UNIFORM PERFECTNESS OF THE LIMIT SETS OF KLEINIAN GROUPS

TOSHIYUKI SUGAWA

Abstract. A compact set C in the Riemann sphere is said to be uniformly perfect if bounded are the moduli of those annuli in the complement which separate C. The limit set of an analytically finite non-elementary Kleinian group is known to be uniformly perfect.

In this note, we shall show that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., satisfies that the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. Indeed, we shall state the result in more precise and quantitative form. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation length in the region of discontinuity.

1. Introduction

In this note, we shall consider uniform perfectness of the limit sets of Kleinian groups. Once one know the limit set is uniformly perfect, the estimation of various quantities involving Kleinian groups becomes easier (see Section 5). Bishop and Jones effectively used this fact in their paper [4].

As soon as a proof of uniform perfectness of the limit sets of Schottky groups appeared in [2], this result was generalized to the case of finitely generated non-elementary Kleinian groups by several specialists (cf. [13] and [14]). Afterward, Canary remarked in [5] that the same result holds for analytically finite Kleinian groups. As for Schottky groups, we should mention the pioneer work of Tsuji [18] (see the comment on Theorem C below).

Recently, Järvi and Vuorinen [6] proved the same result for finitely generated Kleinian groups in higher dimensional case. This is a generalization of Tukia’s result [20] (geometrically finite case). It is noteworthy that their proof does not rely on Ahlfors’ Finiteness Theorem.

In this note, we will present a more general condition for the limit sets of Kleinian groups to be uniformly perfect (Corollary 3.3). Our method also provides a bound for uniform perfectness by some geometric quantity. In practice, it is important to know an explicit bound because the uniform perfectness connects with various quantities involving geometry of the quotient surface (cf. [16]). Indeed, we shall give some applications in Sections 5 and 6. In Section 5, we state results relating to regularity in the sense of Dirichlet and Hausdorff dimension of the limit sets. Section 6 is devoted to an estimate of translation length in terms of the multiplier of a loxodromic element of a Kleinian group.

Date: June 9, 1998.

1991 Mathematics Subject Classification. 30F40.

Key words and phrases. uniformly perfect, Kleinian group, translation length, Hausdorff dimension.

We shall conclude this note by giving several examples of Kleinian groups, one of which has the limit set which is not uniformly perfect.

2. Riemann orbifolds

A 1-dimensional complex orbifold is a Hausdorff topological space locally modeled on the quotient of an open subset of the complex plane ℂ under an action of finite group whose members consist of biholomorphic maps. Note that such a finite group is necessarily a cyclic group, thus determined by its order. By a Riemann orbifold, we will mean a connected 1-dimensional complex orbifold. Thus X can be regarded as a pair of a Riemann surface R and a multiplicity map ν : R → ℕ, where the value of ν corresponds to the order of the finite group, therefore the set of singularities (or, branch points)

b(X) = {x ∈ R; ν(x) > 1} should be discrete. We set X^{∘} = R \ b(R). Then X^{∘} can be regarded as a subdomain of X without singularities. If ν is the constant 1, then X can be naturally indentified with the underlying Riemann surface R. For a precise definition and fundamental properties of Riemann orbifolds, we refer the reader to McMullen’s book [10].

A Riemann orbifold X = (R, ν) is called hyperbolic if R has a holomorphic branched Galois covering map p : ℍ → R from the upper half plane ℍ onto R such that ord_{ζ}p = ν(p(ζ)) for any ζ ∈ ℍ, where ord_{ζ}p denotes the local degree of p at ζ. The map p is called a universal covering map of X.

The covering transformation group Γ = Γ_{X} = {γ ∈ Aut(ℍ); p ∘ γ = p} is called a Fuchsian model of X. Through the universal covering map, the Riemann orbifold X inherits the hyperbolic metric ρ_{X} from ℍ, i.e., ρ_{ℍ} = \frac{|dz|}{2Imz} = p^{∗}(ρ_{X}). We denote by d_{X} the hyperbolic distance on X canonically defined by the hyperbolic metric.

Let C_{X} denote the set of free homotopy classes [α] = [α]_{X} of non-trivial closed curves

α in X^{∘}, where a curve α is said to be non-trivial if it is covered by an element γ ∈ Γ of infinite order, more precisely, there exists a lift α̃ via p such that the initial point is translated to the terminal one by γ, and said to be freely homotopic to another α′ if both are covered by the same element of Γ. (The free homotopy class of a non-trivial curve precisely corresponds to the conjugacy class of an element of Γ which covers the curve and has infinite order.) And let C_{X}^{∗} be the subset of C_{X} consisting of free homotopy classes of curves which are covered by hyperbolic elements of Γ.

