MINKOWSKI DIMENSION AND THE POINCARÉ EXPONENT

CHRISTOPHER J. BISHOP

Abstract. Let G be a non-elementary, analytically finite Kleinian group, Λ(G) its limit set and δ(G) the critical exponent for the Poincaré series. We give a new proof of the fact that if area(Λ(G)) = 0 then δ(G) equals the upper Minkowski dimension of Λ(G). This gives new proofs of the following results:

1. If Λ has zero area then δ = dim(Λ).

2. The Minkowski dimension of Λ exists equals the Hausdorff dimension.

1991 Mathematics Subject Classification. 30F40 ; Secondary 28A78, 30C35, 31A15 .

Key words and phrases. Hausdorff dimension, Minkowski dimension, Kleinian groups, limit sets, Poincaré exponent.

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

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 1

1. Statement of results

If K ⊂ ℝ^{⊭} is compact, let N(K, ϵ) be the minimal number of ϵ balls needed to cover K. We define the upper and lower Minkowski dimension as

\overline{Mdim}(K) = lim sup_{ϵ→0} \frac{log N(K, ϵ)}{log 1/ϵ},
\underline{Mdim}(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 Mdim(K).

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). G is called elementary if it contains a finite index Abelian subgroup. In this paper we will consider only nonelementary 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 always assume Ω is non-empty and that the group is conjugated in PSL(2, ℂ) so ∞ ∈ Ω.

For any Kleinian group, the quotient R = Ω/G is a union of 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). The Ahlfors finiteness theorem says that if G is finitely generated then G is analytically finite.

2 CHRISTOPHER J. BISHOP

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

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

where distance is in the spherical metric. It is easy to show it does not depend on the choice of z_{0}. Usually δ(G) is defined by extending the action of G on S^{2} to a group of isometries of the hyperbolic 3-ball 𝔹^{⊯} ⊂ ℝ^{⊯}, and then considering the series

∑_{G} exp(1 − |g(0)|)^{s}.

However, it is easy to see that this definition gives the same number.

Theorem 1.1. Suppose G is an analytically finite, non-elementary Kleinian group. If area(Λ(G)) = 0 then δ(G) = \overline{Mdim}(Λ(G)).

The assumption that G is non-elementary is needed in Theorem 1.1, for if G is a rank 1, cyclic, parabolic group then δ(G) = 1/2, but Λ is a single point. Define the Hausdorff content

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(K) = inf{α : H_{α}^{∞}(K) = 0}.

This is the Hausdorff dimension of K and it is easy to see that dim(K) ≤ \underline{Mdim}(K). Jones and I proved that δ(G) ≤ dim(Λ) for any nonelementary Kleinian group (Theorem 1.1 of [4]). Combining this result and Theorem 1.1 we easily deduce

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 3

Corollary 1.2. If G is an analytically finite Kleinian group then the Minkowski dimension of Λ exists and equals the Hausdorff dimension.

Corollary 1.3. If G is an analytically finite, non-elementary Kleinian group and Λ(G) has zero area then δ(G) = dim(Λ).

Different proofs of these results are given in [3] and [4] using estimates for the heat kernel on the hyperbolic 3-manifold associated to the Kleinian group G. The proof given here does not require these techniques, i.e., it is a purely “two-dimensional” argument. As such, it may be easier to adapt to other settings, e.g., Julia sets of rational mappings.

G is called geometrically finite if it is finitely generated and there is a finite sided fundamental polyhedron for the action of G on 𝔹. The limit sets of such groups must have zero area [2], so our results apply to them. For geometrically finite groups, Corollary 1.2 was independently established by Stratmann and Urbański in [11]. Corollary 1.3 is also well known in this case, e.g., [12].

The sections of the paper are organized as follows.

Section 2: We define a related critical exponent δ_{Whit} and show δ_{Whit}(K) ≤ \overline{Mdim}(K) for any compact K, with equality if area(K) = 0.

Section 3: We show δ ≤ δ_{Whit} for analytically finite groups with

equality if Ω(G)/G is compact. Section 4: We define good and bad horoballs and prove a lemma

giving some of their properties.

4 CHRISTOPHER J. BISHOP

Section 5: We prove the main theorem when most horoballs of G

are good. Section 6: We prove the theorem in the case dim(Λ) = 2. Section 7: We state a lemma and finish the proof assuming the

lemma and dim(Λ) < 2. Section 8: We prove the lemma. 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 with diam(λS) = λdiam(S).

I thank the referee for carefully reading the manuscript and supplying many suggestions which greatly improved the exposition.

2. Whitney squares and Minkowski dimension A Whitney decomposition of a domain Ω ⊂ ℝ^{⊭} is a collection of

disjoint (except for boundaries) squares {Q_{j}} such that Ω = ∪_{j}Q_{j} and

diam(Q) ≤ dist(Q_{j}, ∂Ω) ≤ 4diam(Q_{j}).

