HAUSDORFF DIMENSION AND KLEINIAN GROUPS

CHRISTOPHER J. BISHOP AND PETER W. JONES

Abstract. Let G be a non-elementary, analytically finite Kleinian group, Λ(G) its limit set and Ω(G) = \overline{ℂ}\Λ(G) its set of discontinuity. Let δ(G) be the critical exponent for the Poincaré series and let Λ_{c} be the conical limit set of G. We prove that

(1) δ(G) = dim(Λ_{c}).

(2) if Λ ≠ S^{2} then G is geometrically infinite iff dim(Λ) = 2. Among the consequences of these results are that for any finitely generated group,

(1) The Minkowski dimension of Λ equals the Hausdorff dimension.

(2) If area(Λ) = 0 then δ(G) = dim(Λ(G)).

(3) δ(G) and dim(Λ(G)) are lower semi-continuous under algebraic convergence.

(4) A simply connected component Ω is either a disk or dim(∂Ω) > 1.

(5) Λ(G) is either totally disconnected, a circle or has dimension > 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 HAUSDORFF DIMENSION AND KLEINIAN GROUPS 1

1. Statement of results

Consider a group G of Möbius transformations acting on the two sphere S^{2}. Such a group G also acts as isometries on the hyperbolic 3-ball 𝔹. The limit set, Λ(G), is the accumulation set (on S^{2}) of the orbit of the origin in 𝔹. We say the group is discrete if it is discrete as a subgroup of PSL(2, ℂ), (i.e., if the identity element is isolated). The ordinary set of G, Ω(G), is the subset of S^{2} where G acts discontinuously, i.e., Ω(G) is the set of points z such that there exists a disk around z which hits itself only finitely often under the action of G. If G is discrete, then Ω(G) = S^{2} \ Λ(G). G is called a Kleinian group if it is discrete and Ω(G) is non-empty (some sources permit Λ = S^{2} in the definition of Kleinian group, but our results are easier to state by omitting it). The limit set Λ(G) has either 0, 1, 2 or infinitely many points and G is called elementary if Λ(G) is finite.

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}. A point x ∈ Λ(G) is called a conical limit point if there is 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). G is called geometrically finite if there is a finite sided fundamental polyhedron for G’s action on 𝔹 and geometrically infinite otherwise. A result of Beardon and Maskit [4] says that G is geometrically finite iff Λ(G) is the union of Λ_{c}(G), the rank 2 parabolic fixed points and doubly cusped rank 1 parabolic fixed points of G. This makes it clear that dim(Λ_{c}) = dim(Λ) and area(Λ) = 0 in the geometrically finite case.

2 CHRISTOPHER J. BISHOP AND PETER W. JONES

For any Kleinian group R = Ω(G)/G is a (branched) Riemann surface. If R is finite type (i.e, a finite union of compact surfaces with at most finitely many punctures and branch points), then G is called analytically finite. These are the groups with which we will work. By the Ahlfors finiteness theorem ([2], [6]), any finitely generated group is analytically finite, so our results hold for finitely generated groups. This case is slightly easier because by Selberg’s lemma any finitely generated discrete matrix group contains a finite index subgroup without torsion. This subgroup must have the same limit set as the original group, so for finitely generated groups it will always be sufficient to assume G has no torsion.

In this paper “circle” will always refer to Euclidean circles or lines (e.g., circles on the sphere). Similarly the terms “disk” or “ball” will always denote spherical balls. The main results of this paper are the following.

Theorem 1.1. If G is a non-elementary, discrete Möbius group on 𝔹 then δ(G) = dim(Λ_{c}(G)).

Theorem 1.2. If G is an analytically finite Kleinian group which is geometrically infinite then dim(Λ(G)) = 2.

Theorem 1.1 uses nothing but the definitions and a few simple properties of Möbius transformations. The direction dim(Λ_{c}(G)) ≤ δ(G) is easy and well known. We have only been able to locate the opposite inequality in the literature under the additional assumption that G is geometrically finite or Fuchsian (e.g., see [52], [55], [44]). More detailed information about the conical limit points can be found in [59] and [60].

We should also note that our proof of Theorem 1.1 works for any discrete group of Möbius transformations acting on the hyperbolic ball in any dimension and in the

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 3

rank 1 symmetric space case in general. In [21] Corlette proved the rank 1 case of Theorem 1.1 for geometrically finite groups without cusps and in [22] Corlette and Iozzi prove it for rank 1 geometrically finite groups with cusps.

It is well known that x ∈ Λ_{c} iff the geodesic ray from 0 ∈ 𝔹 to x corresponds to a geodesic on M = 𝔹/G which returns to some compact subset of M infinitely often. Our proof shows that for any ϵ > 0 there is a subset of Λ_{c}(G) of dimension ≥ δ(G) − ϵ which corresponds to geodesics which never leave the ball of radius R(ϵ) around x. Thus if M is a hyperbolic manifold with finitely generated fundamental group and x ∈ M, then the set of directions corresponding to geodesics rays starting at x which have compact closure has dimension δ(G). This had been proven for geometrically finite Kleinian groups by Fernández and Melián [29] and by Stratmann [49]. Fernández and Melián also show in their paper that this holds for all Fuchsian groups.

Theorem 1.2 was previously known in special cases. Examples of groups with dim(Λ(G)) = 2 were constructed by Sullivan in [53], and Canary [17] proved Theorem 1.2 holds if M = 𝔹/G is a “topologically tame” manifold such that the thin parts have bounded type (in particular, if the injectivity radius is bounded away from zero). Our result shows these extra hypotheses are unnecessary.

Sullivan [55] and Tukia [58] independently showed that if G is a geometrically finite group then dim(Λ(G)) < 2. Thus Theorem 1.2 implies Corollary 1.3. An analytically finite Kleinian group is geometrically finite if and only if dim(Λ(G)) < 2.

The proof of Theorem 1.2 divides into two cases. First, if δ(G) = 2, it follows immediately from Theorem 2.1. Secondly, if δ(G) < 2 and G is geometrically infinite

4 CHRISTOPHER J. BISHOP AND PETER W. JONES

we will show that area(Λ(G)) > 0. Since it is known that δ(G) = dim(Λ(G)) for geometrically finite groups, we also obtain

Corollary 1.4. Suppose that G is a non-elementary, analytically finite generated Kleinian group and that area(Λ(G)) = 0. Then δ(G) = dim(Λ).

For geometrically finite groups Stratmann and Urbanski [50] proved the Hausdorff and Minkowski dimensions of Λ agree. Since Theorem 1.2 clearly implies this for geometrically infinite groups we get

Corollary 1.5. If G is an analytically finite Kleinian group then the Minkowski dimension of Λ exists and equals the Hausdorff dimension.

For alternate proofs of this result which do not need Theorem 1.2 see [8] and [9]. Another corollary of Theorem 1.2 is the following.

Theorem 1.6. If {G_{n}} is a sequence of N-generated Kleinian groups which converges algebraically to G then dim(Λ(G)) ≤ lim inf_{n} dim(Λ(G_{n})).

Using the proof of Theorem 2.1 it is easy to see that δ(G) is lower semi-continuous under algebraic convergence, so using Corollary 1.4, the only case that causes problems is when the {G_{n}} are geometrically finite and G is geometrically infinite with positive area limit set.

As final applications of Theorem 1.2 we have

Corollary 1.7. Suppose G is a finitely generated Kleinian group and Ω is a simply connected, invariant component of Ω(G). Then dim(∂Ω) = 1 iff ∂Ω is a circle.

Corollary 1.8. If G is a finitely generated Kleinian group then its limit set is either totally disconnected, a circle or has Hausdorff dimension > 1.

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 5

These two results were previously known for geometrically finite groups (see [18]), and are clear for geometrically infinite groups since the limit set has dimension 2. This type of result was first formulated by Bowen [12] for quasi-Fuchsian groups with no parabolics. The geometrically finite, cocompact Kleinian group case was proven by Sullivan [52] and by Bramm [13]. See also [54], [44]. There are more elementary proofs of these results which apply to all analytically finite groups at once, and these are given in [10]. The proof given there also shows that in Corollary 1.7, δ(G) = 1 iff Λ is a circle. In particular, this implies that if G is analytically finite, but geometrically infinite then δ(G) > 1.

The Ahlfors conjecture states that the limit set of a finitely generated discrete group of Möbius transformations is either the whole sphere or has zero area. Thus by Corollary 1.4 the Ahlfors conjecture implies δ(G) = 2 for any geometrically infinite group. We do not know an argument for the converse direction, but both results are known to be true for topologically tame groups, [17].

The elementary groups have to be excluded in Theorem 1.1 and Corollary 1.4 because a cyclic group consisting of parabolics has a one point limit set, but δ(G) = 1/2. Also the invariance of Ω is necessary in Corollary 1.7; the boundary of a general component can have dimension strictly less than δ(G).

Very interesting pictures of limit sets can be found in several sources such as [14], [41], [45],

The remaining sections of this paper are organized as follows:

Section 2: We prove Theorem 1.1 and deduce from the proof that δ(G) is lower semi-continuous with respect to algebraic convergence. Section 3: We collect some facts about the convex core of a hyperbolic 3-

6 CHRISTOPHER J. BISHOP AND PETER W. JONES

manifold that we will need in later sections.

Section 4: We record some facts about the heat kernel and prove an estimate on Green’s function which we need to prove Theorem 1.2. Section 5: We prove Theorem 1.2 (geometrically infinite implies Λ has dimension 2). Section 6: We prove Theorem 1.6 (dim(Λ) is lower semi-continuous). Section 7: We deduce some corollaries of our results in the special case when the groups belong to \overline{T (S)}, the closure of the Teichmüller space of a finite type surface S. For example, dim(Λ(G)) is a lower semi-continuous function on \overline{T (S)} and is continuous everywhere except at the geometrically finite cusps, where it is discontinuous.

Many of the auxiliary results we use could probably be found in the literature (or easily deduced from the literature). We have included proofs of most of these results to try to make the paper as accessible to non-experts as possible and to emphasize the elementary nature of our approach.