Remark. It may be more natural to consider all curves in X, but a curve passing a branch point of X cannot be lifted via p in a unique way even if the initial point is assigned, so being freely homotopic fails to be an equivalence relation. To avoid this difficulty, we may assign a sort of “multiplicity data” to the curve. Precisely speaking, we may consider a (regular) curve as the pair (γ, n_{γ}) of a C^{1} map γ : S^{1} → R with γ′(ζ) ≠ 0 and n_{γ} : S^{1} → ℤ such that 0 ≤ n_{γ}(ζ) ≤ ν(γ(ζ)) − 1, where n_{γ}(ζ) indicates the winding number of γ around γ(ζ). But, we will not adopt this formulation for the sake of simplicity.

We write ℓ_{X}(α) = ∫_{α} ρ_{X} and ℓ_{X}[α] = inf_{α′∈[α]} ℓ_{X}(α′). By definition, if γ ∈ Γ covers α, then

ℓ_{X}[α] = l_{γ} = cosh^{−1} (\frac{|trγ|}{2}) , (2.1)

where l_{γ} denotes the translation length of γ and trγ is the trace of a representative of γ in SL(2, ℝ).

Now we set

L(X) = inf_{[α]∈C_{X}} ℓ_{X}[α] = inf_{γ∈Γ_{X}: of infinite order} l_{γ} and
L^{∗}(X) = inf_{[α]∈C_{X}^{∗}} ℓ_{X}[α] = inf_{γ∈Γ_{X}: hyperbolic} l_{γ}.

We call X is modulated if L(X) > 0. We note here that if X is a Riemann surface R, the constant L(R) is nothing but 2I_{R}, where I_{R} is the injectivity radius of R, so we may say that R is of bounded geometry if L(R) > 0 (see [16]).

When the Fuchsian model Γ_{X} of X is finitely generated, the quantity L^{∗}(X) represents the hyperbolic length of the shortest closed geodesic in X. A closed geodesic shortest in the curves in X other than boundary curves is called systole of X. Hence, L^{∗}(X) can be thought as the length of systole of X if Γ is finitely generated and of the first kind. It has been recognized that the length of systole is quite an important function on the moduli space of X (cf. [15]).

Let A_{2}(X) and B_{2}(X) be the complex Banach spaces consisting of holomorphic quadratic differentials φ = φ(z)dz^{2} on X with norms

∥φ∥_{1} = ∫∫ _{X} |φ| = ∫∫ _{X} |φ(z)|dxdy,
∥φ∥_{∞} = sup ρ_{X}^{−2}|φ| = sup ρ_{X}^{−2}(z)|φ(z)|,

respectively. The spaces A_{2}(X) and B_{2}(X) are canonically isomorphic to the spaces A_{2}(ℍ, Γ) and B_{2}(ℍ, Γ) of integrable and bounded holomorphic automorphic forms on ℍ of weight −4 for Γ, respectively. And we set

κ(X) = sup{∥φ∥_{∞}; φ ∈ A_{2}(X) with ∥φ∥_{1} ≤ 1}.

For these spaces, the inclusion problem was first settled by Niebur and Sheingorn [12]. The following strong form is due to Matsuzaki [8].

Theorem 2.1. The space A_{2}(X) is (continuously) included by B_{2}(X) if and only if L^{∗}(X) > 0. Furthermore, there exist universal constants r_{0} and r_{1} such that

\frac{1}{2πL^{∗}(X^{∘})} ≤ κ(X) ≤ max\{\frac{r_{0}}{L^{∗}(X)}, r_{1}\}.

Note that the inclusion map X^{∘} ↪ X induces the restriction maps A_{2}(X) → A_{2}(X^{∘}) and

B_{2}(X) → B_{2}(X^{∘}), which are, respectively, an isometric isomorphism and a bounded linear operator with ∥φ∥_{B_{2}(X^{∘})} ≤ ∥φ∥_{B_{2}(X)} by virtue of the monotonicity of hyperbolic metrics:

ρ_{X} ≤ ρX^{∘}. Therefore, we have κ(X^{∘}) ≤ κ(X). In fact, κ(X^{∘}) ≤ κ(X) ≤ 3κ(X^{∘}) holds (see [17]).

In this article, we will say that X is of Lehner type if L^{∗}(X) > 0. Generally, for a (possibly disconnected) 1-dimensional complex orbifold X, we define L(X) = inf L(X_{0}) and L^{∗}(X) = inf L^{∗}(X_{0}), where the infima are taken over all connected components X_{0} of X, and we say that X is modulated or of Lehner type if L(X) > 0 or L^{∗}(X) > 0, respectively.

A closed set C in the Riemann sphere ℂ̂ with #C ≥ 3 is called uniformly perfect if there exists a constant c > 0 such that C ∩ {z; cr < |z − a| < r} ≠ ∅ for all a ∈ C and 0 < r < diamC. This is equivalent to the condition that the complement R = ℂ̂ \ C is modulated (see Section 5).