The existence of a Whitney decomposition for any open set is a standard fact in real analysis (e.g., Theorem VI.1 of [10]). One can simply take a maximal collection of dyadic squares in Ω such that dist(Q, ∂Ω) ≤ diam(Q).

For any compact set K ⊂ ℝ^{⊭} we can define an exponent of convergence

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

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 5

The sum is taken over all squares in a Whitney decomposition of Ω = K^{c} which are within distance 1 of K (we have to drop the “far away” squares or the series will not converge). It is easy to check that this does not depend on the particular choice of Whitney decomposition.

Lemma 2.1. For any compact set K, δ_{Whit} ≤ \overline{Mdim}(K). If, in addition, area(K) = 0 then δ_{Whit} = \overline{Mdim}(K).

Proof. Suppose {Q_{j}} is a Whitney decomposition of Ω = ℝ^{⋉} \ 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{Mdim}(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{Mdim}(K)−ϵ)} such squares Q, the sum over the whole Whitney collection is greater than

C2^{−ns}2^{n(\overline{Mdim}(K)−ϵ)},

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

6 CHRISTOPHER J. BISHOP

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. By taking r_{j} → 0 very fast, we can get δ_{Whit}(K) = 1 < 2 = \overline{Mdim}(K).

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 7

3. Whitney squares and the Poincaré series In this section we explain the elementary relations between δ 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., [6], 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^{⊭} 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 [7] and [8].)

The limit set of any finitely generated group is uniformly perfect, [9], [5]. In fact, the proof in Canary’s paper [5] 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 the result is still true for analytically finite groups.

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

8 CHRISTOPHER J. BISHOP

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(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 are associated to any given 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(G).

One of the main results of [3] is that δ_{Whit} = δ for any non-elementary analytically finite group. This fact and Lemma 2.1 imply Theorem 1.1,

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 9

but the fact seems harder than the theorem. The purpose of this note is to give a proof of the theorem that does not require proving δ_{Whit} = δ.

10 CHRISTOPHER J. BISHOP

4. Good and bad horoballs

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. We say that B is doubly cusped it there is a another (disjoint) ball B_{1} fixed by the same subgroup.

Suppose R = Ω(G)/G is a finite union of finite type Riemann surfaces R_{1}, …, R_{N}, i.e., each is a compact surface with at most a finite number of punctures. Let {p_{1}, …, p_{m}} be the punctures in R = ∪_{i=1}^{N}R_{i}, and for each p_{i} let B_{i}^{∗} be a neighborhood of p_{i} which lifts to a Euclidean ball B_{i} in Ω which is invariant under some parabolic element of G (see Lemma 1 of [1]). Then X = R \ ∪_{j}B_{j}^{∗} is compact, so we can choose a finite set of points E = {z_{1}, …, z_{P} } ⊂ Ω(G) which project to an 1-dense subset of X (i.e., every point of X is within hyperbolic distance 1 of a point of E). For the reminder of the paper we fix a finite collection of such horoballs B = {B_{|}}, one from each equivalence class of horoballs in Ω. It will also be convenient to assume that B_{i} is contained inside a larger horoball B̂_{i} ⊂ Ω of twice the Euclidean diameter (we can always do this by just taking smaller balls if necessary). If ∞ ∈ Ω then we may also assume that ∞ is not contained in any of the B_{i}. Finally, if a parabolic point is doubly cusped, we require that B contain horoballs of equal size for both “sides” of the parabolic point.

For z ∈ Ω(G) we define d(z) = dist(z, Λ) in the Euclidean metric. Normalize the group so that ∞ ∈ Ω and Λ(G) has diameter 1. Now

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 11

suppose p is a parabolic fixed point of G and that B ⊂ Ω is a horoball at p. By our remarks above, B must be the image of one of our finite collection of horoballs {B_{j}} under some element g of G, i.e., B = g(B_{j}).

We would like to call B good or bad depending on whether the map g : B_{j} → B is “close” to being linear. Using a linear Möbius transformation we can map p to 0 and the horoball B to the disk D(i/2, 1/2). The parabolic element of G fixing p is conjugated by the linear map to a transformation of the form

τ(z) = \frac{z}{1 + ηz} .

We define η(B) = |η|. Given η > 0, we say a horoball B = g(B_{i}) is a “η-bad” horoball if η(B) ≤ η, and is η-good otherwise.

A helpful way to think about good and bad horoballs is as follows. Suppose h is a generator of the parabolic subgroup fixing B. Define

η′(B) = sup_{z∈∂B} \frac{|h(z) − z|}{diam(B)}.

Then its easy to see η′ ≃ η. In other words, B is η-bad if the parabolic subgroup fixing B has a generator which is close to the identity on B. Alternatively, the group G looks less and less discrete on horoballs with smaller and smaller η.

