WIGGLY SETS AND LIMIT SETS

CHRISTOPHER J. BISHOP AND PETER W. JONES

Abstract. We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that if G is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) it is either a circle or has dimension strictly bigger than 1.

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

degenerate groups, geometrically infinite, Teichmüller space, conical limit set, Poincaré exponent, critical exponent.

The first author is partially supported by NSF Grant DMS 92-04092 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF grant DMS-92-13595 WIGGLY SETS AND LIMIT SETS 1

1. Statement of results

The purpose of this paper is to show that a connected, compact set which “oscillates” around every point and at every scale must have dimension strictly larger than 1. Although this seems like a very intuitive result, we do not know any elementary proof and ours is based on the “traveling salesman theorem” of the second author [21]. After proving it, we will apply the result to limit sets of certain Kleinian groups, and eventually prove a generalization of Bowen’s dichotomy: a connected limit set of an analytically finite group is either a circle or has Hausdorff dimension strictly larger than 1.

A dyadic square Q in the plane is one of the form Q = (2^{−n}j, 2^{−n}(j + 1)] × (2^{−n}k, 2^{−n}(k + 1)]. For a positive number λ > 0, we let λQ denote the square concentric with Q but with diameter λdiam(Q), e.g., 2Q is the “double” of Q. Given a set E in the plane and a square Q we define β(Q) as

β(Q) = diam(Q)^{−1} inf_{L∈L} sup_{z∈E∩3Q} dist(z, L),

where L is the set of all lines L intersecting Q. A connected set with the property that there is a β_{0} > 0 such that β_{E}(Q) > β_{0} for every Q with \frac{1}{3}Q ∩ E ≠ ∅ and diam(Q) ≤ diam(E) is called uniformly wiggly. We shall show

Theorem 1.1. Suppose E ⊂ ℝ^{2} is a closed, connected set and is uniformly wiggly with constant β_{0}. Then dim(E) ≥ 1 + Cβ_{0}^{2}, where C is an absolute constant.

This estimate is sharp (except for the choice of C) as can be seen by simple examples. A stochastic version of this theorem with an application to Brownian motion is given in [10]. The theorem is also true in ℝ^{n} (but with different constants).

Figure 1.1. Definition of the β’s

Limit sets of certain Kleinian groups provide a natural class of uniformly wiggly sets. Consider a group G of Möbius transformations acting on the two sphere S^{2}. We say the group is Kleinian if it is discrete as a subgroup of PSL(2, ℂ), (i.e., if the identity element is isolated) and we say it is elementary if it is virtually abelian (i.e., has a finite index Abelian subgroup). Such a group G also acts as a group of isometries on the hyperbolic 3-ball 𝔹^{3}. The limit set, Λ(G), is the accumulation set (on S^{2}) of any point in 𝔹^{3}. The ordinary set of G, Ω(G), is the complement of the limit set. Ω(G)/G is always a union of (possibly branched) Riemann surfaces.

Theorem 1.2. If Ω(G)/G is compact and Λ is connected then Λ is either a Euclidean circle on S^{2} or it is uniformly wiggly (and hence has dimension > 1).

This result for co-compact quasi-Fuchsian groups was obtained by Rohde in [31] who also introduced a version of the uniformly wiggly condition. An alternate version of this condition is also used in [36] and [37]. We we also show that limit sets of another class of groups are uniformly wiggly, namely the degenerate groups. A Kleinian group action extends naturally from S^{2} to a group of isometries of the

WIGGLY SETS AND LIMIT SETS 3

hyperbolic 3-ball 𝔹^{3}. The Poincaré exponent (or critical exponent) of the group is

δ(G) = inf{s : ∑_{G} exp(−sρ(0, g(0))) < ∞},

where ρ is the hyperbolic metric in 𝔹^{3}.

G is called analytically finite if Ω(G)/G is a finite union of finite type surfaces (i.e., each is compact except for a finite number of punctures). By the Ahlfors finiteness theorem [2], [6], all finitely generated Kleinian groups are analytically finite. If G is analytically finite, but Ω(G)/G is not compact, then Λ need not be uniformly wiggly. But even in this case will will prove the following:

Theorem 1.3. Suppose G is analytically finite and Ω is a simply connected, invariant component of Ω(G). Then dim(∂Ω) = 1 iff δ(G) = 1 iff ∂Ω is a circle.

Corollary 1.4. If G is analytically finite then its limit set is either totally disconnected, a circle or has Hausdorff dimension > 1.

Corollary 1.5. If G is analytically finite and geometrically infinite then δ(G) > 1.

In [9] we prove that if G is analytically finite and geometrically infinite then dim(Λ) = 2, but this does not directly imply the corollary (as far as we know).

Theorem 1.3 was first formulated by Bowen [11] for quasi-Fuchsian groups with no parabolics. The geometrically finite, cocompact Kleinian group case was proven by Sullivan [33] and by Braam [12]. See also [34], [26]. The general geometrically finite case was proven by Canary and Taylor in [15]. In this paper we complete the discussion by including the geometrically infinite groups (although our proof covers all cases at once). The fact that the limit set is either a circle or has infinite length (and has tangents almost nowhere) follows from a more general result for divergence

4 CHRISTOPHER J. BISHOP AND PETER W. JONES

type groups by Pommereneke [29]. Pictures of limit sets can be found in several sources such as [13], [25], [28], and [35].

The remaining sections of this paper are organized as follows:

Section 2: We recall some definitions and results.

Section 3: We prove Theorem 1.1. Section 4: We prove Theorem 1.2. Section 5: We recall the basic properties of the Schwarzian derivative. Section 6: We prove Theorem 1.3. Section 7: We consider the case when Ω is a union of disks. Section 8: We prove Corollary 1.4 and Corollary 1.5.

The first author thanks Ed Taylor for explaining the results of [15] and for suggesting the problem of proving the Hausdorff dimension of degenerate limit sets is strictly greater than 1. It was by considering this problem that we were led to the other results in this paper. We also thank the referee for a very careful reading of the manuscript and numerous suggestions which improved it.

2. Background

First we recall the definition of Hausdorff dimension. Given an increasing function φ on [0, ∞), we define

H_{φ}^{δ}(E) = inf{∑ φ(r_{j}) : E ⊂ ∪_{j}D(x_{j}, r_{j}), r_{j} < δ},

and

H_{φ}(E) = lim_{δ→0} H_{φ}^{δ}(E).

This is the Hausdorff measure associated to φ. H_{φ}^{∞} is called the Hausdorff content. It is not a measure, but has the same null sets as H_{φ}. When φ(t) = t^{α} we denote the

WIGGLY SETS AND LIMIT SETS 5

measure H_{φ} by H_{α} and we define

dim(E) = inf{α : H_{α}(E) = 0}.