We thank many people for helpful conversations and suggestions; Michael Anderson, Itai Benjamini, Mladen Bestvina, Dick Canary, Jay Jorgenson, Irwin Kra, Curt McMullen, Bernie Maskit, Yair Minsky, Yuval Peres and Bernd Stratmann among others. The first author particularly thanks Ed Taylor for explaining the results of [18] 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.

Originally this paper was circulated as a preprint which also contained the results of [10]. To improve the readability, the preprint was cut into two shorter papers for publication.

2. The conical limit set and critical exponent

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_{α}. When φ(t) = t^{α} we denote the 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.

8 CHRISTOPHER J. BISHOP AND PETER W. JONES

Next we will prove that δ(G) = dim(Λ_{c}) for any non-elementary, discrete Möbius group. This was previously known for geometrically finite Kleinian groups and all Fuchsian groups, and can be proven in many cases by considering the Patterson measure on the limit set (see [46], [47], [43], [51] and [55]). We shall also build a measure on the conical limit set, but ours is not a group invariant construction. We do not assume that G is finitely generated (much less geometrically finite) and G is allowed to have torsion.

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

Proof. If G has only two limit points, then it is easy to check that it is generated by a loxodromic (with the given fixed points) and a finite group of elliptics. Thus δ(G) = 0 = dim(Λ_{c}). Therefore we may assume G has more than two limit points (and hence infinitely many).

Let δ = δ(G) be the critical exponent for the Poincaré series of G. The direction δ ≥ dim(Λ_{c}) is easy and well known (e.g., Corollary 4.4.3 of [44]). Briefly it goes as follows. Let G = G(0) denote the orbit of 0 ∈ 𝔹 under G. Fix a large number M and for each g ∈ G let B_{g} be the Euclidean ball centered at g(0)/|g(0)| (the radial projection of the orbit point onto the sphere) and radius M(1 − |g(0)|). Let E_{M} be the set of points which are in infinitely many of the balls B_{g}. Since

∑_{g} diam(B_{g})^{δ+ϵ} < ∞,

for any ϵ > 0, we see that dim(E_{M}) ≤ δ(G) for any M. On the other hand, any point of Λ_{c}(G) is in E_{M} for some M. Thus dim(Λ_{c}(G)) ≤ δ(G), as desired.

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 9

To prove the other direction, δ ≤ dim(Λ_{c}), we will construct a subset C ⊂ G of the orbit of 0. We will give C the structure of a tree with root at 0, i.e., we have C = ∪_{n=0}^{∞}C_{n} and each z ∈ C_{n}, n ≥ 1, will have a unique “parent” in C_{n−1} and a collection of “children” C(z) ⊂ C_{n+1}. We will show that we can construct C so that its boundary E ⊂ S^{2} (i.e., the limit points of all sequences {z_{n}} ⊂ C where z_{n+1} ∈ C(z_{n}) for all n) is a subset of Λ_{c} and has dimension as close to δ as we wish.

Let B(z, r) denote the Euclidean ball of radius r around z. We will show that for any ϵ > 0 there are numbers 1 ≤ C, N < ∞ and points C ⊂ G (as above) so that

(1) If w ∈ C(z) then w ∈ B(z, N(1 − |z|)).

(2) If w ∈ C(z), then C^{−1} ≤ (1 − |w|)/(1 − |z|) ≤ 1/2.

(3) For w_{1}, w_{2} ∈ C(z) distinct, B(w_{1}, 2N(1 − |w_{1}|)) and ∩B(w_{2}, 2N(1 − |w_{2}|)) are disjoint.

(4) ∑_{w∈C(z)}(1 − |w|)^{δ−2ϵ} ≥ C^{2}(1 − |z|)^{δ−2ϵ}, First we will show how to construct such a collection of points. We will then show the boundary of the tree on S^{2} has dimension at least δ − 2ϵ and lies inside Λ_{c}.

Figure 2.1. The tree of orbit points

Let {z_{n}} be an enumeration of the orbit of 0. Choose a point x_{0} ∈ S^{2} = ∂𝔹 so that

∑_{j:|z_{j}−x|<r}(1 − |z_{j}|)^{δ−ϵ} = ∞,

for every r > 0 (here |z − x| denotes the Euclidean metric on \overline{𝔹}). We can do this by a simple compactness argument. Since G is non-elementary, x_{0} is not fixed by every element of G. Therefore we can choose an element g ∈ G so that x_{1} = g(x) ≠ x_{0}. Fix r > 0 to be so small that the balls B_{0}, B_{1} on S^{2} (in the Euclidean metric) of radius r around the points x_{0}, x_{1} have disjoint doubles.

Since B_{0} and B_{1} have disjoint doubles they are separated by some positive angle θ when viewed from the origin. In other words, two geodesic rays from 0 landing in B_{0} and B_{1} respectively must make an angle of at least θ at the origin. Using simple properties of hyperbolic geodesics one can check the following:

Property A: There is a N < ∞ (depending only on θ) such that if z ≠ 0 then any geodesic ray starting at z and making angle at least θ/2 with the radial segment [0, z] stays inside the (Euclidean) ball B(z, N(1 − |z|)). This is the N which will work in conditions (1)-(4) above. Property B: There is a 1 ≤ C < ∞ (also depending only on θ) so that if the geodesic segment from z to w and making angle at least θ/2 with [0, z] and has hyperbolic length L then

1 ≤ \frac{(1 − |w|)e^{L}}{1 − |z|} ≤ C. (2.1)

This is the C which works in (1)-(4). Note also that if L > 2 log C then we have 1 − |w| ≤ \frac{1}{2}(1 − |z|).

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 11

Let A_{n} = {z ∈ 𝔹 : 2^{−n−1} ≤ 1 − |z| < 2^{−n}}. We claim that for i = 1, 2,

lim sup_{n→∞} ∑_{j:z_{j}∈B_{i}∩A_{n}}(1 − |z_{j}|)^{δ−2ϵ} = ∞, (2.2)

If not there would be an M < ∞ so that

∑_{j:z_{j}∈B_{i}∩A_{n}}(1 − |z_{j}|)^{δ−2ϵ} ≤ M,

for all n, and hence we obtain the contradiction,

∑_{j:z_{j}∈B_{i}}(1 − |z_{j}|)^{δ−ϵ} ≤ C ∑_{n} 2^{−nϵ} ∑_{j:z_{j}∈B_{i}∩A_{n}}(1 − |z_{j}|)^{δ−2ϵ}
≤ CM ∑_{n} 2^{−nϵ}
< ∞.

We can easily restate (2.2) as

lim sup_{n→∞} \frac{#(z_{j} ∈ B_{i} ∩ A_{n})}{2^{n(δ−2ϵ)}} = ∞, (2.3)

Since the z_{j}’s make up the orbit of a single point, they are uniformly separated in the hyperbolic metric of 𝔹 (i.e., if j ≠ k then ρ(z_{j}, z_{k}) > ϵ_{0} for some ϵ_{0} > 0 independent of j and k). Thus for any A < ∞ we may split the sequence into a finite number B of sequences (depending on A) each of which is separated by at least A in the hyperbolic metric. Moreover,

Property C: Given the N above there is an A so that for w_{1}, w_{2} ∈ A_{n}, ρ(w_{1}, w_{2}) ≥ A implies

B(w_{1}, 3N(1 − |w_{1}|) ∩ B(w_{2}, 3N(1 − |w_{2}|)) = ∅.

By the preceding remark and (2.3) to each point x_{0}, x_{1} ∈ S^{2}, we may associate a collection of points V_{0}, V_{1} ⊂ {z_{j}} such that V_{i} ⊂ B_{i} ∩ A_{n_{i}}, i = 0, 1 and which satisfy

Figure 2.2. V_{0}, V_{1} and g(V_{0}), g(V_{1}).

the following two conditions:

B(w_{1}, 3N(1 − |w_{1}|) ∩ B(w_{2}, 3N(1 − |w_{2}|)) = ∅, (2.4)

for any distinct w_{1}, w_{2} ∈ V_{i} and

∑_{j:z_{j}∈V_{i}}(1 − |z_{j}|)^{δ−2ϵ} ≥ e^{−(δ−2ϵ)L_{i}}#(V_{i}) ≥ C^{4}, (2.5)

where L_{i} is the hyperbolic distance from 0 to V_{i} and C is as above. Since we may take n_{0}, n_{1} as large as we wish, we may assume that L_{i} ≥ 2 log C for i = 0, 1.

We now define the collection C = ∪_{n}C_{n} by induction. Start with C_{0} = {0}. Let C_{1} = V_{0} (V_{1} would work just as well). In general, suppose z = g(0) ∈ C_{n}. Consider the cone of geodesic rays from z to g(B_{0}) and g(B_{1}) respectively. These cones are separated by angle at least θ from each other, so at least one of them is separated by angle at least θ/2 from the radial segment [0, z]. Let i ∈ {0, 1} be an index for which this happens and set C(z) = g(V_{i}). (This is the step of the proof which is not group invariant and the reason why the measure we construct is not group invariant.)

We can now easily verify each of the four desired conditions. Condition 2.4 and

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 13

the fact that the geodesic segments from z to its children have hyperbolic length at least 2 log C prove (1), (2) and (3). Equations 2.1 and 2.5 imply

∑_{w∈C(z)}(1 − |w|)^{δ−2ϵ} ≥ C^{−δ+2ϵ}e^{(δ−2ϵ)L_{i}}(1 − |z|)^{δ−2ϵ}#(V_{i})
≥ C^{2}(1 − |z|)^{δ−2ϵ}.

This is (4).

We now use conditions (1)-(4) to finish the proof of the theorem. Let E be the boundary of the tree on S^{2}, i.e., E is the set of all limit points of sequences z_{1}, z_{2}, … where z_{n+1} is a child of z_{n}. Note that

Property D: If z ∈ S^{2} is in B(z, 2N(1 − |z|)), then z_{n} lies in a cone with vertex z and angle depending only on N.