Lemma 4.1. Suppose G is analytically finite and normalized so that ∞ ∈ Ω(G) and diam(Λ) = 1. Fix a finite collection of horoballs B as above, and assume ∞ is not in any of these horoballs. Then

12 CHRISTOPHER J. BISHOP

1. There is a C_{1} (depending only on η) so that for any η-good horoball B, and any w ∈ ∂B, there is a point z ∈ G(E) such that

C_{1}^{−1}d(w) ≤ d(z) ≤ C_{1}d(w),
C_{1}^{−1}d(w) ≤ |z − w| ≤ C_{1}d(w).

2. There is a η_{2} so that if B is η_{2}-bad then it is singly cusped, i.e., there is not a disjoint horoball also tangent to p.

3. There is a η_{3} > 0 so that if η ≤ η_{3} and B is η-bad and D(x, r) ⊂ Ω with \frac{3}{2}diam(B) ≤ dist(x, B) ≤ diam(B)/(2η), then r ≤ C_{2}ηdist(x, B)^{2}, where the constant C_{2} depends only on G.

4. For any δ > 0 there is a η_{4} > 0 (depending only on δ) such that if B is a η_{4}-bad horoball then there is a disk D ⊂ 3B such that diam(D) ≥ \frac{1}{3}diam(B), and D \ Λ contains no balls of radius ≥ δdiam(D).

5. There is η_{5} > 0 so that if B_{1}, B_{2} are η_{5}-bad horoballs with diam(B_{1}) ≤ diam(B_{2}) then dist(B_{1}, B_{2}) ≥ 100diam(B_{1}).

6. If B_{1}, B_{2} are horoballs with

diam(B_{1}) ≤ diam(B_{2}) ≤ 2diam(B_{1}),

and dist(B_{1}, B_{2}) ≤ Adiam(B_{1}), then both B_{1} and B_{2} are A^{−2}-good.

Proof. To prove (1), note that it is true for all the balls in B by finiteness. If B is η-good then B is the image of some B_{j} in our fixed, finite collection under a map which is the composition of conformal linear map and a map of bounded distortion (depending on η).

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 13

To prove (2), suppose B_{1}, B_{2} ∈ B are paired horoballs at a doubly cusped parabolic point p with parabolic generator h. For i = 1, 2, let z_{i} be the point on ∂B_{i} farthest from p and consider the the cross ratio of z_{1}, h(z_{1}), z_{2}, h(z_{2}). Now suppose B = g(B_{1}) is some η-bad image of B_{1}, and suppose we have chosen g so |g(z_{1})−g(p)| is maximized among elements mapping B_{1} to B. Since cross ratio is preserved by Möbius transformations we can deduce that

|g(z_{1}) − g(h(z_{1}))| ≃ |g(z_{1}) − g(z_{2})| ≃ |g(z_{2}) − g(h(z_{2}))| ≃ ηdiam(B).

Thus all four points are mapped within Cη(B)diam(B) of g(z_{1}). If η is small enough, this is only possible if ∞ ∈ g(B_{2}), contrary to hypothesis.

To prove (3) note that it is enough to do it in the case when p = 0, B = D(i/2, 1/2) and the subgroup fixing p is generated by

h(z) = \frac{z}{1 ± ηz} .

Then

|h(x) − x| = |x − \frac{x}{1 ± ηx}| ≤ 2η|x|^{2} ≤ C_{0}ηdist(x, B)^{2}.

Suppose C_{2} > 100C_{0} and that r ≥ C_{2}ηdist(x, B)^{2}. Then both x and h(x) are in the disk D(x, r) ⊂ Ω so the line segment connecting them projects to a loop on Ω/G of hyperbolic length ≤ 10C_{0}/C_{2}. If C_{2} is large enough (depending only on G), this implies the loop is contained in a neighborhood of a cusp on Ω/G, thus the lift is in a horoball of Ω which is tangent to 0. This horoball is obviously not B since B does not contain x, so there is a second horoball B′ at 0. This contradicts part (2),so we are done.

14 CHRISTOPHER J. BISHOP

The final three statements are all easy consequences of part (3).

We noted in the previous section that if Ω(G)/G is compact then there is a close connection between the Poincaré and Whitney sums. When there are punctures in Ω(G)/G we need to take account of the fact that horoballs contain many Whitney squares, but no orbit points. The next observation is very easy and left to the reader.

Lemma 4.2. Suppose Ω is an open set and B ⊂ Ω is a Euclidean ball such that ∂B ∩ ∂Ω ≠ ∅. Let {Q_{j}} be a Whitney decomposition for Ω. Then if s > 1,

∑_{j:Q_{j}∩B≠∅} diam(Q_{j})^{s} ≃ diam(B)^{s},

where the constants depend only on s.

The following will be useful later.