For α = 1 we sometimes denote H_{1} by ℓ (for “length”). An upper bound for dim(E) can be produced by finding appropriate coverings of the set. We will be more interested in finding lower bounds. The usual idea is the mass distribution principle: construct a positive measure μ on E which satisfies μ(D(x, r)) ≤ Cr^{α}. This implies dim(E) ≥ α since for any covering of E we have

∑_{j} r_{j}^{α} ≥ C^{−1} ∑_{j} μ(D(x_{j}, r_{j}) ≥ C^{−1}μ(E) > 0.

Taking the infimum over all covers gives

H_{α}^{∞}(E) > C^{−1}μ(E) > 0,

which implies dim(E) ≥ α.

A point x ∈ Λ(G) is called a conical limit point if there is a a sequence of orbit points which converges to x inside a (Euclidean) non-tangential cone with vertex at x (such points are sometimes called radial limit points or points of approximation). The set of such points is denoted Λ_{c}(G). The following in Theorem 1.1 of [9].

Theorem 2.1. Suppose G is a discrete group of Möbius transformations with more than one limit point. Then δ(G) = dim(Λ_{c}) ≤ dim(Λ).

Next we want to recall that it is enough to estimate the Poincaré series along an orbit in Ω(G) (rather than along the orbit of 0 ∈ 𝔹^{3}). For z ∈ Ω(G), let d(z) = dist(z, ∂Ω) denote spherical distance.

6 CHRISTOPHER J. BISHOP AND PETER W. JONES

Lemma 2.2. If G is a Kleinian group, 0 is the center of 𝔹^{3} and z_{0} ∈ Ω(G) ⊂ ∂𝔹^{3}, then d(g(z_{0})) ≃ 1 − |g(0)|, for all g ∈ G with constants that depend on z_{0} and G, but not on g. Proof. It clearly suffices to prove this with the origin replaced by any other fixed point x ∈ 𝔹. We may also assume that the group is normalized so that diam(Λ) = diam(S^{2}) = 1. Let D be a disk around z_{0} with radius \frac{1}{2}dist(z_{0}, Λ), let H be the hyperbolic half-plane which meets S^{2} in ∂D and let x ∈ H be the point closest to 0. Then z_{0} is one of the endpoints of the geodesic through x which is perpendicular to H. Thus g(z_{0}) is an endpoint of the geodesic through g(x) perpendicular to g(H). This implies

dist(g(z_{0}), ∂g(D)) ≥ C^{−1}(1 − |g(x)|),

since both endpoints of the geodesic are at least this far away from ∂g(D). Since g(z_{0}) ∈ g(D) and g(D) does not hit Λ, we deduce dist(g(z_{0}), Λ) ≥ C(1 − |g(x)|).

Since Λ is separated from g(z_{0}) by g(H) and Λ is assumed to have large diameter, g(z_{0}) must be on the side of g(H) with smaller diameter. Thus

dist(g(z_{0}), g(x)^{∗}) ≤ C(1 − |g(x)|),

where g(x)^{∗} denotes the radial projection onto S^{2}. Next, note that since x is a fixed hyperbolic distance from the hyperbolic convex hull of Λ, there is a fixed M so that the spherical ball of radius M(1 − |g(x)|) around g(x)^{∗} must hit Λ. Thus dist(g(z_{0}), Λ) ≤ C(1 − |g(x)|), as desired.

3. Big β’s imply dimension > 1

The main tool in the proof of Theorem 1.1 is the second author’s “traveling salesman theorem” from [21]. It states that if E is a set in the plane then the shortest curve Γ which contains E has length comparable (with universal constants) to

diam(E) + ∑_{Q} β(Q)^{2}diam(Q),

where the sum is over all dyadic squares in the plane. The higher dimensional version is given in [27]. Similarly, the length of the shortest curve which passes within ϵ of every point of E has length comparable to

diam(E) + ∑_{diam(Q)≥ϵ} β(Q)^{2}diam(Q),

where the sum is over all dyadic squares bigger than size ϵ.

Proof of Theorem 1.1. Suppose E_{0} is compact, connected and uniformly wiggly. Suppose Q is a square of side length diam(Q) = r = 2^{−N} ≤ diam(E_{0}) with \frac{1}{3}Q∩E_{0} ≠ ∅. Our first objective is to show that for small enough ϵ > 0 (depending only on the β_{0} in the definition of uniformly wiggly) we can find more than 1000ϵ^{−1} subsquares of Q of sidelength ϵdiam(Q) with disjoint doubles and such that each contains a point of E_{0} in its middle third. We will then apply the same argument to each subsquare and use the resulting nested collection of squares to build a measure μ on E_{0}.

Define E = (E_{0} ∩ Q) ∪ ∂Q. Note that E is connected since diam(Q) ≤ diam(E_{0}). Fix some integer n so that 2^{−n} < r (possibly much smaller). Because E is connected, there are more than \frac{1}{3}r2^{n} dyadic subsquares \{Q_{j}\} of \frac{2}{3}Q of size 2^{−n} such that Q_{j}∩E ≠ ∅. To see this, consider concentric “annuli” between \frac{1}{3}Q and \frac{2}{3}Q made of squares of size 2^{−n}; there are \frac{1}{3}r2^{n} such and each must intersect E. See Figure 3.1.

From this we deduce

∑_{Q:diam(Q)=2^{−n}} β_{E}^{2}(Q)diam(Q) ≥ \frac{1}{3}β_{0}^{2}r.

Figure 3.1. Many squares must hit E.

Thus for any integer k ≥ 1 (recall r = 2^{−N}),

∑_{n=N+1}^{N+k} ∑_{Q:diam(Q)=2^{−n}} β_{E}^{2}(Q)diam(Q) ≥ \frac{1}{3}kβ_{0}^{2}r.

Suppose Γ_{k} is the shortest curve in the plane with the property that for each z ∈ E we have dist(z, Γ_{k}) ≤ 2^{−N−k}. It is fairly easy to check that for squares with diam(Q) ≥ 10 · 2^{−N−k}β_{0}^{−1}, we have

β_{Γ_{k}}(Q) ≥ \frac{1}{2}β_{E}(Q) ≥ \frac{1}{2}β_{0}.

Therefore

∑_{n=N+1}^{N+k} ∑_{Q:diam(Q)=2^{−n}} β_{Γ_{k}}^{2}(Q)diam(Q) ≥ \frac{1}{3}(k − 10β^{−1})β_{0}^{2}r.

Choose k > 20β_{0}^{−1}. Then the term on the right is ≥ \frac{1}{6}kβ_{0}^{2}r.

By the second author’s characterization of rectifiable curves in [21], the length of Γ_{k} is at least C_{0}(diam(Γ_{k}) + β_{0}^{2}kr), for some absolute constant C_{0}. We claim that this implies that there are more than (C_{0}β_{0}^{2}kr − 4r)C2^{N+k} boxes {Q_{j}} of side length 2^{−N−k} such that \frac{1}{3}Q_{j} ∩ E ≠ ∅.