Since points in the boundary of tree are in infinitely many such balls, they are vertices of cones containing infinitely many orbit points, i.e., E ⊂ Λ_{c}.

Thus all we have left to do is prove dim(E) ≥ δ − 2ϵ. This is a standard argument which goes as follows.

Let D_{z} = B(z, 2N(1 − |z|)) ∩ S^{2}. For z ∈ C_{n} these are disjoint disks by (2) and (3). Define E_{n} = ∪_{z∈C_{n}}D_{z}. Thus E = ∩_{n}E_{n}. Define a probability measure μ on E by setting μ(E_{0}) = 1, and for z ∈ C_{n} with “parent” z′ ∈ C_{n−1}, set

μ(D_{z}) = \frac{(1 − |z|)^{δ−2ϵ}}{∑_{w∈C(z′)}(1 − |w|)^{δ−2ϵ}} μ(B_{z′}).

It is easy to see by induction that

μ(D_{z}) ≤ (1 − |z|)^{δ−2ϵ} ≤ Cdiam(D_{z})^{δ−2ϵ},

for each z in C. We want to show this inequality is true for any disk D on S^{2}. Let D be any disk and let D_{0} = D_{z} be the lowest generation disk in our construction so

14 CHRISTOPHER J. BISHOP AND PETER W. JONES

that D_{0} ∩D ≠ ∅ but D ⊄ 2D_{0}. Let D_{1} be the parent of D_{0}. By the maximality of D_{0} we have D ⊂ 2D_{1}. Since 2D_{1} is disjoint from any other balls of the same generation,

μ(D) ≤ μ(D_{1})
≤ Cdiam(D_{1})^{δ−2ϵ}
≤ C(NC)^{δ−2ϵ}diam(D_{0})^{δ−2ϵ}
≤ C(2NC)^{δ−2ϵ}diam(D)^{δ−2ϵ}.

This is the desired inequality (the constant in front is larger, but is uniform over all

Figure 2.3. The Cantor set formed by the disks.

disks; the power is the same). We now have

dim(Λ_{c}(G)) ≥ dim(E) ≥ δ − 2ϵ,

by the mass distribution principle. Since ϵ is arbitrary, we get Theorem 2.1. The proof of Theorem 2.1 has the following corollaries.

Corollary 2.2. Suppose r > 0. There are constants C = C(r) and N = N(r) so that the following holds. Suppose G is a non-elementary group of Möbius transformations

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 15

on 𝔹 and suppose there are integers n_{0}, n_{1} and balls B_{0}, B_{1} with disjoint doubles of (Euclidean) radius r and a collection of points F_{i} ⊂ G(0) ∩ A_{n_{i}}, i = 0, 1 which satisfy

z, w ∈ F_{i}, z ≠ w, implies |z − w| ≥ N2^{−n_{i}}, i = 0, 1,

∑_{z∈F_{i}∩B_{i}}(1 − |z|)^{α} ≥ C.

Then δ(G) = dim(Λ_{c}) ≥ α.

Once we have the conditions in the hypothesis, the proof of Theorem 2.1 proves the lemma. The corollary is technical looking, but it shows that getting a lower bound for δ(G) only requires information about a finite number of orbit points. For example, one obvious corollary is that (2.6) δ(G) = sup{δ(G′) : G′ ⊂ G, G′ finitely generated }. This had been proven earlier by Sullivan [53]. If G̃ is another group which is very close to G then G̃ will also satisfy these conditions (since they only involve a finite number of elements in the group). Moreover, our proof of Theorem 2.1 shows that any discrete G with more than one limit point satisfies the conditions of Corollary 2.2 with α = δ(G) − ϵ for any ϵ > 0. Thus

Corollary 2.3. Suppose G is a discrete Möbius group with more than one limit point generated by {g_{1}, …, g_{n}}. Given any δ_{0} > 0 there is an ϵ_{0} > 0 (depending only on δ and G) such that if G̃ is a group containing elements {g̃_{1}, …, g̃_{n}} with ∥g_{i} − g̃_{i}∥ < ϵ_{0} (as elements of PSL(2, ℂ)) then

δ(G̃) ≥ δ(G) − δ_{0}.

Suppose {G_{n}} is a sequence of m-generated Möbius groups each with a specific listing of its generators G_{n} = {g_{1n}, … g_{mn}}. We say that G_{n} converges algebraically

16 CHRISTOPHER J. BISHOP AND PETER W. JONES

to a Kleinian group G with generators {g_{1}, …, g_{m}} if g_{jn} → g_{j} for each 1 ≤ j ≤ m, as elements of PSL(2, ℂ). See [35]. If we identify groups with points in PSL(2, ℂ)^{m}, this is just convergence in the product topology.

Corollary 2.4. Suppose G is discrete and has more than one limit point. If {G_{n}} is a sequence of discrete Möbius groups converging algebraically to G, then

δ(G) ≤ lim inf_{n→∞} δ(G_{n}).

This says that δ(G) is lower semi-continuous with respect to algebraic convergence. Strict inequality is possible even for sequences of Kleinian groups (e.g., we will see later that one can choose a sequence {G_{n}} of geometrically infinite groups (with δ(G_{n}) = 2) in the boundary of Teichmüller space converging to a geometrically finite cusp group G (with δ(G) < 2). The hypothesis that G has 2 or more limit points is also necessary, since hyperbolic cyclic groups (with δ = 0) can converge to a parabolic cyclic group (with δ = 1/2).

Let M = 𝔹/G. If G has no torsion then M is a manifold and otherwise it is an orbifold. Suppose x ∈ M is the point which projects to 0 ∈ 𝔹. It is well known that geodesic rays from 0 to points of Λ_{c} lift exactly to the geodesic rays starting at x which return to some compact set of M infinitely often (the compact set depends on the ray). Thus the unit tangent directions at x corresponding to such geodesic rays has Hausdorff dimension exactly δ(G).

In the proof of Theorem 2.1, the set E ⊂ Λ_{c} which is constructed has an additional property. The geodesic ray from 0 ∈ 𝔹 to each point of the set E corresponds to a geodesic ray in M = 𝔹/G which remains in a bounded part of M. Therefore the proof also shows the following.

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 17

Corollary 2.5. If G is any non-elementary, discrete Möbius group and ϵ > 0 then there is a R = R(ϵ) < ∞ such that given any point x ∈ M = 𝔹/G the set of directions (i.e., unit tangents at x) which correspond to geodesic rays starting at x which never leave the ball of radius R around x has dimension ≥ δ(G) − ϵ = dim(Λ_{c}(G)) − ϵ. In particular, the set of directions at x of bounded geodesic rays has dimension exactly δ(G).

As mentioned in the introduction this was proven for geometrically finite groups by Fernández and Mellián [29] and by Stratmann [49].

The only facts about hyperbolic geometry we used in Theorem 2.1 were Properties (A)-(D). Versions of these facts hold for rank 1 symmetric spaces. Such spaces consist of the usual hyperbolic upper-half spaces over the reals, and the analogous spaces over the complex numbers and quaternions plus one exceptional case corresponding to the Cayley numbers (see [21], [33], [42]). They all have negative curvature bounded and bounded away from zero, so comparison theorems such as Toponogov’s comparison theorem (e.g., [20]) can be used to deduce theses properties. Also see Theorem 2.2 of [21]. Thus Theorem 1.1 also holds in the rank 1 case (the Euclidean metric on the boundary is replaced by a sub-Riemannian metric and Hausdorff dimension on the boundary is computed with respect to the this metric.)

Corollary 2.6. If G is a non-elementary, discrete group of isometries of a rank 1 symmetric space then δ(G) = dim(Λ_{c}(G)).

3. The convex hull of the limit set

A discrete group of Möbius transformations is called geometrically finite if there is a finite sided fundamental polyhedron for its action on 𝔹 and otherwise it is called

18 CHRISTOPHER J. BISHOP AND PETER W. JONES

geometrically infinite. We will make use of several different characterizations of geometrically finite groups. Recall that a rank 1 parabolic fixed point p is called doubly cusped if there are two disjoint balls in Ω(G), tangent at p, and both invariant under the parabolic subgroup fixing p. Beardon and Maskit [4] proved that If G is a finitely generated 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. Geometrical finiteness can also be characterized in terms of the convex hull of the limit set. If K is a compact set on S^{2} = ∂𝔹 we will let C(K) ⊂ 𝔹 denote its convex hull with respect to the hyperbolic metric on 𝔹. If G is a Kleinian group, we let M = 𝔹/G be the hyperbolic 3-manifold (or orbifold) associated to G. Then C(M) = C(Λ(G))/G ⊂ M is called the convex core of M.

For r > 0 we define the radius r neighborhood of C(M) as

C_{r}(M) = {z ∈ 𝔹 : dist(z, C(M)) < r},

where distance is measure in the hyperbolic metric. Although we don’t need it here, we should point out that Thurston has shown that G is geometrically finite iff C_{r}(G) has finite volume for some (all) r > 0. We cannot take r = 0 because if G is any Fuchsian group, then Λ(G) is contained in a circle, so the convex hull of Λ(G) is contained in a hyperplane and hence has zero volume. Thus C(M) has finite volume even for infinitely generated Fuchsian groups. However, for finitely generated groups this is not a problem. If G is a finitely generated Kleinian group and C(M) has finite volume then G is geometrically finite. We define

M_{thin(ϵ)} = {x ∈ M : inj(x) ≤ ϵ},

and denote the complement in M by M_{thick(ϵ)}. Another equivalent definition of geometrically finite is that M_{thick(ϵ)} ∩ C(M) is compact for all ϵ > 0. The equivalence of the many formulations of geometric finiteness is proven in [11].

Figure 3.1. Geometrically finite and infinite manifolds

Suppose E ⊂ S^{2} and let u be the solution of the Dirichlet problem on 𝔹 with boundary values χ_{E}, i.e., u is the hyperbolic harmonic function with boundary values 1 on E and 0 on S^{2} \ E. This function can also be defined by integrating χ_{E} by the appropriate version of the Poisson kernel on S^{2}, as described in Chapter 5 of [44]. For z ∈ 𝔹, we define the harmonic measure of E with respect to z as

ω(z, E, 𝔹) = u(z).