Lemma 4.3. Suppose G is analytically finite and E ⊂ Ω is a finite set with one point in each equivalence class of components. Assume the group has been normalized so ∞ ∈ Ω. Suppose {Q_{j}} is a Whitney decomposition for Ω(G), and let B be a choice of horoballs for G as above. Then if s > 1,

∑_{j:diam(Q_{j})≤1} diam(Q_{j})^{s} ≃ ∑_{z∈E} ∑_{g∈G} dist(g(z), Λ)^{s} + ∑_{B∈B} diam(B)^{s},

where the constants depend on G, s, E, B. If η > 0 then

∑_{j:diam(Q_{j})≤1} diam(Q_{j})^{s} ≃ ∑_{z∈E} ∑_{g∈G} dist(g(z), Λ)^{s} + ∑_{B∈B,η(B)≤η} diam(B)^{s},

where the constants depend on G, s, E, B, η.

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 15

Proof. The first equation is simply the observation that Whitney squares which do not hit any horoballs can be associated (as in the previous section) to orbits of E, whereas the Whitney squares which hit a horoball are controlled by the previous lemma. The second equation is proved using part (1) of Lemma 4.1 to associate to each η-good horoball a nearby orbit point. Thus the part of the horoball sum we are omitting is controlled by the orbit sum.

5. Theorem 1.1 when G has many good horoballs We now start the proof of Theorem 1.1. It is enough to prove

Theorem 5.1. If G is an analytically finite group and area(Λ(G)) = 0 then δ(G) = δ_{Whit}(Λ(G)).

Let D = \overline{Mdim}(Λ) = δ_{Whit}(Λ(G)) and d = dim(Λ). If G has no parabolic elements then Ω(G)/G is compact, and we have already proven this case in Lemma 3.1. Therefore we may assume G has parabolics. This implies δ(G) ≥ 1/2. If D = 1/2, we have nothing to do, so we may assume that D > 1/2.

Suppose ϵ > 0 is so small that D − ϵ > 1/2. Let E = {z_{1}, …, z_{s}} be a finite collection of points in Ω, one projecting to each component of Ω/G. We will show that

∑_{z∈G(E)} d(z)^{D−ϵ} = ∞,

and thus δ(g) ≥ D.

Choose an integer n_{0} so that N(Λ, 2^{−n_{0}}) ≥ 1000 · 2^{n_{0}(D−ϵ/2)}.

16 CHRISTOPHER J. BISHOP

By passing to a subcollection with at least 2^{n_{0}(D−ϵ/2)} elements we claim that we may assume that for any two squares S_{j}, S_{k} we have 9S_{j}∩9S_{k} = ∅. This is easy. Just enumerate the list of squares and inductively remove any square S_{k} for which there is a j < k with 9S_{j} ∩ 9S_{k} ≠ ∅. Since 9S_{j} ∩ 9S_{k} ≠ ∅ implies S_{k} ⊂ 15S_{j}, each S_{j} can cause at most 30^{2} = 900 later squares to be removed. Thus the final list has at least 2^{n_{0}(D−ϵ/2)} elements.

Let r = 3 · 2^{−n_{0}}. Let S = {S_{∥}} be a collection of 2^{n_{0}(D−ϵ/2)} squares of size r so that the triples 3S_{k} are pairwise disjoint and \frac{1}{3}S_{k} ∩ Λ ≠ ∅ for each k.

First we deal with the case when most of the horoballs of size r are good. Let η > 0 (to be fixed later). For each η-good horoball with diam(B) ≥ r/3, let G_{B} be the collection of squares in S which are such that \frac{1}{3}S hits B. Let G be the union of all the G_{B}.

The proof breaks into three cases:

1. #(S ∩ G) ≥ \frac{∞}{∈}#(S) for all large enough n.

2. dim(Λ) = 2.

3. #(S ∩ G) < \frac{∞}{∈}#(S) for infinitely many n and dim(Λ) < 2.

If η is chosen small enough (depending on G) then one can show the last case is impossible (e.g., [4]). However, this is hard result using heat kernel estimates on hyperbolic 3-manifolds and one of our purposes here is to give a self-contained proof that uses only two dimensional techniques.

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 17

Proof of case 1: By part (1) of Lemma 4.1 there is an orbit point z ∈ G(E) ∩ B such that d(z) ≃ diam(B). For this point,

d(z)^{D−ϵ} ≥ C ∑_{S∈G_{B}} diam(S)^{D−ϵ}.

If more than half the squares in S belong to G then this argument shows

∑_{z∈G(E)} d(z)^{D−ϵ} ≥ \frac{1}{2}C2^{n_{0}ϵ/2}.

If this happens for arbitrarily large n_{0} then

∑_{z∈G(E)} d(z)^{D−ϵ} = ∞,

so we have shown that δ(G) ≥ D − ϵ, as desired. This is the end of case (1).

6. Proof of case (2) of Theorem 1.1

Case 2 follows easily from the following.

Lemma 6.1. Suppose G is a analytically finite Kleinian group, normalized so ∞ ∈ Ω(G) and diam(Λ(G)) = 1. If δ_{Whit} = 2 then δ = 2.

Proof. We know the lemma if Ω(G)/G is compact (see Section 2), so we may assume that Ω(G) contains horoballs. Consider the sum over all Whitney squares for Ω,

∑_{j} diam(Q_{j})^{2−2ϵ}.