To prove this claim let {z_{j}} be a collection of points on Γ_{k} so that j ≠ j′ implies |z_{j} − z_{j′}| ≥ 2^{−N−k}, but so that ∪_{j}B(z_{j}, 2^{−N−k+2}) covers E. Let C be the collection

WIGGLY SETS AND LIMIT SETS 9

{∂Q_{k}} of all dyadic squares of size 2^{−N−k} contained in Q which contain some z_{j}. Let Γ = ∪_{Q_{j}∈C}∂Q_{j} ∪ ∪_{j}S_{j} be the union of the boundaries of these squares, together with segments S_{j} which connect ∂Q_{j} with the point z_{j}. Then obviously

ℓ(Γ) ≤ 6 · 2^{−N−k}#(C).

Since Γ has the property that it passes within 2^{−N−k} of every point of E and since Γ_{k} was defined to be the shortest such curve, we must have

6 · 2^{−N−k}#(C) ≥ ℓ(Γ) ≥ ℓ(Γ_{k}) ≥ C_{0}β_{0}^{2}k2^{−N}.

and hence

#(C) ≥ \frac{1}{6}C_{0}β_{0}^{2}k2^{k}.

Now set ϵ = 2^{−k}. If k > 600000β_{0}^{−2}C_{0}^{−1}), then |C| ≥ 100000ϵ^{−1}, and consist of disjoint dyadic squares of size ϵdiam(Q) each of which contains a point of E. Replace each square by its triple. By throwing away 80/81’s of the squares we can assume the remaining ones have disjoint triples, and each hits E in its middle third, as desired.

To finish the proof of the theorem, we build nested generations of squares using the construction above. The initial square Q_{0} forms the first generation. The squares of size ϵdiam(Q_{0}) constructed above form the first generation. In general, given a nth generation square containing a point of E_{0} in its middle third, we construct 1000ϵ^{−1} subsquares as above (with disjoint triples and containing a point of E_{0} in their middle thirds), and put these into the (n + 1)st generation.

We then define a measure μ by assigning each nth generation square equal mass (namely (ϵ/1000)^{n})). Since a nth generation square has size ϵ^{n}diam(Q_{0}), this measure satisfies

μ(Q) ≤ Cdiam(Q)^{α},

10 CHRISTOPHER J. BISHOP AND PETER W. JONES

where

α = \frac{log ϵ − log 1000}{log ϵ} = 1 + \frac{log 1000}{log \frac{1}{ϵ}} > 1,

and Q is an nth generation square. Since ϵ = 2^{−k} and k ≃ β_{0}^{−2} we get log ϵ^{−1} ≃ β_{0}^{−2}. This gives the estimate in the theorem. It only remains to check that this inequality holds for all squares in the plane, but this is a standard argument. Thus by the mass distribution principle dim(E_{0}) ≥ α > 1.

Given a compact set K in the plane let Ω = ℝ^{2} \K be its complement. A Whitney decomposition of Ω is a collection of squares {Q_{j}} which are disjoint, except along their boundaries, and such that there is a C < ∞ such that

\frac{1}{C} dist(Q_{j}, ∂Ω) ≤ diam(Q_{j}) ≤ Cdist(Q_{j}, ∂Ω).

The existence of a Whitney decomposition for any open set is a standard fact in real analysis (e.g., [32]). One can define {Q_{j}} to be the maximal collection of dyadic squares in Ω such that diam(Q) ≤ dist(Q, ∂Ω). This gives

\frac{1}{4}dist(Q, ∂Ω) ≤ diam(Q) ≤ dist(Q, ∂Ω).

Figure 3.2. Whitney squares

WIGGLY SETS AND LIMIT SETS 11

Corollary 3.1. Suppose K is connected and β_{K}(Q) > β_{0} > 0 for every square of the form Q = 10Q′ with Q′ a Whitney square for Ω (the complement of K) and diam(Q′) ≤ diam(K). Then dim(K) > 1.

Proof. To prove this we simply note that the hypotheses imply that β_{K}(Q) ≥ β_{0}/C for every square and some absolute C < ∞. If β_{K}(Q) > 1/100 there is nothing to do (take C = 100). If β_{K}(Q) ≤ 1/100 then Q contains a Whitney square Q′ for Ω of comparable size with 10Q′ ⊂ Q and so β_{K}(Q) ≥ β_{K}(10Q′)/C ≥ β_{0}/C.

We should point out that in the previous result, one needs the estimate for Whitney squares in all the complementary components of K. One can construct a Jordan domain so that the β’s for Whitney squares of the bounded component are all bounded away from zero, but so that the boundary has dimension 1.

4. Limit sets which are uniformly wiggly

Proof of Theorem 1.3 : It suffices to show that if Λ is not uniformly wiggly, then Λ is a circle. Consider a square Q with diam(Q) ≤ diam(Λ), \frac{1}{3}Q∩Λ ≠ ∅ and β = β_{Λ}(Q) very small. Since Λ is connected we may pass to a subsquare if necessary and assume that Λ ∩ β^{−1/2}Q is contained in a strip S of width ≤ Cβ^{1/2}diam(Q) and that it “crosses” Q. Choose a point z_{1} ∈ Ω ∩ 3Q with distance diam(Q) from Λ. Since Ω(G)/G is compact, we can find a group element g ∈ G so that dist(g(z_{1}), Λ) > δ (in Euclidean metric) for some δ depending on G, but not on z_{1}. (This would be false if G had rank 1 parabolics.) Since Λ is G-invariant, g(Λ) = Λ. On the other hand, g(Λ) is contained in g(S) ∪ g(S^{2} \ β^{−1/2}Q).

Since lines map to circles under Möbius transformations g(S) is the region between two circles and the distance between these circles looks like at most Cβ^{1/2}. To see this

12 CHRISTOPHER J. BISHOP AND PETER W. JONES

let L be the line through z_{1} which is perpendicular to the strip, S and let z_{2} be the point on L midway between z_{1} and S. By Koebe’s distortion theorem dist(g(z_{2}), Λ) is also bounded uniformly away from 0. Let z_{3}, z_{4} be the points where L intersects the two sides of the strip S. The cross ratios of (z_{1}, z_{2}, ∞, z_{3}) and (z_{1}, z_{2}, ∞, z_{4}) clearly differ by only a factor bounded by Cβ^{1/2}. Thus the same is true after we map by g. Since g(L) is perpendicular to the two circles bounding g(S) and g(z_{1}), g(z_{2}) are bounded away from these circles, we see that g(z_{3}) and g(z_{4}) are bounded away from g(∞). Thus the maximum width of g(S) is bounded by Cβ^{1/2}, as desired.