This also represents the probability that a (hyperbolic) Brownian motion started at z will hit the sphere at infinity at a point of E.

20 CHRISTOPHER J. BISHOP AND PETER W. JONES

For z ∈ 𝔹 define

w(z) = max_{D⊂Ω(G)} ω(z, D, 𝔹),

where the max is over all round disks in Ω(G). Then C(Λ) = {z ∈ 𝔹 : w(z) ≤ 1/2}. Note that ω is G-invariant and so is well defined on M. It is also G invariant, and so defines a function on M.

Next we want to show that the convex core can be separated from the geometrically finite ends by a “nice” surface of finite hyperbolic area. To prove this we need to recall the notion of uniformly perfect sets. For z ∈ Ω(G) define

d(z) = dist(z, ∂Ω(G)) = dist(z, Λ),

where “distance” means spherical distance.

A compact set K is called uniformly perfect if there is an ϵ > 0 such that for any x ∈ K and r < diam(K) there exists y ∈ K such that

ϵr ≤ |x − y| ≤ r.

There are several well known equivalent formulations of this condition (e.g., [28], [30]). Suppose K is compact and Ω is its complement. Then the following are known to be equivalent:

(1) K is uniformly perfect.

(2) There is a positive lower bound for the length of the shortest closed hyperbolic geodesic in Ω.

(3) There is a constant C < ∞ so that

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

The following was proved in [48]. Also see [15] and [34].

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 21

Lemma 3.1. If G is a finitely generated, non-elementary Kleinian group then Λ is uniformly perfect. In particular, if ρ is the hyperbolic metric on a component Ω of Ω(G), then

|dρ(z)| ∼ \frac{|dz|}{d(z)}.

Our next goal is to see that C(M) can be separated from Ω(G) by a finite area surface. It is true that ∂C(M) itself has finite area ([27]), but the following is much easier to prove and is sufficient for our purposes. Lemma 3.2. Suppose G is an analytically finite Kleinian group and M = 𝔹/G. Let {Ω_{j}}_{1}^{N} be the conjugacy classes of components of Ω(G) (i.e., the geometrically finite ends of M). For each Ω_{j} there is a locally Lipschitz surface S_{j} in M so that the following holds.

(1) dist(S_{j}, C(M)) > 2.

(2) Each S_{j} has hyperbolic finite area.

(3) The function f(y) = (vol(B(y, 1))^{−1/2} is integrable over S_{j}, i.e.,

∫_{S_{j}}(vol(B(y, 1))^{−1/2}dA(y) < ∞. (B(y, 1) is the ball in M of hyperbolic radius 1 around y, vol denotes hyperbolic volume in M and dA is hyperbolic area on S_{j}).

(4) The surfaces S_{j} separate C(M) from the geometrically finite ends of M, i.e., there is an ϵ_{0} > 0 so that w(z) ≤ 1 − ϵ_{0} on the component M_{1} of M \ ∪_{j}S_{j} containing C(M).

(5) There is a constant C < ∞ so that the Hausdorff distance between S and

∂C(M) is less than C. In other words S ⊂ {x ∈ M : dist(x, ∂C(M)) < C},

22 CHRISTOPHER J. BISHOP AND PETER W. JONES

and

∂C(M) ⊂ {x ∈ M : dist(x, S) < C}.

Proof: Fix a fundamental domain F ⊂ Ω(G) for G. Since ∂Ω = Λ is uniformly perfect, we know dρ ∼ d(z)^{−1}|dz|, where d(z) and |dz| denote the spherical metric. Take ϵ ≪ 1/10 small F is at most hyperbolic distance ϵ from and let D_{z} = {w ∈ S^{2} : |z − w| ≤ d(z_{j})/10\}. Note that F ⊂ ∪_{z∈F} \frac{1}{2}D_{z}. Let H_{z} be the hyperbolic half-space in 𝔹 which meets S^{2} in the disk D_{z}. Let L be the boundary in 𝔹 of the region ∪_{g∈G} ∪_{z∈F} g(H_{z}). This is clearly a G invariant surface in 𝔹 which hits S^{2} exactly in Λ. Moreover, for any z ∈ F, L ∩ B_{z} is always contained in the top half of B_{z}. Thus L is locally Lipschitz.

Simple geometric arguments and Lemma 3.1 show that if ϵ is small enough then S is at least hyperbolic distance 2 from C(Λ). Furthermore, the radial projection onto S^{2} is clearly bi-Lipschitz from S to Ω(G) (where the metric on S is the hyperbolic metric on 𝔹 restricted to S and the metric on Ω is the usual hyperbolic one). For each component Ω_{j} we simply take S_{j} to be the intersection of a fundamental polygon in 𝔹 for G with S. Thus the hyperbolic area of S is bounded by at most a constant times the area of a fundamental region in Ω. This is (2). To prove (3) we split the integral over S_{j} into pieces: one for each part of S_{j} lying over a horoball in Ω and the remaining compact part of S_{j}. The integral over the compact part is clearly bounded. The integral over each of the cusps is easily bounded using the fact that the injectivity radius decreases exponentially in the cusps (in terms of hyperbolic distance) and the observation that for y ∈ S, vol(B(y, 1)) ∼ inj(y). Here inj(y) denotes the injectivity radius of M at y. Details are left to the reader. Conditions (1) and (4) are easy to check if ϵ_{0} is small enough.

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 23

To check (5), we note that any point x ∈ ∂C(Λ) is on the boundary of some hyperbolic halfspace which meets S^{2} in a disk contained in Ω, and the spherical radius of this disk is comparable to the spherical distance from its center to Λ. Thus the point of L over the center of the disk is a bounded spherical distance from x (depending only on ϵ in the definition of L). Conversely, given any point z on L we can find a point w_{1} ∈ Λ with dist(z, w_{1}) ∼ dist(z, Λ) (distances are spherical, as above). By uniform perfectness of Λ, we can find a second point w_{2} ∈ Λ with dist(w_{1}, w_{2}) ∼ dist(z, w_{1}). Thus the top of the geodesic from w_{1} to w_{2} is a point of C(Λ) which is a bounded distance from z. This completes the proof.

We want to use the fact that if G is geometrically infinite then we can find a sequence of points x ∈ C(M) tending to infinity and with inj(x) bounded away from zero. This follows from the Margulis lemma ( see, e.g., [5]) and an alternate definition of geometric finiteness. According to the Margulis lemma (e.g., page 134 of [5]) there is an ϵ > 0 (independent of G) so that if y ∈ 𝔹, then the group G_{ϵ}(x) generated by

{g ∈ G : ρ(y, g(y)) ≤ ϵ},

is almost-nilpotent and hence elementary. Let

T_{ϵ}(G) = {x ∈ 𝔹 : G_{ϵ}(x) is infinite }.

According to Definition GF4 in [11], G is geometrically finite iff C(M) \ T_{ϵ}(G) is compact for some ϵ less than the Margulis constant. If G has no elliptics this is the same as saying that C(M) \ M_{thin}(ϵ) is compact where

M_{thin(ϵ)} = {x ∈ M : inj(x) ≤ ϵ}.

Thus if G is geometrically infinite and has no elliptics (which we may assume if G is finitely generated by Selberg’s lemma), we obtain the desired sequence.

24 CHRISTOPHER J. BISHOP AND PETER W. JONES

If G has elliptics then the only difference between the conditions is the possibility of points with small injectivity radius due to the action of finite subgroups of G. By considering the possible finite elementary groups (as listed in [36] or Section 5 of [3] for example), we see that for groups of bounded order we can always move unit hyperoblic distance and reach a point with injectiviity radius bounded uniformly away from zero. The only finite Kleinian groups of high order are the cyclic groups ℤ_{n} and the corresponding dihedral groups.

Thus if G is geometrically infinite we can find points x_{n} ∈ C(M) tending to infinity with either the injectivity radius bounded away from zero or the points are in the thin parts of finite cyclic groups of arbitrarily high order. If the latter case occurs then choose lifts of the points in a convex fundamental domain for G in 𝔹 and assume the points accumulate to a single point of S^{2} (which we may assume by passing to a subsequence if necessary). Then the geodesic connecting two of the points x_{n}, x_{m} lies in the fundamental domain and C(M), and has large distance from 0 (depending on n, m). Moreover, if x_{n}, x_{m} are in the thin parts of different finite groups, there is a point on the geodesic which is on the boundary of the thin part of one of the groups and not in any other thin part (otherwise the group associated to that point would not be elementary). This point has injectivity radius bounded uniformly away from zero, and so we can constuct the desired sequence.

Note that is argument also implies that for any geodesic ray in 𝔹 which lands in the limit set either landing point is fixed by some element of G or the ray leaves every thin part (and hence injectivity radius is bigger than some ϵ > 0 on the ray arbitarily close to the landing point).

We will need the following lemma in the proof of Theorem 1.6. Note that if we

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 25

replaced C(M) by a r-neighborhood of itself, we would simply be stating one of the equivalent definitions of geometrically finite.

Lemma 3.3. If G is a Kleinian group (not necessarily finitely generated) and Λ(G) has positive area then C(M) has infinite volume.

Note that we need the hypothesis that G is Kleinian (i.e., Λ ≠ S^{2}), since it is possible for G to be a discrete group with Λ(G) = S^{2}, but C(M) = M to have finite volume (e.g., if G is co-compact). Proof. We will show there is a sequence {x_{n}} ∈ C(M) with dist(x_{n}, ∂C(M)) → ∞, and inj(x_{n}) > ϵ for all n. Thus C(M) contains infinitely many disjoint balls of fixed volume, proving the lemma.

As above, for z ∈ 𝔹 define

w(z) = max_{D⊂Ω(G)} ω(z, D, 𝔹),

where the max is over all round disks in Ω(G). It is easy to see that for any R > 0 there is an ϵ so that w(x) < ϵ implies dist(x, ∂C(M)) > R.