3. Kleinian groups and main results

Let G be a Kleinian group acting on the Riemann sphere ℂ̂, i.e., G is a discrete subgroup of PSL(2, ℂ) whose region of discontinuity Ω(G) in ℂ̂ is not empty. We denote by Λ(G) the limit set of G, i.e., Λ(G) = ℂ̂ \ Ω(G).

In the following, we always assume that G is non-elementary, in ohter words, #Λ(G) ≥ 3. Then Λ(G) is known to be a perfect set. The quotient space X(G) = Ω(G)/G has a natural hyperbolic 1-dimensional complex orbifold structure with which the canonical projection π : Ω(G) → X(G) is a holomorphic covering map.

Let X be a connected component of X(G) and Ω be a connected component of π^{−1}(X). If q : ℍ → Ω is a holomorphic universal covering map, then clearly p = π ∘ q is a holomorphic universal covering map of X. Let H = H_{Ω} be the component subgroup of G corresponding to Ω, i.e., H = Stab_{G}(Ω) = {g ∈ G; g(Ω) = Ω}. And denote by Γ and Γ̃ the covering transformation groups of q and p, respectively. Then, we have a natural exact sequence

1 ⟶ Γ ⟶ Γ̃ ⟶^{χ} H ⟶ 1

of group homomorphisms, here χ can be described as q ∘ γ = χ(γ) ∘ q for all γ ∈ Γ̃. For each h ∈ H, we denote by l_{h,Ω} the translation length of h in Ω :

l_{h,Ω} := inf_{z∈Ω} d_{Ω}(z, h(z)). (3.1)

Set

H_{e} = {h ∈ H; h has a fixed point in Ω} and
H_{p} = {h ∈ H; h corresponds to a puncture of X = Ω/H}

Precisely speaking, an element h of H belongs to H_{p} if and only if there exist an element h_{0} ∈ H of infinite order and a subdomain ω of Ω with the following properties:

1. h = h_{0}^{n} for a non-zero integer n,

2. h_{0}(ω) = ω and g(ω) ∩ ω = ∅ for all g ∈ H \ ⟨h_{0}⟩, and

3. ω/⟨h_{0}⟩ is conformally equivalent to the punctured disk.

We remark that there may exist an elliptic or parabolic element h of H with h ∉ H_{e} or h ∉ H_{p}, respectively (see Example 7.2 below). (Of course, an element of H_{e} \ {1} or H_{p} is necessarily elliptic or parabolic, respectively.) Now we define λ_{H,Ω}, λ_{H,Ω}^{∗} as follows:

λ_{H,Ω} = inf_{h∈H\H_{e}} l_{h,Ω}, and λ_{H,Ω}^{∗} = inf_{h∈H\(H_{e}∪H_{p})} l_{h,Ω}.

Noting that an elliptic or parabolic element in a Fuchsian group always represents a branch point or a puncture of the quotient surface, similarly we set

λΓ̃ = inf_{γ∈Γ̃\Γ̃_{e}} l_{γ} = inf_{Γ̃∋γ: of infinite order} l_{γ} and
λ∗ ; Γ̃ = inf_{γ∈Γ̃\(Γ̃_{e}∪Γ̃_{p})} l_{γ} = inf_{Γ̃∋γ: hyperbolic} l_{γ},

where l_{γ} is the translation length of γ in ℍ, i.e., l_{γ} = cosh^{−1}(|trγ|/2). We can define λ_{Γ} , λ_{Γ}^{∗} in the same manner, however λ_{Γ} = λ_{Γ}^{∗} holds since Λ(G) has no isolated points. Let N_{H,Ω} be the number defined by N_{H,Ω} = sup_{h∈H_{e}} ordh. And we define the number N_{H,Ω}^{∗} as follows: If H_{p} is non-empty, then N_{H,Ω}^{∗} = +∞, otherwise we set N_{H,Ω}^{∗} = N_{H,Ω}.

For these constants, the next result is fundamental for our present aim.

Lemma 3.1.

(3.2)  min\{\frac{1}{N_{H,Ω}} λ_{Γ} , λ_{H,Ω}\} ≤ λΓ̃ ≤ min\{λ_{Γ} , λ_{H,Ω}\}, and
(3.3)  min\{\frac{1}{N_{H,Ω}^{∗}} λ_{Γ} , λ_{H,Ω}^{∗}\} ≤ λ∗ ; Γ̃ ≤ min\{λ_{Γ} , λ_{H,Ω}^{∗}\}.

By the relation (2.1), we obtain that λ_{Γ} = L(Ω) and λΓ̃ = L(X) and the similar relations for λ^{∗} and L^{∗} hold, hence the above theorem is equivalent to the following:

(3.4)  min\{\frac{1}{N_{H,Ω}} L(Ω), λ_{H,Ω}\} ≤ L(X) ≤ min\{L(Ω), λ_{H,Ω}\}, and
(3.5)  min\{\frac{1}{N_{H,Ω}^{∗}} L(Ω), λ_{H,Ω}^{∗}\} ≤ L^{∗}(X) ≤ min\{L(Ω), λ_{H,Ω}^{∗}\}.