Similarly, g(S^{2} \ β^{−1/2}Q) is contained in a small disk. Thus Λ is contained in a neighborhood of a circle on the sphere where the width of the region can be estimated in terms of β_{Λ}(Q). Thus if β_{Λ}(Q) is not bounded below, Λ must be circle.

Figure 4.1. Co-compact groups have big β’s

Its easy to see that if Ω/G is compact then Λ is uniformly perfect (e.g., [14], [30]). Using this, the proof above gives Corollary 4.1. Suppose Ω/G is compact. Then Λ is either contained in a circle or β_{Λ}(Q) ≥ β_{0} for every square Q such that \frac{1}{3}Q ∩ Λ ≠ ∅ and diam(Q) ≤ diam(Λ).

WIGGLY SETS AND LIMIT SETS 13

Next we will consider another class of groups which have uniformly wiggly limit sets. Suppose G is a finitely generated degenerate group, i.e., Ω(G) = Ω has a unique component and this component is simply connected. Such a group must be geometrically infinite, so results from [9] imply we actually have dim(Λ) = 2, but this is much harder, and does not directly imply the set is uniformly wiggly.

Theorem 4.2. If G is a finitely generated degenerate group then Λ(G) is uniformly wiggly.

Proof. If the Fuchsian equivalent Ĝ = Φ ∘ G ∘ Φ^{−1} has no parabolics, then we already know this (e.g., Theorem 1.2). Therefore we may assume Ĝ does have parabolics. Fix a small number δ and for each parabolic fixed point of Ĝ, choose a generator ĝ_{j} of the parabolic group fixing that point and let B_{j}^{1} be the set of points which are moved less than hyperbolic distance δ by ĝ_{j}. Similarly, let B_{j}^{2} be the points moved by less than δ/2 by ĝ_{j}. Let X = 𝔻 \ ∪_{j}B_{j}^{2}. Then X/Ĝ is compact.

The image, Φ(B_{j}^{1}) must be a quasidisk with constant depending only on δ. To see this, suppose g ∈ G corresponds to ĝ and conjugate g so its fixed point is ∞, i.e., g is a translation. Let γ be an arc of length δ on ∂B_{j}^{1}. By the Koebe distortion theorem its image is a smooth arc and Φ(∂B_{j}^{1}) consists of translations of this arc joined end to end, and hence is a quasicircle. In particular, it satisfies the three point condition: there is an M < ∞ so that if z_{1}, z_{2}, z_{3} are any three points on the curve with z_{1} on the shorter arc between z_{2} and z_{3}, then

\frac{|z_{3} − z_{1}|}{|z_{2} − z_{1}|} ≤ M,

(e.g., Theorem IV.4, [3]). Since composing with Möbius transformations preserves quasicircles, any image of a horoball conjugate to B_{j}^{1} is a quasidisk with the same

Figure 4.2. Degenerate limit sets can’t have small β’s.

constant.

Now suppose Λ is not uniformly wiggly, i.e., suppose we have a square Q with \frac{1}{3}Q ∩ Λ ≠ ∅, diam(Q) ≤ diam(Λ) and β = β_{Λ}(Q) is small. Let S be a strip of width βdiam(Q) centered on a line L containing Λ ∩ Q. Choose a point z ∈ \frac{1}{3}Q which is about distance β^{1/2} from L. If we can choose z ∈ Φ(X) then the proof of theorem 1.3 gives a contradiction, if β is small enough.

So suppose z ∉ Φ(X). Then z is in the image of some horoball B_{j}^{2} and is moved less than hyperbolic distance δ/2 by some element g of G. Let z′ be the reflection of z across L. By normal families, it is easy to see that if β is small enough then g moves z′ by less than δ in the hyperbolic metric on Ω (because in the limit as β → 0, the transformation tends to one preserving L and is symmetric with respect to L). This means that z′ ∈ Φ(B_{j}^{1}).

Thus z and z′ are both in Φ(B_{j}^{1}) ⊂ Ω which is a quasidisk with constant depending on G but not on z, Q or β. The boundary of this quasidisk must intersect the segment connecting z to z′ in at least two points and must leave Q between these two points. Thus if β is small enough the three point condition must be violated, giving a contradiction. This proves the result.

WIGGLY SETS AND LIMIT SETS 15

5. The Schwarzian derivative

The Schwarzian derivative of a locally univalent function f is defined by

S(f)(z) = [\frac{f′′(z)}{f′(z)}]′ − \frac{1}{2}[\frac{f′′(z)}{f′(z)}]^{2}

Recall that S(f) ≡ 0 iff f is a Möbius transformation and that S satisfies the composition law

S(f ∘ g) = S(f)(g′)^{2} + S(g).

In particular, if g is Möbius then

S(f ∘ g) = S(f)(g′)^{2} and S(g ∘ f) = S(f).

In particular, if G is a Kleinian group and Ω is a simply connected invariant component of Ω(G) and Φ : 𝔻 → Ω is a Riemann mapping then |S(Φ)|(1−|z|^{2})^{2} is constant along orbits of the Fuchsian equivalent Ĝ = Φ^{−1} ∘ G ∘ Φ. Thus either Φ is Möbius or |S(Φ)|(1 − |z|^{2})^{2} bounded away from zero along some orbit of Ĝ.

The following facts are standard (e.g., see [17].) Suppose f is analytic on 𝔻. If f is univalent and φ = log f′ then

|S(f)(z)| ≤ \frac{6}{(1 − |z|^{2})^{2}} and |φ′(z)| ≤ \frac{6}{1 − |z|^{2}} .

The following is from [8] but we give a proof for completeness.

Lemma 5.1. There is a C < ∞ such that if Φ : 𝔻 → Ω is univalent and ∂Ω is rectifiable (i.e., has finite 1-dimensional measure) then

∫∫_{𝔻} |Φ′(z)||S(Φ)(z)|^{2}(1 − |z|^{2})^{3}dxdy < Cℓ(∂Ω).

Proof. The lemma is an application of the following version of Green’s formula. Suppose that F is holomorphic on 𝔻 and satisfies

∫∫_{𝔻} |F′(z)|^{2}(1 − |z|)dxdy < ∞.

Then F ∈ H^{2}(𝔻) (the Hardy space) and

∫_{∂𝔻} |F (z)|^{2}ds(z) ≃ |F (0)|^{2} + ∫∫_{𝔻} |F′(z)|^{2}(1 − |z|)dxdy.

Setting F (z) = ∑_{n} a_{n}z^{n}, then the proof is an easy caculation using the fact that the right hand side is ∑_{n} |a_{n}|^{2} and that the left hand side is

|a_{0}|^{2} + ∑_{n=1}^{∞} n^{2}|a_{n}|^{2} ∫_{0}^{1} r^{2(n−1)}(1 − r)rdr = |a_{0}|^{2} + ∑_{n=1}^{∞} n^{2}|a_{n}|^{2}(\frac{1}{2n} − \frac{1}{2n + 1})
≃ ∑_{n=0}^{∞} |a_{n}|^{2}

A similar calculation shows

∫_{∂𝔻} |F (z)|^{2}ds(z) ≃ |F (0)|^{2} + |F′(0)|^{2} + ∫∫_{𝔻} |F′′(z)|^{2}(1 − |z|)^{3}dxdy.

