ},

which converges if and only if

s > 2 − \frac{1 − 2γ + α}{δ + γ} = 2 − \frac{δ_{inrad}}{δ_{area}} (2 − δ_{area}).

Thus

δ_{Whit}(K_{3}) = 2 − δ_{inrad}(K_{3})(2 − δ_{area}(K_{3}))/δ_{area}(K_{3}).

Finally for any value δ_{Whit}(K_{3}) ≤ η ≤ 2 we let K_{4} be a Jordan arc with upper Minkowski dimension η (it is easy to build such sets by the usual self-similar constructions if η < 2 and only slightly more complicated if η = 2). By Lemma 2.1 we have δ_{Whit} = η, and since K_{4} has no bounded complementary components the other exponents are all 0.

It is easy to see that if we take copies of K_{1}, K_{3}, K_{4} in disjoint disks and set K = K_{1} ∪ K_{3} ∪ K_{4} then

δ(K) = max(δ(K_{1}), δ(K_{3}), δ(K_{4})),

where δ represents any on the critical exponents under consideration. Thus by ap-

propriate choices of α, β, γ, δ and η we can make

δ_{diam} = \frac{1}{β} , δ_{area} = \frac{1 + α}{γ},

δ_{inrad} = \frac{1 + α}{δ + γ}, δ_{Whit} = η,

take on any values consistent with the inequalities in Theorem 13.1.

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C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651

E-mail address: bishop@math.sunysb.edu