Now we define λ(G), λ^{∗}(G), N(G) and N^{∗}(G) by inf_{Ω} λ_{H,Ω}, inf_{Ω} λ_{H,Ω}^{∗}, sup_{Ω} N_{H,Ω} and sup_{Ω} N_{H,Ω}^{∗}, respectively, where Ω runs over all components of Ω(G) and H = H_{Ω}. Remark here that the constants λ_{H,Ω}, λ_{H,Ω}^{∗}, N_{H,Ω} and N_{H,Ω}^{∗} depend only on the conjugacy class of Ω under the action of G, i.e., on the component X = Ω/H of X(G). Then, we immediately obtain the following

Theorem 3.2. For a non-elementary Kleinian group G it follows that

min\{\frac{1}{N(G)}L(Ω(G)), λ(G)\} ≤ L(X(G)) ≤ min\{L(Ω(G)), λ(G)\}, and
min\{\frac{1}{N^{∗}(G)}L(Ω(G)), λ^{∗}(G)\} ≤ L^{∗}(X(G)) ≤ min\{L(Ω(G)), λ^{∗}(G)\}.

Corollary 3.3. Thus we have L(Ω(G)) ≥ L^{∗}(X(G)). In particular, if X(G) is of Lehner type then Λ(G) is uniformly perfect.

Corollary 3.4. For a torsion-free non-elementary Kleinian group G, the complex orbifold X(G) = Ω(G)/G satisfies

L(X(G)) = min{L(Ω(G)), λ(G)}.

The counterpart L^{∗}(X(G)) = min{L(Ω(G)), λ^{∗}(G)} does not hold in general (see Example 7.3 below).

Remark. To guarantee that λ(G) > 0 it is sufficient to assume that sup_{z∈Ω(G)} ι(z) < ∞, where ι(z) denotes the injectivity radius of Ω(G) at z (oral communication with Prof. Matsuzaki). But, this condition seems to be hard to check.

In the case that G is analytically finite, i.e., X(G) consists of a finite number of Riemann orbifolds of finite type, it is easily verified that L^{∗}(X(G)) > 0 thus we have the following collorary.

Corollary 3.5 (Canary [5]). For an analytically finite non-elementary Kleinian group G, the limit set Λ(G) is uniformly perfect.

We should remark that by Ahlfors’ Finiteness Theorem this result produces the finitely generated case.

4. Proof of Lemma 3.1

Under the situation of the lemma, first we prove the next elementary

Lemma 4.1. For an element h ∈ H, we have

inf_{γ∈χ^{−1}(h)} l_{γ} = l_{h,Ω}. (4.1)

Proof. Let γ ∈ χ^{−1}(h), then q ∘ γ = h ∘ q by definition. For a ζ ∈ ℍ we put z = q(ζ). Then, by the definition of metrics and the Schwarz-Pick lemma, we have

d_{ℍ}(ζ, γ(ζ)) ≥ d_{Ω}(q(ζ), q(γ(ζ))) = d_{Ω}(z, h(z)) ≥ l_{h,Ω}.

Since ζ is arbitrary, it follows that l_{γ} ≥ l_{h,Ω}.

Now we prove the reverse inequality. Take a point z in Ω, and let α be a geodesic arc joining z and h(z) in Ω such that d_{Ω}(z, h(z)) = ∫_{α} ρ_{Ω}. Choose a point ζ ∈ ℍ with q(ζ) = z, and let β be a lift of α via q with initial point ζ, then the terminal point of β can be written by γ(ζ) for some γ ∈ Γ̃. We note here that χ(γ) = h by definition. Therefore, we have

d_{Ω}(z, h(z)) = ∫_{α} ρ_{Ω} = ∫_{β} ρ_{ℍ} ≥ d_{ℍ}(ζ, γ(ζ)),

thus d_{Ω}(z, h(z)) ≥ l_{γ} ≥ inf_{γ∈χ^{−1}(h)} l_{γ}. Since z is arbitrary, we have l_{h,Ω} ≥ inf_{γ∈χ^{−1}(h)} l_{γ}. Now the proof is completed.

Now we prove Lemma 3.1. First, we note that if γ ∈ Γ̃ is elliptic, then h = χ(γ) ∈ H_{e} \ {1}. Note also that χ^{−1}(1) = Γ. Therefore we conclude that

(Γ \ {1}) ∪ χ^{−1}(H \ H_{e}) ⊂ Γ̃ \ Γ̃_{e},

and this and the above lemma immediately yield the right-hand side inequality in (3.2).