The same thing is true for Lipschitz domains ([16], [20]). Suppose 0 ∈ D, D is Lipschitz with constant M, and that dist(0, ∂D) ∼ diam(D) = 1. Then

∫_{∂D} |F (z)|^{2}ds(z) ≃ |F (0)|^{2} + ∫∫_{D} |F′(z)|^{2}(1 − |z|)dxdy,
∫_{∂D} |F (z)|^{2}ds(z) ≃ |F (0)|^{2} + |F′(0)|^{2} + ∫∫_{D} |F′′(z)|^{2}(1 − |z|)^{3}dxdy.

We apply this with F = (Φ′)^{1/2}. Then if φ = log(Φ′), F′ = \frac{1}{2}(Φ′)^{1/2}(φ′), and

F′′ = \frac{1}{2}(Φ′)^{1/2}(φ′′ + \frac{1}{2}(φ′)^{2}) = \frac{1}{2}(Φ′)^{1/2}(S(Φ) + (φ′)^{2}).

WIGGLY SETS AND LIMIT SETS 17

Thus

∫∫_{𝔻} |Φ′(z)||S(Φ)|^{2}(1 − |z|)^{3}dxdy
≤ C ∫∫_{𝔻} |Φ′(z)||S(Φ)(z) + (φ′(z))^{2}|^{2}(1 − |z|)^{3}dxdy

+C ∫∫_{𝔻} |Φ′(z)||φ′(z)|^{4}(1 − |z|)^{3}dxdy

≤ C ∫∫_{𝔻} |F′′(z)|^{2}(1 − |z|)^{3}dxdy

+C sup_{z} |φ′(z)|^{2}(1 − |z|^{2}) ∫∫_{𝔻} |F′(z)|^{2}(1 − |z|)dxdy.

Φ is univalent so sup_{z} |φ′(z)|^{2}(1 − |z|^{2}) is uniformly bounded and so by the formulas above, each of the two terms on the right is bounded by C ∫ |Φ′|dθ ≤ Cℓ(∂Ω).

Although we will not need it in this paper, we should mention that there is a close relationship between the Schwarzian of a Riemann mapping Φ and the β’s of the the corresponding ∂Ω. In [7] the following is proven: suppose E is compact and connected , Ω = \overline{ℂ}\E and Φ : 𝔻 → Ω the Riemann map. Then

|S(Φ)(w)|(1 − |w|^{2})^{2} ≤ C ∑_{n=0}^{∞} β_{E}(2^{n}Q)2^{−μn}

where r = dist(Φ(w), ∂Ω) The number μ satisfies 0 < μ < 1 but can be taken as close to 1 as we wish. The constant C depends only on the choice of μ. There is also a version of this for disconnected E using the universal covering map on the complement. Thus large Schwarzian implies large β’s, although the converse need not be true.

A set {z_{n}} ⊂ 𝔻 is called non-tangentially dense if almost every point of ∂𝔻 is a non-tangential limit point of the set. Although in this definition the angle associated to each boundary point may be different, this is not really a restriction; if we fix any positive angle and only consider convergence within cones of that angle then almost

18 CHRISTOPHER J. BISHOP AND PETER W. JONES

every boundary point is still a limit point (e.g., see the proof of Theorem IX.5.1 of [19]). Thus if we fix any 0 < C < ∞ and let I_{n} be the interval on ∂𝔻 with center z_{n}/|z_{n}| of length C(1 − |z_{n}|) we see that {z_{n}} is non-tangentially dense iff almost every point of the circle is in arbitrarily small I_{n}’s.

The orbit of a Fuchsian group is non-tangentially dense iff the group is divergence type, i.e.,

∑_{g∈G}(1 − |g(0)|) = ∞.

See, for example, Theorem 6.3.3 of [26]. If G is analytically finite with an invariant simply connected component Ω, then the Fuchsian equivalent Ĝ is divergence type. The following is a quantitative version of Theorem 4 in [29].

Lemma 5.2. Suppose G has an invariant component Ω which is not a disk and that the Fuchsian equivalent Ĝ of G is divergence type. Let z_{0} be in the Ĝ orbit of 0 and let I_{0} be the interval centered at z_{0}/|z_{0}| of length 10(1 − |z_{0}|). Let Φ be a Riemann mapping onto Ω such that ϵ = |S(Φ)(0)| > 0 and for I ⊂ I_{0} let tI = {tz : z ∈ I}. Then

\frac{ℓ(Φ((1 − r|z_{0}|)I))}{dist(Φ(z_{0}), ∂Ω)} → ∞,

as r → 0 with estimates that only depend on Ĝ, |I|/|I_{0}| and |S(Φ)(0)|.

Proof. By rescaling we may assume dist(Φ(z_{0}), ∂Ω) = 1. Suppose that Φ((1−r|z_{0}|)I) has length ≤ L for all 0 < r < 1. Then Φ′ is in the Hardy space H^{1} for the Carleson square with base I, so by the non-tangential maximal theorem there is a compact set E ⊂ I with |E| ≥ \frac{1}{2}|I| so that for each x ∈ E there a cone (say of angle π/2) in which

\frac{1}{M} ≤ |Φ′| ≤ M,

where M depends only on L. By taking the union of these cones we construct a “sawtooth” domain W ⊂ 𝔻 such that M^{−1} ≤ |Φ′| ≤ M on W.

Figure 5.1. The sawtooth domain

For each orbit point z_{n} ∈ W, let D_{n} denote the intersection of W with the disk centered at z_{n} of radius \frac{1}{2}(1 − |z_{n}|). By the Koebe 1/4 theorem |Φ′| ≥ C|Φ′(z_{n})| on all of D_{n} and by subharmonicity,

∫_{D_{n}} |S(Φ)|^{2}dxdy ≥ Carea(D_{n})|S(Φ)(z_{n})|^{2}.

Thus

∫∫_{D_{n}} |Φ′(z)||S(Φ)(z)|^{2}(1 − |z|^{2})^{3}dxdy ≥ Cϵ^{2} ∫∫_{D_{n}} |Φ′(z)|(1 − |z|)^{−1}dxdy
≥ CM^{−1}ϵ^{2}(1 − |z_{n}|).

Therefore by Lemma 5.1,

L ≥ C ∫∫_{W} |Φ′(z)||S(Φ)(z)|^{2}(1 − |z|^{2})^{3}dxdy
≥ CM^{−1}ϵ^{2} ∑_{n:z_{n}∈W} 1 − |z_{n}|.