Since Λ(G) has positive area the Lebesgue density theorem gives us a point of density z_{0} ∈ Λ(G). We may also assume z_{0} is not one of the (countable many) points fixed by some element of G. Let γ be the hyperbolic geodesic connecting the origin to z_{0} (i.e., a radius of 𝔹) and consider points x ∈ γ converging to z_{0}. Clearly w(x) → 0, so dist(x, ∂C(M)) → ∞ as x → z_{0}. Therefore we only have to show that the injectivity radius of x is ≥ ϵ along some subsequence converging to z_{0}.

If this were false then eventually γ would remain in one of the thin components for all points close enough to the boundary. This implies z_{0} is fixed by some element of G, and this contradiction completes the proof.

26 CHRISTOPHER J. BISHOP AND PETER W. JONES

We will use the following result in the proof of Theorem 1.2 (but it is not essential; we will also sketch a proof which does not require it).

Lemma 3.4. Suppose G is analytically finite and geometrically infinite. Then there is a sequence {x_{n}} ∈ C(M) with dist(x_{n}, ∂C(M)) → ∞ and inj(x_{n}) > ϵ for all n.

Proof. By our earlier remarks, we can find ϵ > 0 and {x_{n}} ⊂ C(M) tending to infinity with inj(x) > ϵ. All we need to do is check that there is a subsequence with dist(x_{n}, ∂C(M)) → ∞.

If not, there is some A < ∞ such that dist(x_{n}∂C(M)) < A for all n. By Lemma 3.2 these points are also a bounded distance A′ from the surface S we constructed. Therefore we can construct a sequence {y_{n}} ⊂ S such that ρ(x_{n}, y_{n}) < A′. For each of the finitely many cusps on S, there is a neighborhood of the cusp so that any point in M within hyperbolic distance A′ of has injectivity radius < ϵ/2 (just consider the parabolic fixing the cusp point on S^{2}). Thus the {y_{n}} must all lie in a compact region of S. This is clearly a contradiction since x_{n} → ∞ implies y_{n} → ∞, proving the lemma.

4. The heat kernel

In this section we recall some facts about the heat kernel K(x, y, t) on M = 𝔹/G and prove a simple estimate which we will need in the proof of Theorem 1.2. 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

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 27

a Brownian motion started at the point x.

We will also need the fact that the Green’s function G(x, y) for M may be obtained by integrating the heat kernel, i.e.,

G(x, y) = ∫_{0}^{∞} K(x, y, t)dt.

We are mainly interested in the Green’s function, but it is easier to deal with the heat kernel because of the semi-group associated to K and the fact that this kernel can be easily expressed in terms of the eigenfunctions of the Laplacian on M.

The estimate we need is known. For example, in [23] E.B. Davies shows that

Theorem 4.1. If δ > 0 then there is a constant C = C(δ) such that

0 ≤ K(x, y, t) ≤ Cvol(B(x, t^{1/2}))^{−1/2}vol(B(y, t^{1/2}))^{−1/2}e^{−ρ(x,y)^{2}(4+δ)^{−1}t^{−1}},

for 0 < t < 1, where ρ(x, y) denotes the hyperbolic distance between x and y in M, and vol(B(x, t)) denotes the volume (in M) of the ball of radius t around x. Furthermore,

0 ≤ K(x, y, t) ≤ Cvol(B(x, 1))^{−1/2}vol(B(y, 1))^{−1/2}e^{(δ−λ_{0})t}e^{−ρ(x,y)^{2}(4+δ)^{−1}t^{−1}},

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

See also [25], [26] and [32] and their references. For completeness, we include the proof of the following estimate which is weaker than Davies’, but which is sufficient for our purposes. We state our result in three dimensions only, but a similar estimate holds in any dimension.

Lemma 4.2. Let G be any discrete Möbius group and let M = 𝔹/G. Then there is an absolute C < ∞ so that for any x, y ∈ M and t > 0 we have

K(x, y, t) ≤ \frac{C(1 + ρ(x, y))t^{−3/2}e^{−t}e^{− \frac{1}{8} t^{−1}ρ(x,y)^{2}}}{vol(B(x, 1))^{1/2}vol(B(y, 1))^{1/2}} , (4.1)

for 0 < t ≤ \frac{1}{8}ρ(x, y), and

K(x, y, t) ≤ \frac{Ce^{−λ_{0}t}}{vol(B(x, 1))^{1/2}vol(B(y, 1))^{1/2}} , (4.2)

for t ≥ 1. Here λ_{0} denotes the base eigenvalue of M.

Using these estimates and integrating over t we obtain the estimate on Green’s function which we will use in the next section.

Corollary 4.3. Let G be any discrete Möbius group and let M = 𝔹/G. Suppose the base eigenvalue λ_{0} is non-zero. Then there are constants C < ∞ and c > 0 (depending only on λ_{0}) so that for any x, y with ρ(x, y) ≥ 8, we have

G(x, y) ≤ Cvol(B(x, 1))^{−1/2}vol(B(y, 1))^{−1/2}e^{−cρ(x,y)}.

Proof. We set R = \frac{1}{8}ρ(x, y) and split the integral

G(x, y) = ∫_{0}^{∞} K(x, y, t)dt = ∫_{0}^{R} K(x, y, t)dt + ∫_{R}^{∞} K(x, y, t)dt.

Now use the estimates from the lemma (we drop the volume terms since they do not involve t) to get

≤ C[∫_{0}^{R}(1 + ρ(x, y))t^{−3/2}e^{−t}e^{− \frac{1}{8} t^{−1}ρ(x,y)^{2}} + ∫_{R}^{∞} e^{−λ_{0}t}]
≤ C[(1 + ρ(x, y)) ∫_{0}^{R} t^{−3/2}e^{−t}e^{− \frac{1}{16} t^{−1}ρ(x,y)^{2}}e^{− \frac{1}{16} t^{−1}ρ(x,y)^{2}} + ∫_{R}^{∞} e^{−λ_{0}t}]
≤ C[(1 + ρ(x, y))e^{−ρ(x,y)/2} ∫_{0}^{R} t^{−3/2}e^{− \frac{1}{16} t^{−1}} + \frac{1}{λ_{0}} e^{−λ_{0}R}]
≤ Ce^{−ρ(x,y)/4} + \frac{C}{λ_{0}} e^{−λ_{0} \frac{1}{8} ρ(x,y)}
≤ \frac{C}{λ_{0}} e^{−cρ(x,y)},

where c = min(\frac{1}{8}λ_{0}, \frac{1}{4}).

This estimate on Green’s function is not as strong as possible, but is more than we need. All we will use is that

G(x, y) ≤ Cvol(B(x, 1))^{−1/2}vol(B(y, 1))^{−1/2}η(ρ(x, y)),

where η(t) → 0 as t → ∞.

Proof of Lemma 4.2: We begin with the proof of estimate (4.1). We recall that if k(w, z, t) is the heat kernel on the hyperbolic ball 𝔹, then ([24], 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,

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

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).

Let

S_{n} = {z_{j} : n ≤ ρ(w, z_{j}) − ρ(x, y) < n + 1},

so that {z_{j}} = ∪_{n}S_{n}. Notice that each S_{n} can be covered by at most Ce^{2(n+ρ(x,y))} hyperbolic balls of radius 1. (This is a very weak estimate; with more work the 2 can be replaced by δ(G) + ϵ for any ϵ > 0.)

We claim that if B ⊂ 𝔹 is some hyperbolic ball of radius 1, then the number points of {z_{j}} which can lie in B is at most

Cvol(B(y, 1))^{−1}. (4.4)

To prove this let U be a connected, simply connected neighborhood of y ∈ M which lies in the ball of radius 1 around y and has the same volume as this ball. We associate to each z_{j} the component U_{j} of the lift of U which contains z_{j}. If z_{j} ∈ B then U_{j} is in the double of B (same center, radius 2). Since different U_{j} are disjoint and each has volume vol(B(y, 1)), we see that there are at most Cvol(B(y, 1))^{−1} of them, where C is the volume of a ball of radius 2 in 𝔹.

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 31

Combining estimates (4.4) and (4.3) we obtain for 0 < t < \frac{1}{8}ρ(x, y),

K(x, y, t) = ∑_{n=0}^{∞} ∑_{z_{j}∈S_{n}} k(w, z_{j}, t)
≤ Cvol(B(y, 1))^{−1} ∑_{n=0}^{∞} e^{2n+2ρ(x,y)}t^{−3/2}(1 + ρ(x, y))

exp(−t − ρ(x, y) − n − \frac{(ρ(x, y) + n)^{2}}{4t})

≤ Cvol(B(y, 1))^{−1}t^{−3/2}e^{−t}(1 + ρ(x, y))

∑_{n=0}^{∞} exp(ρ(x, y) + n − \frac{ρ(x, y)^{2} + 2nρ(x, y) + n^{2}}{4t})

The sum is bounded by (recall t ≤ \frac{1}{8}ρ(x, y)),

≤ e^{\frac{−ρ(x,y)^{2}}{8t}} ∑_{n=0}^{∞} exp(ρ(x, y) + n − \frac{1}{8t}ρ(x, y)^{2} − \frac{1}{2t}nρ(x, y) − \frac{n^{2}}{4t})
≤ e^{\frac{−ρ(x,y)^{2}}{8t}} ∑_{n=0}^{∞} exp(ρ(x, y) + n − ρ(x, y) − 4n − 0)
≤ Ce^{\frac{−ρ(x,y)^{2}}{8t}} .

Thus

K(x, y, t) ≤ Cvol(B(y, 1))^{−1}(1 + ρ(x, y))t^{−3/2}e^{−t}e^{\frac{−ρ(x,y)^{2}}{8t}} .

A standard fact about the heat kernel is its symmetry, i.e.,

K(x, y, t) = K(y, x, t).

Thus

K(x, y, t) = (K(x, y, t) · K(y, x, t))^{1/2}.

Estimate (4.1) now follows from this and the applying the estimate above twice.