In order to get the left-hand side inequality, we consider an element γ of the residual part χ^{−1}(H_{e} \ {1}) \ Γ̃_{e}. Since h = χ(γ) is of finite order, say n, we have γ^{n} ∈ χ^{−1}(1) = Γ, thus nl_{γ} = l_{γ^{n}} ≥ λ_{Γ} . Hence l_{γ} ≥ λ_{Γ} /n ≥ λ_{Γ} /N_{H,Ω}. By this observation, we are convinced the validity of the left-hand side of (3.2). Noting that any parabolic element of Γ̃ is mapped to H_{p} by the homomorphism χ, we can show the inequality (3.3) in the same way as above.

5. Some consequences

In this section, we shall exhibit several applications of our theorems. We denote by M_{Ω(G)} (M_{Ω(G)}^{∘}) the supremum of the moduli of annuli (round annuli, respectively) separating Λ(G), where the modulus of an annulus is defined here as the number m when this annulus is conformally equivalent to the round annulus {z ∈ ℂ, 1 < |z| < e^{m}} and the round annulus means a bounded annulus with boundary consisting of concentric circles, and we say that an annulus A separates Λ(G) if A ∩ Λ(G) = ∅ and if both components of ℂ̂ \ A intersects Λ(G). If ∞ ∈ Λ(G), then we can define another constant C_{Ω(G)} by inf_{z∈Ω(G)} δ(z)ρ_{Ω(G)}(z), where δ(z) denotes the Euclidean distance from z to Λ(G). Here it should be noted that δ(z)ρ_{Ω(G)}(z) ≤ 1 is always true. For these constants, we know the following estimates.

Theorem A (cf. [16]). For a non-elementary Kleinian group G, we obtain

L ≤ \frac{π^{2}}{M_{Ω(G)}} ≤ min\{Le^{L}, \frac{1}{2}L^{2} coth^{2}(L/2)\}, and

\frac{1}{2}M_{Ω(G)} − K_{0} ≤ M_{Ω(G)}^{∘} ≤ M_{Ω(G)},

where L = L(Ω(G)) and K_{0} is an absolute constant ≤ 1.7332 …. Moreover if ∞ ∈ Λ(G), we also have

M_{Ω(G)} − K_{1} ≤ M_{Ω(G)}^{∘} ≤ M_{Ω(G)}, and

\frac{tanh L/2}{4} ≤ C_{Ω(G)} ≤ \frac{\sqrt{3}L}{\sqrt{π^{2} + 4L^{2}}} , (5.1)

where K_{1} is an absolute constant ≤ 2.8911 ….

In particular, L(Ω(G)) > 0 if and only if M_{Ω(G)} < ∞. The first inequality in the above partly follows from the next result, which is an improvement of Maskit’s one [7] and will be used later.

Theorem B (cf. [16]). Let R be a hyperbolic Riemann surface. For the free homotopy class [α] of a non-trivial loop α in R, we have the following estimate.

ℓ_{R}[α] ≤ \frac{π}{2} E_{R}[α] ≤ ℓ_{R}[α]e^{ℓ_{R}[α]}.

In the above, E_{R}[α] denotes the extremal length of the curve family [α], more precisely, E_{R}[α] = sup_{σ}(∫∫_{R} σ(z)^{2}dxdy)^{−1}, where the supremum is taken over all Borel measurable conformal metrics σ satisfying that ∫_{α′} σ(z)|dz| ≥ 1 for any α′ ∈ [α] (such a metric σ is called admissible for [α]).

Furthermore, Pommerenke has given a remarkable characterization of uniform perfectness in terms of capacity density.

Theorem C (Pommerenke [13]). Λ(G) is uniformly perfect if and only if there exists a constant c ∈ (0, 1] such that Cap(Λ(G) ∩ B(a, r)) ≥ cr for any a ∈ Λ(G) and 0 < r < diam(Λ(G)), where Cap denotes the logarithmic capacity and B(a, r) the closed disk centered at a with radius r.

Here we mention the work of Tsuji. He proved in [18] that any point of the limit set of a non-elementary (finitely generated) Schottky group has a positive capacity density (with a uniform bound). In view of the above theorem, this is a substantial proof of uniform perfectness of it. Earlier than this, Myrberg [11] showed that any non-elementary Kleinian group has the limit set of positive capacity.

We also note that Cap(B(a, r)) = r and 2^{−7}e^{−M_{Ω(G)}^{∘}} can be taken as the constant c in the above statement (see [16]). In particular, by virtue of Wiener’s criterion, we have the following

Corollary 5.1. If L(Ω(G)) > 0 the limit set Λ(G) is regular in the sense of Dirichlet.

For general Kleinian groups, at least we can state the following

Corollary 5.2. Let G be a non-elementary Kleinian group. Any loxodromic or parabolic fixed point of G is a regular point of Λ(G) in the sense of Dirichlet.