We want to see that the last term grows to infinity as r → 1 with an estimate which is independent of W, giving a contradiction. Suppose this was false, i.e., there are sequences of sawtooth domains \{W_{n}\} and radii \{r_{n}\} so that |E_{n}| = |∂W_{n}∩r_{n}I| ≥ \frac{1}{2}|I|

20 CHRISTOPHER J. BISHOP AND PETER W. JONES

and

∑_{j:z_{j}∈W_{n}} 1 − |z_{j}| ≤ C < ∞,

for all n. Passing to the limit (in the Hausdorff metric) of some subsequence we get a sawtooth domain W so that |E| = |∂W ∩ I| ≥ \frac{1}{2}|I| and

∑_{n:z_{n}∈W} 1 − |z_{n}| ≤ C < ∞.

(We leave it as an easy exercise to verify that there is a limiting domain W with these properties.)

To show this is impossible we use the Vitali covering lemma: if E is a set and {I_{j}} is a collection of intervals such that each point of E is contained in intervals in {I_{j}} of arbitrarily small length, then there is a disjoint subcollection which covers almost every point of E (e.g., see [38]).

By assumption, Ĝ has non-tangentially dense orbits, so if we associate to each orbit point z_{n} the interval I_{n} of length 1 − |z_{n}| centered and z_{n}/|z_{n}|, then almost every point of the circle is in infinitely many of the intervals F = {I_{n}}. Let F be the subset of ∂W ∩ I which is Vitali covered by \{I_{n}\}. Then |F | ≥ \frac{1}{2}|I| and we can use the Vitali covering lemma to obtain a disjoint covering of almost every point of F by intervals in {I_{n}}. In fact, by repeated applications of the covering lemma we can find infinitely many collections F_{k} = {I_{j}^{k}} ⊂ F each of which is a disjoint covering of almost all of F, and so that no interval belongs to more than one collection. Thus

∑_{n:z_{n}∈W} 1 − |z_{n}| ≥ ∑_{k} ∑_{I_{n}∈F_{k}} |I_{n}| = ∑_{k} \frac{1}{2}|I| = ∞.

Therefore Φ(rI) must have length tending to ∞ as r → 1 with estimates depending only on I, the group Ĝ and |S(Φ)(0)|.

WIGGLY SETS AND LIMIT SETS 21

If we divide I_{0} into N equal intervals, we can apply the proof to each interval and then by taking the minimum growth rate for ℓ(Φ(rI)), get an estimate which is valid for all N intervals. Since any interval of length ≥ 4π/N contains at least one of the above intervals, the growth rate also holds for such an interval. This finishes the proof.

The hypothesis in this result that the Fuchsian equivalent be divergence type is necessary. Astala and Zinsmeister [5] have shown that any convergence group (i.e., ∑_{γ∈G}(1 − |γ(0|) < ∞) has a quasiconformal deformation to a Kleinian group whose limit set is a rectifiable curve but not a circle.

6. Proof of Theorem 1.3

Proof of Theorem 1.3: If the limit set is a circle then clearly dim(∂Ω) = 1 and hence by Theorem 2.1 δ ≤ 1. Since it is easy to see that δ ≥ 1 under the hypotheses of the theorem, we get δ = 1. Thus to complete the proof we need only show that if Ω is not a disk then δ > 1.

So assume Ω is not a disk. Let Φ : 𝔻 → Ω be a Riemann mapping and and normalize so that |S(Φ)(0)| ≠ 0. Let Ĝ = Φ^{−1} ∘ G ∘ Φ denote the Fuchsian equivalent of G. Let {z_{j}} be the orbit of 0 under Ĝ and for each j, let I_{j} ⊂ ∂𝔻 be the interval on the boundary centered at z_{j}/|z_{j}| with length 1 − |z_{j}|. We let S_{j} denote the Carleson square with base I_{j},

S_{j} = {z : z/|z| ∈ I_{j}, 1 − |I_{j}| ≤ |z| < 1}.

Let

d_{j} = d(Φ(z_{j})) = dist(Φ(z_{j}), ∂Ω) ≃ (1 − |z_{j}|)|Φ′(z_{j})|.

We will show

22 CHRISTOPHER J. BISHOP AND PETER W. JONES

Lemma 6.1. There is a C < ∞ (depending only on G) such that if g ∈ Ĝ \ {Id}, and z_{0} = g(0) then there is a collection of orbit points {z_{k}} = {g_{k}(0)} ⊂ S_{0} such that

(1) ∑_{k} d_{k} ≥ 2d_{0},

(2) \frac{1}{2}(1 − |z_{0}|) ≥ (1 − |z_{k}|) ≥ (1 − |z_{0}|)/C.

(3) the intervals {I_{k}} are disjoint and I_{k} ⊂ I_{0} for all k.

Note that conditions (1) and (2) imply ∑_{k} d_{k}^{1+ϵ} ≥ d_{0}^{1+ϵ}, if ϵ is small enough (depending on C). Using condition (3), we can break the orbit of 0 ∈ 𝔻 into generations G_{n} so that

∑_{z_{k}∈G_{n}} d_{k}^{1+ϵ} ≥ ∑_{z_{k}∈G_{n−1}} d_{k}^{1+ϵ} ≥ 1,

which implies

∑_{z_{k}∈Ĝ(0)} d_{k}^{1+ϵ} = ∞.

By Lemma 2.2, this proves δ(G) > 1.

Thus it suffices to prove Lemma 6.1. Let Φ : 𝔻 → Ω be a Riemann map, normalized so that S(Φ)(0) ≠ 0. Conjugate G so that Φ(0) = ∞ and diam(∂Ω) = 1. If Ω/G is a surface with punctures, then we can find a G invariant collection of disjoint balls B_{1} = {B_{j}^{1}} in Ω, each invariant under a parabolic element of G and so that (Ω\ ∪_{j} B_{j}^{1})/G is compact (i.e., we are taking a neighborhood of each puncture on Ω/G and lifting it to Ω). This is the only place in the proof where we use analytic finiteness.

Each B_{j}^{1} is thus conjugate to one of a finite subcollection and each ball has a parabolic fixed point of G on its boundary. To each ball in B_{1} we associate smaller invariant balls B_{j}^{2} ⊂ B_{j}^{1} so that the hyperbolic distances between ∂B_{j}^{1} and ∂B_{2}^{j} is 1.

WIGGLY SETS AND LIMIT SETS 23

Also any point of Ω which is outside ∪_{j}B_{j}^{2} is within a bounded hyperbolic distance of the orbit of ∞. Let C_{1} denote this bound.

If Ω/G has no punctures, just replace the collections B_{j} by the empty set in the proof that follows. Note that in this case every point of Ω is within a bounded hyperbolic distance of the orbit of ∞.

For each t > 0 consider the level line of Green’s function in the disk

Γ_{t} = {z : |z| = 1 − t},

and for each orbit point z_{j} ∈ Ĝ(0) let

Γ_{t}^{j} = Γ_{t(1−|z_{j}|)} ∩ S_{j}.