By the definition of δ_{Whit}, this diverges. Using Lemma 4.3 we can split the sum into two pieces; one corresponding to all squares Q_{j} which hit some horoball and the other corresponding to all Whitney squares which miss every horoball, i.e.,

∑_{j:diam(Q_{j})≤1} diam(Q_{j})^{2−2ϵ} ≃ ∑_{z∈E} ∑_{g∈G} dist(g(z), Λ)^{2−2ϵ} + ∑_{B∈B} diam(B)^{2−2ϵ}.

If the first sum on the right diverges then δ > 2 − 2ϵ and we are done. Thus we may assume the second sum on the right diverges for all ϵ > 0. Thus if

B_{\} = {B_{|} : ∈^{−\−∞} ≤ diam(B_{|}) < ∈^{−\}},

and N_{n} = #B_{\}, we must have N_{n} ≥ 2^{n(2−2ϵ)}, for infinitely many values of n. Fix a value of n_{0} where this holds and note that for at least half the balls B in B_{\} there is a second ball B′ ∈ B_{\} such that

dist(B, B′) ≤ 2^{−n(1−ϵ)} ≤ diam(B)2^{nϵ},

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 19

otherwise we would have so many disjoint balls of radius 2^{−n(1−ϵ)} that they could not all be contained in a bounded neighborhood of Λ (recall that diam(Λ) = 1).

By part (4) of Lemma 4.1, this implies that for such a pair both B and B′ are 2^{−2ϵn}-good horoballs. Let G_{\_{′}} ⊂ B_{\_{′}} be the subcollection of 2^{−2ϵn}-good horoballs. For any η-good horoball B let z be the point given in part (1) of Lemma 4.1 such that d(z) ≥ Cηdiam(B). Let H be the parabolic subgroup fixing B_{j}. Then an easy calculation shows that there are at least Cη^{−1} orbits of z under H with distance ≥ Cηdiam(B) from Λ. Thus

∑_{h∈H} d(h(z))^{α} ≥ Cdiam(B)^{α}η^{α−1},

for any 1 < α ≤ 2 and some C depending on G and α.

Thus if z_{j} is the good point in B_{j} given by (1) of Lemma 4.1,

∑_{z∈G(E)} d(z)^{α} ≥ C ∑_{B_{j}∈G_{\′}} ∑_{k∈ℤ}d(h^{k}(z_{j}))^{α}
≥ C ∑_{B_{j}∈G_{\′}} diam(B)^{α}2^{−2ϵn_{0}(α−1)}
≥ C2^{n_{0}(2−ϵ)}2^{−n_{0}α}2^{−2ϵn_{0}(α−1)}.

Taking ϵ = 0 and solving for α we see this diverges for small enough ϵ if α < 2. Thus δ ≥ 2, as desired.

7. Theorem 1.1 when G has few good horoballs We now do case 3, i.e., we assume that fewer than half the elements

of S are in G for all large enough n_{0} (i.e., we assume most horoballs are bad) and that dim(Λ) < 2. We use a stopping time construction which is described by the following lemma. Recall that d = dim(Λ) and D = \overline{Mdim}(Λ).

Lemma 7.1. Suppose ϵ > 0 and r > 0. There is a constant C_{0} (depending only on G and ϵ) and constants η_{0} > 0 and ν_{0} (depending on G, ϵ and r) such that the following holds: Suppose we have a square S such that \frac{1}{3}S ∩ Λ ≠ ∅ and S does not intersect any η_{0}-good horoball with diameter ≥ diam(S)/3. Then either

∑_{z∈S∩G(E)} d(z)^{D−ϵ} ≥ ν_{0}diam(S)^{D−ϵ},

or there is a collection of subsquares C(S) = {S_{|}} ⊂ S with 1. diam(S_{j}) ≤ rdiam(S) for all j,

2. {3S_{j}} ⊂ S and are pairwise disjoint,

3. S_{j} \ Λ does not contain a ball of radius diam(S_{j})/10,

4. ∑_{j} diam(S_{j})^{2} ≥ C_{0}diam(S)^{2}. We shall refer to the two possibilities as alternatives one and two respectively.

Let us assume that the lemma holds and see how to finish the proof. We will prove the lemma in the next section.

Suppose ϵ > 0. Let C_{0} = C_{0}(ϵ, G) be as given by the lemma. Let r = C_{0}^{1/ϵ} and let η_{0}, ν_{0} be as given by the lemma.

We define generations of nested squares S_{′}, S_{∞}, … as follows. Let S_{′} = S, be the collection considered in the previous section; i.e., S_{′} is a collection of \frac{1}{2} · 2^{n_{0}(D−ϵ/2)} squares of size r = 2^{−n_{0}} which have disjoint

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 21

triples, so that \frac{1}{3}S ∩ Λ ≠ ∅ and such that S does not hit any η-good horoball of size ≥ diam(S)/3.