In fact, if z_{0} is a fixed point of a loxodromic or parabolic element γ of G, then γ is contained in a finitely generated non-elementary subgroup G_{0} of G. Since Λ(G_{0}) is uniformly perfect, by Theorem C, we see that \overline{lim}_{r→0}Cap(Λ(G) ∩ B(z_{0}, r))/r ≥ \overline{lim}_{r→0}Cap(Λ(G_{0}) ∩ B(z_{0}, r))/r > 0, which implies the regularity of Λ(G) at z_{0} (see, for example, Tsuji [19] p.104). Further, we should note that the set of loxodromic fixed points of a non-elementary Kleinian group is dense in the limit set.

Another application of unifom perfectness is concerned with the Hausdorff dimension. This sort of result is essentially due to Järvi-Vuorinen [6]. The following quantitative form follows from a result in [16].

Theorem D.The Hausdorff dimension H-dim(Λ(G)) of Λ(G) can be estimated from below as follows.

H-dim(Λ(G)) ≥ \frac{log 2}{log(2e^{M_{Ω(G)}^{∘}} + 1)} (≥ \frac{log 2}{M_{Ω(G)}^{∘} + log 3}) .

As an immediate consequence of this, we can see that any non-elementary Kleinian group has the limit set of positive Hausdorff dimemsion.

6. Estimate of translation length

As an application of (5.1), we present here an estimate of the translation length of a loxodromic element of Kleinian group in the region of discontinuity in terms of the trace or the multiplier. Before stating our result, we refer to a general result on translation length, which is suggested to the author by K. Matsuzaki. The original idea is due to Bers [3]. See also the proof of Proposition 6.4 in [9].

Let G be a non-elementary Kleinian group and H a component subgroup of G corresponding to a component Ω of the region of discontinuity Ω(G). Let h be a loxodromic element of H, i.e., η = tr^{2}(h) ∈ ℂ\[0, 4]. By the Möbius invariance of l_{h,Ω} and tr^{2}(h), we may assume that h has the form h(z) = λz with |λ| > 1 where η = (\sqrt{λ}+\sqrt{λ}^{−1})^{2} = λ+λ^{−1}+2. We note that 0, ∞ ∈ Λ(G) under this assumption. For an arbitrary z_{0} ∈ Ω, let α be a geodesic arc joining z_{0} and h(z_{0}) in Ω such that d_{Ω}(z_{0}, h(z_{0})) = ℓ_{Ω}(α). Without loss of generality, we may further assume that z_{0} = 1. Now we consider the quotient map p : ℂ^{∗} → ℂ^{∗}/⟨h⟩ =: T, and set R = p(Ω) = Ω/⟨h⟩ ⊂ T. Then β = p_{∗}α = p(α) is a closed geodesic in R, where R is endowed with the hyperbolic metric. We note here that ℓ_{Ω}(α) = ℓ_{R}(β). In view of Theorem B, we have \frac{π}{2}E_{R}[β] ≤ ℓ_{R}[β]e^{ℓ_{R}[β]}. Since [β] = [β]_{R} ⊂ [β]_{T} , by the monotonicity of extremal length (cf. [1]), it turns out that E_{R}[β] ≥ E_{T} [β]_{T} . Thus it is sufficient to compute E_{T} [β]_{T} .

The function q(z) = exp(2πiz) is a universal covering map of ℂ^{∗} from the complex plane ℂ. Let α̃ be a lift of α via q with initial point 0 and τ denote the terminal point of α̃. In particular, exp(2πiτ) = λ, so we may write τ = \frac{1}{2πi} log λ. Further note that p ∘ q : ℂ → T is a quotient map of ℂ by the lattice generated by 1 and τ over ℤ. And it follows that any curve in [β]_{T} is lifted to an arc with initial point a in [0, 1] and terminal point a + τ, and vice versa. Thus, by the standard length-area method (cf. [1]), one can easily see that the extremal admissible metric for [β]_{T} is given by the projection σ of |τ|^{−1}|dz| under p ∘ q, hence E_{T} [β]_{T} = (∫∫_{T} σ^{2})^{−1} = (|τ|^{−2} · |Imτ|)^{−1} = |τ|^{2}/|Imτ| = |Im(1/τ)|^{−1}. Thus we have

ℓ_{R}[β]e^{ℓ_{R}[β]} ≥ \frac{π|τ|^{2}}{2|Imτ|} = \frac{| log λ|^{2}}{4 log |λ|}.

The quantity log λ is sometimes called the complex length of h (cf. [9]). Here, denote by Logz the principal branch of log z, i.e., Logz is the branch of the logarithm determined by −π < ImLogz ≤ π. Since | log λ| ≥ |Logλ| and z_{0} is arbitrary, we have the following

Proposition 6.1. Let G be a non-elementary Kleinian group and H its component subgroup which corresponds to a component Ω of Ω(G). For any loxodromic element h of H with multiplier λ we have an estimate of the translation length of h in Ω as in the following:

l_{h,Ω}e^{l_{h,Ω}} ≥ \frac{|Logλ|^{2}}{4 log |λ|} (≥ log |λ|/ 4).