The first thing we want to see is that the image of Γ_{t}^{j} is very long if t is small enough (independent of j).

Figure 6.1. Level lines for Green’s function.

Lemma 6.2. For any M > 0 there is a t_{0} such that if t ≤ t_{0} then

ℓ(Φ(Γ_{t}^{j})) ≥ Md_{j}.

24 CHRISTOPHER J. BISHOP AND PETER W. JONES

Proof. This is just Lemma 5.2.

We would like to take the orbits for Lemma 6.1 to be near the curve Γ_{t}^{j}. More precisely, we will break Γ_{t}^{j} into unit hyperbolic segments {γ_{k}} and to each segment associate the closest orbit point z_{k}. If there were no parabolic points then each orbit point would be associated to a bounded number of segments, say N (depending on only G) and by the standard distortion theorems for conformal maps,

ℓ(Φ(γ_{k})) ≃ d(Φ(z_{k})).

Thus by throwing out repeated segments we would have a collection of points z_{k} ∈ S_{j} with

∑_{k} d_{k} ≥ C \frac{M}{N} d_{j},

and

d_{k} ≥ Cd_{j},

(where C depends on t and constants in certain distortion theorems for conformal maps). This proves the Lemma 6.1 when there are no parabolics.

If the surface Ω/G has punctures, then there may be points of Γ_{t}^{j} which are very far from the closest orbit of 0, and we need to replace such pieces by new curves which are closer to orbit points. The idea is that if Γ_{t}^{j} passes though the “bottom half” of a horoball B in the unit disk with tangent point p on the unit circle, then we replace Γ_{t}^{j} ∩ B by the arcs

Γ_{B} = (∂B ∩ {z : 1 − |z| ≤ t(1 − |z_{j}|)}) \ B(p, t(1 − |z_{j}|)).

See Figure 6.2. Then the Euclidean distance of Γ_{B} from the unit circle is greater than ϵ_{B}t(1 − |z_{j}|) where ϵ_{B} is a positive constant depending only on the conjugacy class of B.

Figure 6.2. Changing Γ inside horoballs.

The arcs Γ_{B} remain within a bounded hyperbolic distance of the orbit of 0. Most importantly, Beurling’s projection theorem implies that the length of Φ(Γ_{B}) is bounded below by a universal constant times the length of Φ(Γ_{t}^{j} ∩ B). To see this, recall that Beurling’s projection theorem (e.g., [4]) says that if Ω is simply connected and z_{0} ∈ Ω then

ω(z_{0}, ∂Ω ∩ D(x, r), Ω) ≤ C(\frac{r}{dist(z_{0}, ∂Ω)})^{1/2},

for any disk D(x, r). In particular, if a subset of ∂Ω has large harmonic measure with respect to z_{0}, it must have diameter bounded away from zero by a multiple of dist(z_{0}, ∂Ω).

Cut Γ_{t}^{j} ∩ B into unit hyperbolic segments {γ_{j}} and project each, except the center one, vertically onto an arc γ_{j}^{∗} of Γ_{B}. Then γ_{j}^{∗} has harmonic measure bounded uniformly away from zero with respect to w_{j}, the center of γ_{j}. Thus by Beurling’s theorem, Φ(γ_{j}^{∗}) must have length bounded below by a universal constant times dist(Φ(w_{j}), ∂Ω), which by Koebe’s distortion theorem, is comparable to ℓ(Φ(γ_{j})). The center piece of Γ_{t}^{j} ∩ B is handled by using Koebe’s theorem to compare it to adjacent arcs.

Thus we have proved

26 CHRISTOPHER J. BISHOP AND PETER W. JONES

Lemma 6.3. For each z_{j} there is an arc Γ̃_{t}^{j} (consisting of pieces of Γ_{t}^{j} and arcs of horoballs such that

(1) Γ̃_{t}^{j} ⊂ S_{j} ∩ {|z| ≤ 1 − ϵ_{t}(1 − |z_{j}|)}.

(2) ℓ(Φ(Γ̃_{t}^{j})) ≥ Md_{j}.

(3) every component of Γ̃_{t}^{j} has hyperbolic length at least 1.

(4) Each point of Γ̃_{t}^{j} has at most hyperbolic distance C_{1} from G(0).

We can now finish the proof of Lemma 6.1 just as in the case without parabolics described above. This finishes the lemma and hence completes the proof of Theorem 1.3.

7. Groups with round components

Suppose G is a finitely generated Kleinian group and Ω(G) contains only components which are round disks. Then either Ω(G) consists of exactly two components or has infinitely many components which are disks. In the first case, G is an extended Fuchsian group and the limit set is a circle. In the second case Λ must have dimension > 1 by a more general result of Larman [23]: there is an ϵ > 0 so that if {D_{j}} is any collection of three or more disjoint open disks in the plane, then dim(ℝ^{2} \ ∪_{j}D_{j}) ≥ 1 + ϵ.

Theorem 7.1. Suppose G is an analytically finite Kleinian group and Ω(G) contains infinitely many components which are disks. Then dim(Λ(G)) ≥ δ(G) > 1.

This is slightly different than Larman’s result because we do not insist that every component be a disk. This is known result; it is a special case of Theorem 1 of [15] which Canary and Taylor prove using a result of Furusawa [18]. It is also contained

WIGGLY SETS AND LIMIT SETS 27

in results of Sullivan and of Patterson. Some very interesting pictures of this type of limit set appear in [13]. Further results on such limit sets and the corresponding groups are given in [22].

We include a short proof of Theorem 7.1 for completeness. It is a fairly standard computation involving Hausdorff measures. Let D_{1}, D_{2} be distinct round components with disjoint closures. Denote the stabilizers by G_{1} and G_{2}. Since the orbit of D_{1} under G_{2} accumulates densely on ∂D_{2} and vice versa, we may assume (by choosing new disks if necessary) that

dist(D_{1}, D_{2}) ≥ 1,
diam(D_{1}) ≤ 1/1000,

and the double of each disk in contained in a fundamental polygon of the other group (so the translates of the doubles are disjoint).

Fix values of δ > 0 and N < ∞. Suppose we construct a Cantor set E by an iterative construction in which a disk D is replaced by at most N disks {D_{j}} such that

(1) D_{j} ⊂ 2D, and {2D_{j}} are disjoint.

(2) δdiam(D) ≤ diam(D_{j}) ≤ diam(D)/100.

(3) ∑_{j} diam(D_{j}) ≥ 2diam(D). Then it is easy to see that dim(E) ≥ α(δ, N) > 1.