Suppose S_{\} has been defined. Let A_{∞}^{\} ⊂ S_{\} be the subcollection of squares for which alternative one holds and A_{∈}^{\} be the subcollection for which alternative two holds. Then we define

S_{\+∞} = ⋃_{S∈A_{∈}^{\}} C(S).

In other words, given S_{\}, we define S_{\+∞} by throwing away all the squares where alternative one of Lemma 7.1 holds and for each square S ∈ S_{\} where the second alternative holds we replace it by the collection C(S) of subsquares satisfying (1)-(4) in Lemma 7.1.

To each square S ∈ S_{∞} = ∪_{\}S_{\} we associate a positive number μ(S) as follows. For S ∈ S_{′} let

μ(S) = diam(S)^{D−ϵ}.  (7.1)

For S ∈ S_{\}, n ≥ 1, there is a unique S_{0} ∈ S_{\−∞} containing S (i.e., S_{0} is its “parent”) and by definition alternative two holds for S_{0}. Set

μ(S) = \frac{diam(S)^{2}}{∑_{S′∈C(S_{′})} diam(S′)^{2}} μ(S_{0}).  (7.2)

Note that ∑_{S′∈C(S_{′})} μ(S′) = μ(S_{0}). Let μ(n) = ∑_{S∈S_{\}} μ(S). It is clear that {μ(n)}_{n=0}^{∞} is non-increasing.

Lemma 7.2. If μ(n) ↛ 0 then dim(Λ) ≥ 2 − ϵ. (Recall μ depends on ϵ.)

Proof. Suppose μ(n) ↛ 0. Then the numbers {μ(S)} define a measure positive measure on the set Y = ∩_{n} ∪_{S∈S_{\}} S ⊂ Λ. We claim that

μ(S) ≤ Cdiam(S)^{2−ϵ},

for every square with diam(S) ≤ r where C is a constant that depends on the choice of n_{0}, but not on S.

We first verify this by induction for squares in S_{∞}. If S ∈ S_{′}, the claim is true by definition with C = A_{0} = 2^{n_{0}(2−D)}.

If S ∈ S_{\} is contained in S_{0} ∈ S_{\−∞}, then part (4) of Lemma 7.1 implies

μ(S) = \frac{diam(S)^{2}}{∑_{S′∈C(S_{′})} diam(S′)^{2}} μ(S_{0}) ≤ A_{0} \frac{diam(S)^{2}}{C_{0}diam(S_{0})^{2}} diam(S_{0})^{2−ϵ}

and hence

μ(S) ≤ A_{0}C_{0}^{−1}diam(S)^{2}diam(S_{0})^{−ϵ}
≤ A_{0}C_{0}^{−1}r^{−ϵ}diam(S)^{2−ϵ}
≤ A_{0}diam(S)^{2−ϵ},

as desired. Also note (for future use) that this proves the weaker estimate

μ(S) ≤ diam(S)^{D−ϵ}.  (7.3)

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 23

Now consider a general square S ⊂ S_{0} ∈ S_{′}. Let S_{1} be the smallest square in S_{∞} containing S. Suppose S_{1} ∈ S_{\−∞}. Then by (7.2),

μ(S) = \frac{∑_{S′∈S_{\},S′⊂S} diam(S′)^{2}}{∑_{S′∈S_{\},S′⊂S_{∞}} diam(S′)^{2}} μ(S_{1})
≤ A_{0} \frac{diam(S)^{2}}{C_{0}diam(S_{1})^{2}} diam(S_{1})^{2−ϵ}
≤ A_{0}C_{0}^{−1}diam(S)^{2}diam(S_{1})^{−ϵ}
≤ A_{0}C_{0}^{−1}diam(S)^{2−ϵ}(\frac{diam(S_{1})}{diam(S)})^{−ϵ}
≤ A_{0}C_{0}^{−1}diam(S)^{2−ϵ}.

Thus the inequality holds for general squares with the constant C = A_{0}C_{0}^{−1}.

Thus if {S_{j}} was any covering of Λ we would have

0 < μ(Λ) ≤ ∑_{j} μ(S_{j}) ≤ C ∑_{j} diam(S_{j})^{2−ϵ}.

Therefore, dim(Λ) ≥ 2 − ϵ, as desired.

By the previous lemma if ϵ < 2−dim(Λ), then we must have μ(n) → 0. Assume this is the case.

Recall that A_{∞}^{\} is the collection of all squares in S_{\} where alternative one held (i.e., the construction above stopped) and A_{∈}^{\} are the remaining squares where alternative two held. Note that ∪_{n}A_{∞}^{\} is a union of disjoint squares. For S ∈ A_{∈}^{\−∞}, let A_{∞}(S) be the collection of all squares in C(S) ⊂ S_{\} where alternative one holds and let A_{∈}(S) be the squares where alternative two holds. For S′ ∈ A_{∞}(S) ∈ A_{∞}^{\}, define

ν(S′) = ∑_{z∈S′∩G(E)} d(z)^{D−ϵ}.