Now we turn to the proof of (4.2). We start by setting x = y and t = 1 in the equality derived above to get

(4.5)  K(x, y, 1) ≤ Cvol(B(x, 1))^{−1}.

32 CHRISTOPHER J. BISHOP AND PETER W. JONES

We now use an argument shown to us by Jay Jorgenson to handle the case t ≥ 1. Let {Ω_{n}} be an exhaustion of M = 𝔹/G by compact submanifolds (or suborbifolds) with smooth boundaries and let x, y ∈ Ω_{n} for all n. Denote by K_{n}(x, y, t) the heat kernel on Ω_{n}. Then we have

K_{n}(x, y, t) → K(x, y, t),

as n → ∞. Fix some n and recall that L^{2}(Ω_{n}) has a complete orthonormal basis of eigenfunctions for the Laplacian, {φ_{j}^{n}}_{j=0}^{∞}, with eigenvalues {λ_{j}^{n}} chosen so that λ_{j}^{n} ≤ λ_{j+1}^{n} (see [19]). Then

K_{n}(x, y, t) = ∑_{j=0}^{∞} φ_{j}^{n}(x)φ_{j}^{n}(y)e^{−λ_{j}^{n}t}
= e^{−λ_{0}^{n}t} ∑_{j=0}^{∞} φ_{j}^{n}(x)φ_{j}(y)e^{−(λ_{j}^{n}−λ_{0}^{n})t}.

Now if y = x we notice that each term of the series, φ_{j}^{n}(x)^{2}e^{−(λ_{j}^{n}−λ_{0}^{n})t} is a decreasing function of t, and hence taking n → ∞,

(4.6)  K(x, x, t) ≤ e^{−λ_{0}t} lim_{n→∞} ∑_{j=0}^{∞} φ_{j}^{n}(x)^{2}e^{−λ_{j}+λ_{0}} = e^{λ_{0}−λ_{0}t}K(x, x, 1).

The Cauchy-Schwarz inequality implies

K(x, y, t) ≤ (K(x, x, t) · K(y, y, t))^{1/2},

so combining this with the estimates (4.5) and (4.6) gives for t ≥ 1,

K(x, y, t) ≤ Cvol(B(x, 1))^{−1/2}vol(B(y, 1))^{−1/2}e^{−λ_{0}t},

as desired.

It may be worth pointing out that Davies’ estimate gives an interesting bound on the counting function of a Kleinian group. Given x, y ∈ M = 𝔹/G, assume x

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 33

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 Davies’ estimate implies that for any ϵ > 0,

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

where λ_{0} = δ(2 − δ). See [9] for details.

5. Geometrically infinite groups have dimension 2

In this section we will prove,

Theorem 5.1. If G is analytically finite and geometrically infinite then

dim(Λ(G)) = 2.

If δ(G) = 2 then this follows from Theorem 1.1. Therefore we may assume δ(G) < 2. In this case Theorem 5.1 follows from

Theorem 5.2. If G is an analytically finite, geometrically infinite group and δ(G) < 2 then Λ(G) has positive area.

Proof of Theorem 5.2: Let Λ = Λ(G) be the limit set of G. Let δ = δ(G) be the critical index for the Poincaré series and λ_{0} the base eigenvalue for the Laplacian on M = 𝔹/G. The Elstrodt-Patterson formula relating δ and the base-eigenvalue by

λ_{0} = \begin{cases}1, & if δ < 1 \\ δ(2 − δ), & if δ ≥ 1 \end{cases}

(e.g., Theorem 2.18 of [56]). In particular, δ(G) < 2 implies λ_{0} > 0.

34 CHRISTOPHER J. BISHOP AND PETER W. JONES

Let C(Λ) be the convex hull of the limit set in 𝔹 and C(M) = C(Λ)/G ⊂ M be the convex core of M (see Section 3). Let S = ∪_{j}S_{j} be the surface given by Lemma 3.2 which separates C(M) from the geometrically finite ends, Ω(G). Let ω(x, Ω(G)) denote the harmonic measure of the geometrically finite ends with respect to the point x ∈ M. More precisely, let u be the solution of the Dirichlet problem on 𝔹 with boundary values 1 on Ω and 0 on Λ (e.g., see Chapter 5 of [44] for a description of u in terms of the Poisson kernel of χ_{Ω}). Then u is a G-invariant harmonic function on M so defines a harmonic function on M which we denote ω(x, Ω(G)). One easily see that

u(0) = \frac{area(Ω)}{area(S^{2})}.

If x ∈ C(M) then the surface S separates x from Ω(G) so the Gauss theorem yields

ω(x, Ω(G)) = C ∫_{S} \frac{∂G}{∂n}(x, y)dA(y), (5.1)

where \frac{∂}{∂n} is the normal derivative to S and dA is area measure on S. This equality is easier to see when lifted to 𝔹. Let S̃ denote the lift of S to 𝔹 and let B_{N} denote the hyperbolic ball of radius N in 𝔹 centered at 0. Let B̃_{N} be the subdomain of B_{N} consisting of all points separated by S̃ from 0, and let R_{N} = ∂B̃_{N} ∩ ∂B_{N}. Then ∂B̃_{n} = R_{N} ∪ S̃_{N}, where S̃_{N} = S̃ ∩ B_{N}. Now

u(0) = lim_{N→∞} C ∫_{R_{N}} u\frac{∂g}{∂n}dA,

because x ∈ S^{2} ∩ Ω if and only if for all large N, Γ_{N}x ∈ R_{N}, where Γ_{N}x is the inward projection of x onto ∂B_{N}. On the other hand, Gauss’ theorem yields (g is harmonic on B̃_{N}),

0 = ∫B̃_{N} ΔgdV = ∫S̃_{N} \frac{∂g}{∂n}dA − ∫_{R_{N}} \frac{∂g}{∂n}dA,

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 35

where in both terms we take the derivatives in the normal direction, “pointing towards 0”. Taking limits in N in our last two equations yields

u(0) = C ∫S̃ \frac{∂g}{∂n}dA.

Since S̃ is group invariant and G(x) is obtained by summing Green’s function g over an orbit in the ball,

∫_{S} \frac{∂G}{∂n}dA = ∫S̃ \frac{∂g}{∂n}dA,

completing the proof of (5.1).

Suppose ρ(x, y) ≥ 1. By lifting from M to the hyperbolic ball and using Harnack’s inequality we obtain the estimate G(x, z) ≤ CG(x, y) for all z ∈ B_{ρ}(y, 1/2). To see this, let Π denote the projection from 𝔹 to M and let u(w) = G(x, Π(w)). Then if Π(0) = y and w ∈ B_{ρ}(0, 1) we see that u is a positive harmonic function, so we can apply Harnack’s inequality to deduce u(w) ≤ Cu(0). Projecting back to M gives the desired estimate.

Next we want to estimate the gradient |∇_{y}G(x, y)| ≤ CG(x, y), (5.2) as long as ρ(x, y) ≥ 1. This follows because harmonic functions satisfy the mean value property on balls, i.e.,

u(x) = ∫_{B(x,1)} u(w)dw.

Thus

|u(x) − u(z)| = ∫_{B(x,1)ΔB(z,1)} |u(w)|dw
≤ ∥u∥_{∞}vol(B(x, 1)ΔB(z, 1))
≤ C∥u∥_{∞}|x − z|,

36 CHRISTOPHER J. BISHOP AND PETER W. JONES

where EΔF = (E \ F) ∪ (F \ E). This proves the estimate if we take u(x) = G(x, y). Thus by (5.1) and (5.2),

ω(x, Ω(G)) = C ∫_{S} \frac{∂G}{∂n}(x, y)dA(y)
≤ C ∫_{S} G(x, y)dA(y).

By Corollary 4.3 this gives

ω(x, Ω(G)) ≤ C ∫_{S} vol(B(x, 1))^{−1/2}vol(B(y, 1))^{−1/2}e^{−cρ(x,y)}dA(y).

By Lemma 3.4 we can choose a point x ∈ C(M) where the injectivity radius inj(x) > ϵ > 0 is uniformly bounded below and

dist(x, S) ≥ dist(x, ∂C(M)) ≥ R,

is as large as we wish. For such a point we get

ω(x, Ω(G)) ≤ Cϵ^{−3/2}e^{−cR} ∫_{S} vol(B(y, 1))^{−1/2}dA(y).

By Lemma 3.2, the integral is bounded, so

ω(x, Ω(G)) ≤ Cϵ^{−3/2}e^{−cR} ≤ 1/2,

if R is large enough. Thus we can find a point x ∈ 𝔹 at which the harmonic measure of Ω(G) is strictly less than 1. This means that Ω(G) has less than full area measure on the sphere, so the limit set must have positive area. This proves the theorem.

Sketch of proof using Brownian motion: The preceding argument can be rewritten in terms of Brownian motion on the manifold M as follows. To show that Λ(G) has positive area, it suffices to show that a Brownian motion in the ball has a positive probability of first hitting the boundary in Λ(G). Projecting to M = 𝔹/G, it suffices

Figure 5.1. Brownian motion escapes in the convex core.

to find a point x ∈ C(M) so that a Brownian motion started at x has a positive probability of going to infinity in M without ever crossing the surface S.

Let U be a unit neighborhood of S. The expected time a Brownian motion started at x spends in U is expressed in terms of the heat kernel by

∫_{0}^{∞}[∫_{U} K(x, y, t)dy]dt.

The heat kernel estimates of the previous section imply that if we choose x ∈ C(M) with injectivity radius bounded away from zero and with dist(x, U) very large, then the expected time a Brownian motion started at x spends in U is a small as we wish.

However, the expected time it takes a Brownian motion started at a point y of S to first leave U (i.e., to travel unit distance from S) is bounded away from zero independent of the starting point y. This is because the expected time to travel distance 1 in M is greater than or equal the expected time to travel this distance in the covering space 𝔹, and this is bounded away from zero. Let t_{0} > 0 be a lower bound for the expected time to travel distance 1. Thus the expected time a Brownian motion started at x spends in U is at least the probability that it ever

38 CHRISTOPHER J. BISHOP AND PETER W. JONES

hits S multiplied by the bound t_{0}. From this it is easy to see that the chance that a Brownian motion started at x ever hits S is small if dist(x, S) is large. Thus with positive probability the Brownian path never crosses S, as desired.