For D_{1} we can choose δ, N and such disks {D_{j}^{1}} by taking part of the orbit of D_{2} under G_{1} (using the fact that G_{1} is divergence type). Similarly for D_{2} and {D_{j}^{2}}. At a general step in the construction suppose we have a disk D′ which is a “child” of D′′ (i.e, D′ ⊂ 2D′′ and diam(D′) ≤ diam(D′′)/100). Then D′ corresponds to either D_{1} or D_{2} under the action of G_{1} ∗ G_{2}, so assume it is D_{1}. Then there is an element

Figure 7.1. Constructing a Cantor set.

g ∈ G_{1} ∗ G_{2} so that g(D_{1}) = D′ and g(D_{2}) = D′′. It’s easy to check that g has bounded distortion on 2D_{1} (since it corresponds to the much smaller disk D′). Thus {g(D_{j}^{1})} satisfy the desired conditions with respect to D′ and the construction may be continued (the constants may be different, but we have uniform bounds). The bounded distortion also shows that the orbit of a single point in the first generation disk stays near the center of each higher generation disk and this shows δ(G) > 1 as well.

8. Proof of Corollary 1.4 and Corollary 1.5

Proof of Corollary 1.4: Suppose G has a connected limit set. Then any component Ω of Ω(G) is simply connected. The subgroup fixing any component Ω of Ω(G) is a finitely generated Kleinian group G_{Ω} and Ω is an invariant component of its ordinary set. Thus by Theorem 1.3 either dim(Λ(G)) > 1 or every component of Ω(G) is a disk. If the latter case holds then either Ω(G) has two components or infinitely many. If it has two then Λ(G) is a circle. Otherwise Λ(G) = \overline{ℂ}\ ∪_{j} D_{j} for some infinite collection of disjoint open disks. Thus dim(Λ(G)) > 1 by Larman’s theorem or Theorem 7.1.

It follows from the Klein-Maskit combination theorems that either Λ(G) is totally disconnected or Λ(G) contains a connected component which is itself the limit set

of finitely generated subgroup (see e.g. [1], [24]). If this component is not a circle

then we are done by Theorem 1.3. If the component is a circle but Λ(G) is not then

there are infinitely many circular components and the argument of Section 7 shows

dim(Λ) > 1.

It follows from the Klein-Maskit combination theorems that if G is finitely gen-

erated and Λ(G) is totally disconnected then G is geometrically finite and hence

δ(G) = dim(Λ_{c}(G)) = dim(Λ(G)). Thus Corollary 1.3 implies Corollary 1.5.

References

1. W. Abikoff and B. Maskit. Geometric decompositions of Kleinian groups. Amer. J. of Math., 99:687–697, 1977.

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

3. L.V. Ahlfors. Lectures on quasiconformal mappings. Math. Studies 10. Van Nostrand, 1966.

4. L.V. Ahlfors. Conformal invariants: topics in geometric function theory. McGraw-Hill, 1973.

5. K. Astala and M. Zinsmeister. Mostow rigidity and Fuchsian groups. Comptes Rendu Acad. Sci., Paris, 311:301–306, 1990.

6. L. Bers. Inequalities for finitely generated Kleinian groups. J. d’Analyse Math., 18:23–41, 1967.

7. C.J. Bishop and P.W. Jones. Harmonic measure and arclength. Ann. Math., 132:511–547, 1990.

8. C.J. Bishop and P.W. Jones. Harmonic measure, L^{2} estimates and the Schwarzian derivative. J. d’Analyse Math., 62:77–113, 1994.

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

10. C.J. Bishop, P.W. Jones, R. Pemantle, and Y. Peres. The dimension of the Brownian frontier is greater than 1. J. Funct. Analysis, 143:309–336, 1997.

11. R. Bowen. Hausdorff dimension of quasicircles. Publ. I. H. E. S., 50:11–25, 1979.

12. P. Braam. A Kaluza-Klein approach to hyperbolic three-manifolds. L’Einseign. Math., 34:275– 311, 1988.

13. S. Bullet and G. Mantica. Group theory of hyperbolic circle packings. Nonlinearity, 5:1085–1109, 1992.

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

15. R.D. Canary and E. Taylor. Kleinian groups with small limit sets. Duke Math. J., 73:371–381, 1994.

16. R. Coifman, P. Jones, and S. Semmes. Two elementary proofs of the L^{2} boundedness of Cauchy integrals on Lipschitz graphs. J. Amer. Math. Soc., 2:553–564, 1989.

17. P. Duren. Univalent functions. Springer-Verlag, 1983.

18. H. Furusawa. The exponent of convergence of Poincaré series of combination groups. Tôhoku Math. J., 43:1–7, 1991.

19. J.B. Garnett. Bounded analytic functions. Academic Press, 1981.

20. D.S. Jerison and C.E. Kenig. Hardy spaces, A_{∞} and singular integrals on chord-arc domains. Math. Scand., 50:221–248, 1982.

21. P. W. Jones. Rectifiable sets and travelling salesman problem. Invent. Math., 102:1–15, 1990.

22. L. Keen, B. Maskit, and C. Series. Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets. J. Reine Angew. Math., 436:209–219, 1993.

23. D.H. Larman. On the Besicovitch dimension of the residual set of arbitrary packed disks in the plane. J. London Math. Soc., 42:292–302, 1967.

24. B. Maskit. Kleinian groups. Springer-Verlag, 1988.

25. G. McShane, J.R. Parker, and I. Redfern. Drawing limit sets of Kleinian groups using finite state automata. Experimental Math., 3:153–170, 1994. (also appeared as Warick preprint 1994/24).

26. P.J. Nicholls. The ergodic theory of discrete groups. LMS Lecture Notes 143. Cambridge University Press, 1989.

27. K. Okikiolu. Characterizations of subsets of rectifiable curves in R^{n}. J. London Math. Soc., 46:336–348, 1992.

28. J.R. Parker. Kleinian circle packings. Topology, 34:489–496, 1995. (also appeared as Warick preprint 1994/23).

29. Ch. Pommerenke. Polymorphic functions for groups of divergence type. Math. Ann., 258:353– 366, 1982.

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

31. S. Rohde. On conformal welding and quasicircles. Michigan Math. J., 38:111–116, 1991.

32. E. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1971.

33. D. Sullivan. The density at infinity of a discrete group of hyperbolic motions. Publ. I.H.E.S., 50:172–202, 1979.

34. D. Sullivan. Discrete conformal groups and measureable dynamics. Bull. Amer. Math. Soc., 6:57–73, 1982.

35. R. Tomaschitz. Quantum chaos on hyperbolic manifolds: a new approach to cosmology. Complex Systems, 6:137–161, 1992.

36. J. Väisälä. Bilipschitz and quasisymmetric extension properties. Ann. Acad. Sci. Fenn., 11:239– 274, 1986.

37. J. Väisälä, M. Vuorinen, and H. Wallin. Thicj sets and quasisymmetric maps. Nagoya Math. J., 135:121–148, 1994.

38. 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

P.W. Jones, Mathematics Department, Yale University, New Haven, CT 06520