24 CHRISTOPHER J. BISHOP

Since alternative one of Lemma 7.1 holds for S′, ν(S′) ≥ ν_{0}diam(S′)^{D−ϵ}, so by (7.3) ν(S′) ≥ ν_{0}μ(S′), (μ(S′) must be defined because alternative two holds for its parent). Thus,

∑_{S′∈A_{∞}(S)} ν(S′) ≥ ν_{0} ∑_{S′∈A_{∞}(S)} μ(S′) = ν_{0}[μ(S) − ∑_{S′∈A_{∈}(S)} μ(S′)].

Therefore,

∑_{S′∈A_{∞}^{\}} ν(S′) ≥ ν_{0}[ ∑_{S′∈A_{∈}^{\−∞}} μ(S′) − ∑_{S′′∈A_{∈}^{\}} μ(S′′)]
= ν_{0}[μ(n − 1) − μ(n)].

Hence, since μ(n) → 0, a telescoping series argument gives

∑_{n=0}^{∞} ∑_{S′∈A_{∞}^{\}} ν(S′) ≥ ν_{0} ∑_{n=0}^{∞}(μ(n − 1) − μ(n)) = ν_{0}μ(0).

Thus by (7.1),

∑_{n=0}^{∞} ∑_{S′∈A_{∞}^{\}} ν(S′) ≥ ν_{0} ∑_{S∈S_{′}} diam(S)^{D−ϵ}
≥ ν_{0}#(S_{′})∈^{−\_{′}(D−ϵ)}
= ν_{0}2^{n_{0}(D−ϵ/2)}2^{−n_{0}(D−ϵ)}
= ν_{0}2^{n_{0}ϵ/2}

Thus, since the squares in ∪_{n}A_{∞}^{\} are all disjoint

∑_{z∈G(E)} d(z)^{D−ϵ} ≥ ∑_{n} ∑_{S∈A_{∞}^{\}} ν(S) ≥ ν_{0}2^{n_{0}ϵ/2}.

Taking n_{0} → ∞ proves δ(G) ≥ D − ϵ. Taking ϵ → 0 shows δ(G) ≥ D = \overline{Mdim}(Λ), as desired.

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 25

8. Proof of Lemma 7.1

The idea of the proof is that the two alternatives for S in the lemma simply depend on whether S contains many bad horoballs or not. If it does not then the first alternative holds, but if it does then the second is true.

Let η_{0} > 0 (we will show the lemma is correct if η_{0} is small enough, depending only on G, ϵ and r). Let H_{g} be the union of the η_{0}-good horoballs and let H_{b} be the union of all the η_{0}-bad horoballs. Let U = Ω \ (H_{b} ∪ H_{g}) (this is part of Ω(G) that projects to the “compact part” of Ω(G)/G). The proof divides into three cases depending on the relative sizes of S ∩ H_{b}, S ∩ H_{g} and S ∩ U. Fix ϵ > 0 and let E ⊂ Ω be a finite set so that points in G(E) are at least hyperbolic distance ϵ apart, but so that every point of U is within 10ϵ of some point of G(E).

Case A: First suppose that “a lot” of S corresponds to the compact part of Ω(G)/G. More precisely, let U_{S} = \frac{1}{3}S ∩ U and assume

area(U_{S}) ≥ \frac{1}{2000}area(S).

We will show that the first alternative holds for S. If ϵ in the definition of E is small enough (depending only of the uniformly perfect constant of Λ), then for each point w ∈ U there is a point z ∈ G(E) such that |z − w| ≤ d(z). Moreover, each z ∈ G(E) is the center of a disk D_{z} = D(z, \frac{ϵ}{100}d(z)). These disks are all disjoint, but they cover a fixed fraction of the area of U. This is because given any point of U we can move hyperbolic distance ≤ 10ϵ and reach one of the ϵ-disks. Moreover,

26 CHRISTOPHER J. BISHOP

since Λ∩ \frac{1}{3}S ≠ ∅, moving hyperbolic distance 10ϵ does not allow leaving S from starting inside \frac{1}{3}S (if ϵ is small enough).

Hence,

∑_{z∈G(E)∩U_{S}} area(D_{z}) ≥ A area(U_{S}) ≥ \frac{A}{2000}area(S).

Thus,

∑_{z∈G(E)∩ \frac{1}{3} S} d(z)^{2} ≥ Cdiam(S)^{2}.

Then since d(z) ≤ diam(S) for all z ∈ G(E) ∩ S,

∑_{z∈G(E)∩ \frac{1}{3} S} d(z)^{D−ϵ} ≥ Cdiam(S)^{D−ϵ},

and we are done.