Sketch of proof using doubling: It actually suffices to use a weaker estimate on the heat kernel and less information on the convex hull. All we need to know is that Brownian motion is transient on a complete, connected, infinite volume Riemannian manifold with lowest eigenvalue bounded away from zero. For example, the estimate

K(x, y, t) ≤ C_{1}e^{−C_{2}t},

for some fixed x and y and all t > T_{0} would be sufficient for this.

We pay for the less precise estimate by a more involved construction on the manifold. Let S be the surfaces described by Lemma 3.2 which separate the convex core from the geometrically finite ends. Cut M along S and let M_{1} be the component containing C(M). Glue two copies of M_{1} along S. We claim that the resulting manifold N (the double of M_{1}) has lowest eigenvalue bounded away from zero. If so then the heat kernel estimates apply to the new manifold N and we deduce that the expected time a Brownian motion spends in U (the unit radius neighborhood of the S) is finite. By the Borel-Cantelli lemma this says that there are points x in M from which the probability of ever hitting S is strictly less than 1 (in fact, it is as small as we wish). Thus there is a point x ∈ N \ U from which the chance of ever hitting U is less than 1/2. But the two components of N \ S are both exactly M_{1}. Thus Brownian motion in M_{1} must have a positive probability of tending to infinity without ever hitting S.

This proves the result, except for verifying that N has first eigenvalue bounded away from 0. We will not verify this in detail, but simply note that since M has constant negative curvature and lowest eigenvalue > 0, Buser’s inequality (e.g. [16])

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 39

implies the Cheeger constant for M is bounded away from zero. From this one proves that the Cheeger constant for the manifold with boundary M_{1} is non-zero, and hence that the Cheeger constant for the doubled manifold N is non-zero. Then Cheeger’s estimate says the first eigenvalue for N is non-zero, as desired.

6. Lower semi-continuity of Hausdorff dimension

In this section we will prove,

Theorem 6.1. If G is a finitely generated Kleinian group and {G_{n}} is a sequence of Kleinian groups converging algebraically to G then

dim(Λ(G)) ≤ lim inf_{n} dim(Λ(G_{n})).

In particular, if G is geometrically infinite then this result and Theorem 1.2 imply that lim_{n} dim(Λ(G_{n})) = 2.

If G is elementary then dim(G) = 0, and the result is trivial so we may assume G is non-elementary. Similarly, if lim inf dim(Λ(G_{n})) = 2, there is nothing to do, so assume (after passing to a subsequence) that lim dim(Λ(G_{n})) exists and is strictly less than 2. In particular, we may assume all the G_{n}’s are geometrically finite and hence δ(G_{n}) = dim(Λ(G_{n})). If G is geometrically finite then δ(G) = dim(Λ(G)), so Theorem 6.1 follows from Corollary 2.4. Therefore we may also assume that G is geometrically infinite. The following result shows this is impossible and completes the proof of Theorem 6.1.

Theorem 6.2. If {G_{n}} is a sequence of geometrically finite Kleinian groups which converges algebraically to a finitely generated, geometrically infinite discrete group G then δ(G_{n}) → 2.

40 CHRISTOPHER J. BISHOP AND PETER W. JONES

This result follows from two known results

Theorem 6.3 (Canary). If G is a n-generated, geometrically finite group then

λ_{0} ≤ \frac{A_{n}}{vol(C(M))},

where A_{n} is a constant that only depends on the number of generators of G.

Proof: This is essentially Theorem A of [16] except that there Canary proves

λ_{0} ≤ A\frac{χ(∂C(M))}{vol(C(M))},

where A is an absolute constant and where χ denotes the Euler characteristic. However, the Euler characteristic of ∂C(M) is the same as that of Ω(G)/G, because there is always a homeomorphism between the two (e.g., see Epstein and Marden’s paper [27]). By the Bers inequality [6] (a quantitative version of the Ahlfors finiteness theorem) the area, and hence the Euler characteristic, of Ω(G)/G can be bounded in terms of n, the number of generators of the group G. Thus Canary’s results says

λ_{0} ≤ \frac{A_{n}}{vol(C(M))},

where A_{n} depends only on the number of generators.

Theorem 6.4. Suppose {G_{n}} are geometrically finite Kleinian groups such that

sup_{n} vol(C(M)) < ∞.

If the sequence {G_{n}} converges algebraically to a finitely generated, geometrically infinite, discrete group G then Λ(G) has zero area.

Proof: This is an easy case of a result obtained by E. Taylor in [57] and is probably well known. Here we will sketch a proof which follows an argument given by Jørgensen and Marden in [35].

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 41

Suppose Λ(G) has positive area. We will derive a contradiction. Suppose {G_{n}} is a sequence of Kleinian groups converging algebraically to the Kleinian group G. The set of compact subsets of a compact metric space is itself compact with the Hausdorff metric

d(E, F) = max_{z∈E} dist(z, F) + max_{w∈F} dist(w, E),

so by passing to a subsequence (which we also denote {G_{n}}) we may assume the sets Λ_{n} = Λ(G_{n}) converge in the Hausdorff metric to a compact set Λ_{∞}.

We say a sequence {G_{n}} converges polyhedrally to a group H if H is discrete and for some x_{0} ∈ 𝔹, the fundamental polyhedra (the Dirichlet polyhedron)

P (G_{n}) = {z ∈ 𝔹 : ρ(z, x_{0}) ≤ ρ(z, g(x_{0})) for all g ∈ G_{n}},

converge to P (H) uniformly on compact subsets of 𝔹. By Proposition 3.8 of [35] any algebraically converging subsequence has a polyhedrally convergent subsequence and the polyhedral limit group contains the algebraic limit group (but they need not be equal).

A third notion of convergence of groups is geometric convergence. Given a sequence of groups {G_{n}} we define

Env{G_{n}} = {g ∈ PSL(2, ℂ) : g = lim_{n} g_{n}, g_{n} ∈ G_{n}},

and we say G_{n} → H = Env{G_{n}} geometrically if for every subsequence {G_{n_{j}}}, Env{G_{n_{j}}} = Env{G_{n}}. Proposition 3.10 of [35] says that G_{n} converges geometrically to H if and only if it converges polyhedrally to H.

Thus we may assume that we have groups G ⊂ H such that

(1) G_{n} → G algebraically,

(2) G_{n} → H polyhedrally and geometrically,

42 CHRISTOPHER J. BISHOP AND PETER W. JONES

(3) Λ_{n} → Λ_{∞} in the Hausdorff metric.

If we can show that the convex core of 𝔹/H has finite volume, then we can obtain a contradiction as follows. Since Λ(G) ⊂ Λ(H), the latter set also has positive area. If Λ(H) ≠ S^{2} then this contradicts Lemma 3.3. On the other hand, if Λ(H) = S^{2} then we get vol(𝔹/H) < ∞, which implies the thick part of 𝔹/H is compact, which implies the thick part of 𝔹/G_{n} is eventually compact, another contradiction.

Thus it suffices to prove the convex core of 𝔹/H has finite volume.

We first claim that Λ(H) ⊂ Λ_{∞}. If Λ_{∞} = S^{2} there is nothing to do, so we may assume Λ_{∞} is not the whole sphere. In this case we follow the proof of Proposition 4.2 of [35]. Let Ω = S^{2} \ Λ_{∞} and suppose K, K′ are compact sets such that

K ⊂ int(K′) ⊂ K′ ⊂ Ω.

Suppose h ∈ H. Because {G_{n}} converges geometrically to H we can write h = lim_{n} g_{n} with g_{n} ∈ G_{n}.

We claim that h(K) ⊂ Ω. If not then h(K) intersects Λ_{∞}, so int(h(K′)) also hits Λ_{∞}. This implies that int(h(K′) intersects Λ_{n} for all large enough n, say n ≥ N_{1}. Therefore int(g_{n}(K′)) hits Λ_{m} for all m ≥ N_{1}, for all sufficiently large n, say n ≥ N_{2}. So if N_{3} = max(N_{1}, N_{2}), then n ≥ N_{3} implies

int(g_{n}(K′)) ∩ Λ_{n} ≠ ∅.

Therefore

int(K′) ∩ g_{n}^{−1}(Λ_{n}) = int(K′) ∩ Λ_{n} ≠ ∅.

This is a contradiction, so we must have h(K) ⊂ Ω. This implies h(Ω) ⊂ Ω. Since the same argument applies to h^{−1}, we see that h(Ω) = Ω, or equivalently, h(Λ_{∞}) = Λ_{∞}. Since Λ_{∞} is a closed set which is invariant under the group H we must have

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 43

Λ(H) ⊂ Λ_{∞} as desired (recall that the limit set is the smallest closed H-invariant set if H is non-elementary).

Since Λ(H) ⊂ Λ_{∞} the convex hull C(Λ(H)) of Λ(H) in 𝔹 is contained in the convex hull C(Λ_{∞}) of Λ_{∞}. The convex hulls C(Λ_{n}) of the sets Λ_{n} converge, uniformly on compacta, to C(Λ_{∞}). Thus for any R < ∞,

vol(C(Λ(H)) ∩ P (H) ∩ B(x_{0}, R)) ≤ lim inf_{n} vol(C(Λ_{n}) ∩ P (G_{n}) ∩ B(x_{0}, R)) ≤ M,

by the lemma below. Thus C(Λ(H)) ∩ P (H) has finite volume, as desired.

The above inequality uses the the following lemma about the convergence of convex hulls.

Lemma 6.5. Suppose {K_{n}} ⊂ S^{2} are compact sets which converge in the Hausdorff metric to K. Then for any R < ∞,

lim_{n→∞} vol(C(K_{n}) ∩ B(0, R)) = vol(C(K) ∩ B(0, r)).