The left-hand side of this inequality is of exponential order, but in general it seems to be difficult to improve this order. But, if the limit set Λ(G) is uniformly perfect, we have an estimate of linear order.

Theorem 6.2. Let H be a component subgroup of a non-elementary Kleinian group G corresponding to a component Ω of Ω(G), and suppose that C = \frac{1}{4} tanh(L(Ω)/2) > 0. Then, for a loxodromic element h ∈ H, the translation length l_{h,Ω} can be estimated as

l_{h,Ω} ≥ C|Logλ|,

where λ is the multiplier of h, i.e., tr^{2}(h) = λ + λ^{−1} + 2.

Remarks. Noting that C ≥ \frac{1}{4} tanh(L(Ω(G))/2), we can see C > 0 if Λ(G) is uniformly perfect.

In case Ω is simply connected, L(Ω) = ∞ by definition, so we have l_{h,Ω} ≥ |Logλ|/4. We further remark that, in general, we cannot estimate the translation length from

above by the trace or multiplier. This can be understood by the existence of accidental parabolic transformations.

Proof. We shall prove the theorem under the exactly same normalizations and notation in the previous sentences. If we denote by δ(z) the distance from z to Λ(G), by (5.1), we have δ(z)ρ_{Ω}(z) ≥ C for z ∈ Ω and δ(z) ≤ |z| because 0 ∈ Λ(G). Then, we compute

d_{Ω}(z_{0}, h(z_{0})) = ∫_{α} ρ_{Ω}(z)|dz| ≥ C ∫_{α} \frac{|dz|}{δ(z)} ≥ C ∫_{α} \frac{|dz|}{|z|}
= C ∫α̃ |dζ| ≥ C |∫α̃ dζ| = C| log λ| ≥ C|Logλ|.

Now the theorem is proved.

7. Examples

In this section, we present simple examples of Kleinian groups of Schottky type. The first construction provides an example of an infinitely generated Kleinian group whose boundary is not uniformly perfect. Looking at Theorem 3.2, one may guess that it is sufficient to construct a Kleinian group whose quotient orbifold has arbitrarily short geodesics which are lifted to closed curves in the region of discontinuity. In fact, such an example can be given by infinitely generated Schottky groups as Pommerenke indicated in [13].

Example 7.1. Let a_{j}, b_{j} ∈ ℂ be sequences tending to ∞, and r_{j} > 0 and α_{j} ∈ ℂ with |α_{j}| = 1 be given so that all closed disks A_{j} = B(a_{j}, r_{j}), B_{j} = B(b_{j}, r_{j}) are disjoint (j = 1, 2, …). We set g_{j}(z) = b_{j} − \frac{α_{j}r_{j}^{2}}{z−a_{j}} , then A_{j} and B_{j} are the isometric circles for g_{j} and g_{j}^{−1}, respectively, thus G = ⟨g_{1}, g_{2}, … ⟩ is an infinitely generated Schottky group with a fundamental domain ℂ \ ∪_{j}(A_{j} ∪ B_{j}) ⊂ Ω(G).

Now we set r̃_{j} = dist(a_{j}, (∪_{k≠j}A_{k}) ∪ (∪_{k}B_{k})) > r_{j}. Then, we can directly see that M_{Ω(G)}^{∘} ≥ sup_{j} r̃_{j}/r_{j}, hence Λ(G) is not uniformly perfect if sup_{j} r̃_{j}/r_{j} = ∞.

The second construction serves an infinitely generated Kleinian group G which contains a parabolic element h which does not represent any puncture of Ω(G)/G.

Example 7.2. Let h be the transformation z ↦ z + 2i. For j ∈ ℤ, we take a_{j}, b_{j} ∈ ℝ and 0 < r_{j} < 1 in such a way that b_{j}−a_{j} > 2r_{j} and a_{j+1}−b_{j} > r_{j}+r_{j+1} and lim_{j→±∞} a_{j} = ±∞. Taking a sequence α_{j} in the unit circle, we set g_{j}(z) = b_{j} − \frac{α_{j}r_{j}^{2}}{z−a_{j}} . Then the domain ω = {z ∈ ℂ; |Imz| < 1}\∪_{j}(A_{j}∪B_{j}) is a fundamental domain of the Kleinian group G with free generators h, g_{j}(j ∈ ℤ), where A_{j} = B(a_{j}, r_{j}) and B_{j} = B(b_{j}, r_{j}). In particular, one can observe that the Riemann surface Ω(G)/G has no punctures. Let β_{j} : [−1, 1] → Ω(G) be the curve given by β_{j}(t) = ti + (a_{j} + b_{j})/2. Since δ(β_{j}(t)) ≥ (b_{j} − a_{j})/2 − r_{j}, we have