Case B: Now we assume “a lot” of S lies in H_{g}. Let V_{S} = \frac{1}{3}S ∩ H_{g}, and suppose

area(V_{S}) ≥ \frac{1}{2000}area(S),

(i.e., the part of S in good horoballs has large area). By hypothesis, the only η_{0}-good horoballs hitting \frac{1}{3}S have diameter ≤ \frac{1}{3}S, and so we can associate to each such horoball B a point z ∈ G(E) ∩ CS so that d(z) ≃ ϵdiam(B). Thus, as above, ∑_{z∈G(E)∩CS} d(z)^{2} ≥ C ∑_{good horoballs in S} diam(B)^{2} ≥ Cdiam(S)^{D−ϵ} ≃ area(S). Since D − ϵ < 2, this and bounded overlaps of the squares CS imply

∑_{z∈G(E)∩CS} d(z)^{D−ϵ} ≥ Cdiam(S)^{D−ϵ}.

Thus alternative one holds in this case also.

Case C: Now assume area(U_{S}) + area(V_{S}) ≤ \frac{1}{1000}area(S).

MINKOWSKI DIMENSION AND THE POINCARÈ EXPONENT 27

Since Λ has zero area, the “bad” part of S must have large area, i.e.,

area(\frac{1}{3}S ∩ H_{b}) ≥ \frac{99}{100}area(S).

Next we want to show that we may assume that \frac{1}{3}S does not hit any bad horoball of comparable size.

Suppose that \frac{1}{3}S hits a η_{0}-bad horoball with diam(B) ≥ \frac{1}{10}diam(S). If η_{0} is small enough then part (3) of Lemma 4.1 implies that B can be the only η_{0}-bad horoball hitting \frac{1}{3}S with diameter ≥ \frac{1}{50}diam(S). Thus we can find another square S′ ⊂ S with diam(S′) = \frac{1}{3}(S) which only hits small horoballs. More precisely, we can choose S′ ⊂ S so that

area(S′ ∩ H_{b}) ≥ \frac{1}{100}area(S′),

and such that S′ does not hit any η_{0}-bad horoballs with diameter ≥ \frac{1}{10}diam(S′).

So by replacing S by S′ if necessary, we may now assume that S is a square such that

area(S ∩ W_{g}) ≥ \frac{1}{100}area(S),

and such that S does not hit any η_{0}-bad horoballs with diameter ≥ \frac{1}{10}diam(S′). Let \{B_{j}\} be the collection of bad horoballs which hit \frac{1}{3}S. Since they all have diameter ≤ diam(S)/10, they are contained in S and their areas sum to be at least Carea(\frac{1}{3}S).

The horoballs are all disjoint, so by the simple Vitali covering lemma (e.g., Lemma 7.4 of [13]) there is a subcollection of these balls {B̂_{j}} which have disjoint triples and whose areas sums to be more that C′area(S).

By part (2) of Lemma 4.1, to each of these balls we can associate a

square S_{j} ⊂ 3B̂_{j} such that

diam(S_{j}) = \frac{1}{2}diam(B̂_{j}),

and so that S_{j} \Λ contains no balls of radius δdiam(S_{j}). Here δ may be

as small as we wish, assuming we take η_{0} small enough. Choose η_{0} so

small that δ < r/1000 (where r is the number given in the statement

of Lemma 7.1). Inside S_{j} choose a collection {S_{jk}} of r^{−2} squares of

diameter r/100 which have disjoint triples. Then ∪_{j} ∪_{k} S_{jk} obviously

has all the desired properties.

References

[1] L. Ahlfors. Finitely generated Kleinian groups. Amer. J. of Math., 86:413–429, 1964.

[2] L. Ahlfors. Fundamental polyhedrons and limit point sets of Kleinian groups. Proc. Nat. Acad. Sci., 55:251–254, 1966.

[3] C.J. Bishop. Geometric exponents and Kleinian groups. 1995. preprint.

[4] C.J. Bishop and P.W. Jones. Hausdorff dimension and Kleinian groups. 1994. Stony Brook IMS preprint 1994/5, to appear in Acta Math.

[5] R.D. Canary. The Poincaré metric and a conformal version of a theorem of Thurston. Duke Math. J., 64:349–359, 1991.

[6] L. Carleson and T.W. Gamelin. Complex Dynamics. Universitext. Springer Verlag, 1994.

[7] J.L. Fernández. Domains with strong barrier. Revista Mat. Iberoamericana, 5:47–65, 1989.

[8] M.J. González. Uniformly perfect sets, Green’s function and fundamental domains. Revista Mat. Iberoamericana, 8:239–269, 1992.

[9] Ch. Pommerenke. On uniformly perfect sets and Fuchsian groups. Analysis, 4:299–321, 1984.

[10] E. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1971.

[11] B. Stratmann and M. Urbanski. The box counting dimension for geometrically finite Kleinian groups. to appear, Fundamenta Matematika.

[12] D. Sullivan. Entropy, Hausdorff measures new and old and limit sets of geometrically finite Kleinian groups. Acta. Math., 153:259–277, 1984.

[13] R. Wheeden and A. Zygmund. Measure and integral. Marcel Dekker, 1977.

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

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