Proof. Let χ_{n} denote the characteristic function of C(K_{n}) and χ_{K} the characteristic function of K. We first claim that

lim_{n→∞} χ_{n}(z) = χ_{K}(z),

for all z ∉ ∂C(K). First suppose z ∈ int(C(K)). Then there are four points z_{1}, …, z_{4} ∈ int(C(K)), so that the convex hull of these points contains a neighborhood of z. Since K_{n} → K in the Hausdorff metric it is easy to see that for any ϵ > 0, and all large enough n, C(K_{n}) will contain points within ϵ of each of the four points, and thus will contain z (if ϵ is small enough).

On the other hand, suppose z ∉ C(K). Then there is a closed hyperbolic half-space which contains z but does not hit C(K). This half space hits S^{2} in closed disk which

44 CHRISTOPHER J. BISHOP AND PETER W. JONES

is a positive distance from K. Thus for all large enough N the disk does not hit K_{n} and so z ∉ C(K_{n}). this proves the claimed convergence.

Finally, since each point of ∂C(K) is on the boundary of an open half-space which misses ∂C(K), the Lebesgue density theorem implies ∂C(K) has zero volume. Thus the Lebesgue Dominated Convergence Theorem implies the lemma.

Now that we have the two results, we can can finish the proof of Theorem 6.2. Suppose G is a finitely generated, geometrically infinite discrete group and {G_{n}} are geometrically finite groups converging to G algebraically. If Λ(G) has zero area then δ(G) = dim(Λ(G)) = 2 by Theorem 1.4 and δ(G_{n}) → 2 by Corollary 2.4. Thus we may assume Λ(G) has positive area. By Theorem 6.4 we must have vol(C(G_{n})) → ∞, so by Theorem 6.3 we get λ_{0}(G_{n}) → 0. Thus by the Elstrodt-Patterson formula δ(G_{n}) → 2, as n → ∞. This completes the proof of Theorem 6.2.

7. Teichmüller spaces

In this section we shall consider dim(Λ(G)) as a function on the closure of the Teichmüller space T (S) of a finite type hyperbolic surface S.

Given a finite type surface S (compact with a finite number of punctures, possible none), the Teichmüller space T (S) is the set of equivalence classes of quasiconformal mappings of S to itself. Each such is represented by a Beltrami differential μ̃ which may be lifted to a Beltrami differential μ on the the upper halfplane, ℍ. Let Γ be a Fuchsian group acting on ℍ such that ℍ/Γ = S. There is a quasiconformal mapping F of the plane which fixes 0, 1, ∞ and such that \overline{∂}F/∂F = μ on ℍ and so that F is conformal on the lower half plane. On the lower half-plane the Schwarzian derivative

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 45

S(F) satisfies

∥S(F)∥ = sup_{z} |Im(z)|^{2}|S(F)(z)| ≤ 6 < ∞.

This realizes T (S) as a bounded subset of a Banach space and gives a metric on T (S). The closure of T (S) with respect to this metric is denoted \overline{T (S)} and the boundary by ∂T (S). Points of \overline{T (S)} may be identified with certain Kleinian groups which are isomorphic to Γ. Moreover, convergence in the Teichmüler metric implies algebraic convergence of the groups. A group is called degenerate if Ω(G) has exactly one component and this component is simply connected. Such groups must be geometrically infinite by a result of Greenberg [31]. G ∈ ∂T (S) is called a cusp if there is a hyperbolic element in Γ which becomes parabolic in G. Bers showed ∂T (S) consists entirely of degenerate groups and cusps and that degenerate groups form a dense G_{δ} set in ∂T (S) in [7] (in fact, the cusps lie on a countable union of real codimension 2 surfaces). McMullen [40] proved there is a dense set of geometrically finite cusps in ∂T (S).

Recall that Theorem 1.6 says that if {g_{n}} converges algebraically to G then

dim(Λ(G)) ≤ lim inf_{n→∞} dim(Λ(G_{n}))).

One special case where this holds is for G ∈ \overline{T (S)}, the closure of the Teichmüller space of a finite type hyperbolic Riemann surface S. Since Λ(G) is at most 2 and is lower semi-continuous, it is continuous whenever it takes the value 2 (i.e., at the geometrically infinite groups). Since these points are dense on the boundary of Teichmüller space, this function must be discontinuous at the geometrically finite cusps on the boundary. Thus

46 CHRISTOPHER J. BISHOP AND PETER W. JONES

Corollary 7.1. Suppose S is a hyperbolic Riemann surface of finite type. Then dim(Λ(G)) is lower semi-continuous on \overline{T (S)}, and is continuous everywhere except at the geometrically finite cusps in ∂T (S) (where it must be discontinuous).

This also shows that equality in Corollary 2.4 and Theorem 1.6 need not occur, because a geometrically finite cusp (δ(G) < 2) can be approximated by degenerate groups (so δ(G_{n}) → 2). The discontinuity at the geometrically finite cusps had been proved earlier by Taylor in [57]; he showed that for each geometrically finite cusp G there is a sequence G_{n} → G algebraically, but G_{n} → H geometrically where H is a geometrically finite group containing G and

dim(Λ(G_{n})) → dim(Λ(H)) > dim(Λ(G)).

If f is lower semi-continuous then {f ≤ α} is closed. Thus,

Corollary 7.2. The set E_{α} = \{G ∈ \overline{T (S)} : dim(Λ(G)) ≤ α\} is closed in \overline{T (S)}. The set F_{α} = {G ∈ ∂T (S) : dim(Λ(G)) ≤ α < 2} is a closed, nowhere dense subset of ∂T (S).

Since a lower semi-continuous functions takes a minimum on a compact set, dim(Λ) attains a minimum value on ∂T (S). We prove in [10] that Λ is either totally disconnected, a circle or has dimension > 1. Limit sets corresponding to points of ∂T (S) cannot be circles or totally disconnected, so the third option holds for them. Thus

Corollary 7.3. dim(Λ(G)) takes a minimum value on ∂T (S) and this minimum is strictly larger than 1.

It is not clear where the minimum occurs. Canary has suggested it might occur at the cusp group corresponding to shrinking a minimum length geodesic on S to a

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 47

parabolic, since this requires the “least” deformation of the Fuchsian group (in some sense). Since dim(Λ(G)) takes a minimum on ∂T (S) which is > 1, any group in \overline{T (S)} with small enough dimension must be quasi-Fuchsian. Thus

Corollary 7.4. Suppose {G_{n}} is a sequence of quasiconformal deformations of a Fuchsian group G (i.e., {G_{n}} is a sequence in T (S), S = 𝔻/G). If dim(Λ(G_{n})) → 1, then G_{n} → G.

Proof. This is immediate from Theorem 6.1 and the fact (deduced from Theorem 1.7) that G is the only point in \overline{T (S)} where dim(Λ) = 1. (Actually, since the groups involved are all geometrically finite we could use Corollary 2.4 and geometrically finite case of Theorem 1.4 instead of the more difficult result Theorem 6.1.)

Corollary 7.5. Suppose G is a finitely generated and has a simply connected invariant component Ω_{0} (possibly not unique). Let Ω_{0}/G = S. Then for any ϵ > 0 there is a δ, depending only on S and ϵ, such that dim(Λ(G)) < 1 + δ implies G is a ϵ-quasiconformal deformation of a Fuchsian group.

Proof. If G is a quasi-Fuchsian group this follows from the previous result. Maskit [38] proved that a finitely generated Kleinian group with two invariant components is quasi-Fuchsian, so we may now assume G is a b-group (i.e., has exactly one simply connected, invariant component). If G is geometrically infinite then dim(Λ(G)) = 2, which contradicts our assumption. Therefore, G must be geometrically finite. Abikoff [1] proved that every geometrically finite b-group covering S is on the boundary of the Teichmüller space T (S), and so its dimension is bounded away from 1 by Corollary 7.3.

48 CHRISTOPHER J. BISHOP AND PETER W. JONES

It is not true that the ϵ in Corollary 7.5 can be taken to depend only on the topological type of S (e.g., the number of generators of G). For example, given a surface with punctures S, it is possible to use the combination theorems to construct a b-group G with dim(Λ(G)) as close to one as we wish and so that Ω_{0}/G is homeomorphic to (though not conformally equivalent to) S.

Larman showed that there is an ϵ_{0}, such that if {D_{j}} is a collection of three or more disjoint open disks then the dimension of \overline{ℂ} \ ∪_{j}D_{j} is larger than 1 + ϵ_{0}. A careful reading of Larman’s paper [37] shows that his proof gives

Theorem 7.6. There is an ϵ_{0} > 0 such that if {D_{j}} is any collection of three or more disjoint open ϵ_{0}-quasidisks, then dim(ℂ \ ∪_{j}D_{j}) > 1 + ϵ_{0}.

Recall that a web group is a finitely generated Kleinian groups each of whose component subgroups is quasi-Fuchsian. Suppose G is web group. If Ω(G) has only two components then G is quasi-Fuchsian [39]). So suppose G has three or more components and let {G_{1}, …, G_{n}} be representatives of each conjugacy class of component subgroups. By the last corollary either one of these subgroups has a limit set with dimension > 1 + δ or all the limit sets are ϵ_{0}-quasicircles. In the latter case, Larman’s theorem implies that Λ(G) has dimension bigger than ϵ_{0}. In either case the dimension is bounded away from 1 by a number which only depends on the conformal structure of Ω(G)/G.

Corollary 7.7. Suppose G is a finitely generated web group which is not quasi-Fuchsian. Then dim(Λ) > 1 + ϵ where ϵ > 0 depends only on the conformal types of the components of Ω(G)/G.

It is not true that the dimension of limit sets of proper web groups (i.e., not

HAUSDORFF DIMENSION AND KLEINIAN GROUPS 49

quasi-Fuchsian) is bounded uniformly away from 1. Canary, Minsky and Taylor have constructed examples of proper web groups (with a fixed number of generators) whose limits sets have dimension arbitrarily close to 1 (personal communication).

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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

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