ℓ_{Ω(G)}(β_{j}) ≤ ∫_{β_{j}} \frac{|dz|}{δ(z)} ≤ \frac{4}{b_{j} − a_{j} − 2r_{j}} ,

where δ(z) denotes the Euclidean distance from z to Λ(G). Since h(β_{j}(−1)) = β_{j}(1), we see that l_{h,Ω(G)} ≤ inf_{j∈ℤ} ℓ_{Ω(G)}(β_{j}). Therefore, if sup_{j}(b_{j} −a_{j}) = ∞, then we obtain l_{h,Ω(G)} = 0. On the other hand, if r_{j} = r_{0}, α_{j} = α_{0}, a_{j} = a_{0} + 2j(b_{0} − a_{0}) and b_{j} = b_{0} + 2j(b_{0} − a_{0}) for all j ∈ ℤ, then G is a normal subgroup of the Kleinian group G̃ = ⟨h, h^{∗}, g_{0}⟩, where h^{∗}(z) = z + 2(b_{0} − a_{0}). The property G ◁ G̃ implies Ω(G̃) = Ω(G). Since Y = Ω(G)/G̃ is a compact Riemann surface of genus 2, thus L(Y) > 0, we have λ(G) ≥ L(X(G)) ≥ L(Y) > 0. In particular, l_{h,Ω} > 0 in this case.

Finally we give a family of finitely generated torsion-free Kleinian groups which shows that the counterpart for L^{∗} in Corollary 3.4 does not hold.

Example 7.3. Let 0 < r < 1 and α ∈ ℂ with |α| = 1 be fixed. Set h(z) = z + 2i and g_{t}(z) = 2t − αr^{2}/z, where t > 1 is a real parameter. Then G_{t} = ⟨h, g_{t}⟩ is a free Kleinian group with fundamental domain ω = ω̂ \ (B ∪ B′), where ω̂ = {z ∈ ℂ; |Imz| < 1}, B = {|z| ≤ r} and B′ = {|z − 2t| ≤ r}. Since h represents a pair of punctures of the quotient surface, the element h is contained in G_{p}. We now prove the following claim: There exist constants K < +∞ and c > 0 depending only on r such that

c ≤ min{L(Ω(G_{t})), λ^{∗}(G_{t})} and (7.1)

L^{∗}(X(G_{t})) ≤ \frac{K}{t} . (7.2)

Let us write g = g_{t}, G = G_{t} and Ω = Ω(G_{t}). We begin with a few preliminary observations. We set B_{n} = h^{n}(B) and B′_{n} = h^{n}(B′) for n ∈ ℤ. Then we know that Λ(G) ∩ ℂ ⊂ ω̃ := ℂ \ ∪_{n∈ℤ}(B_{n} ∪ B′_{n}). Since ∞ ∈ Λ(G), it follows that 0 = g^{−1}(∞) and 2t = g(∞) belong to Λ(G). For later use, we estimate the quantity d = sup{|z|; z ∈ B ∩ Λ(G)}. First we note

d = sup_{w∈Λ(G)\B′} |g^{−1}(w)| = sup_{w∈Λ(G)\B′} \frac{r^{2}}{|w − 2t|} = r^{2}/dist(2t, Λ(G) \ B′).

Since 2 − r ≤ dist(2t, Λ(G) \ B′) ≤ 2, we obtain r^{2}/2 ≤ d ≤ r^{2}/(2 − r) < r^{2}. In particular, we see

dist(∂B, B ∩ Λ(G)) ≥ 1 − r^{2}. (7.3)

Noting g^{−1}(2t ± 2i) = ±αr^{2}/2 ∈ Λ(G), we also have diamΛ_{0} ≥ r^{2}.

For simplicity, we further assume that 0 < r ≤ 1/4 in the sequel. First we prove L(Ω) ≥ c_{1} > 0. To see this, it suffices to show that M_{Ω}^{∘} is uniformly bounded by Theorem A. Let A be an arbitrary round annulus in Ω which separates Λ(G). Set ρ = exp(−m(A)), where m(A) denotes the modulus of A. We note here the following elementary lemma.

Lemma 7.1. For the round annuli A = \{ρ < |z| < 1\}, we set A_{0} = \{ρ\frac{3+ρ}{1+3ρ} < |z| < \frac{1+3ρ}{3+ρ}\}. Then, if a Möbius transformation T maps A into ℂ, we can take a round annulus A′ in T (A) in such a way that T (A_{0}) ⊂ A′. In particular, m(A′) ≥ log 1/ρ + 2 log \frac{1+3ρ}{3+ρ} ≥ m(A) − 2 log 3.

Now we take the annulus A_{0} in A with the property similar to the above. And choose an element f ∈ G such that f(A_{0}) ∩ ω ≠ ∅. Then f(A) contains a round annulus A′ = {z; r_{0} < |z − a| < r_{1}} with f(A_{0}) ⊂ A′. We note here that log r_{1}/r_{0} = m(A′) ≥ m(A_{0}) ≥ m(A) − 2 log 3. Now we estimate r_{1}/r_{0} from above. Since A′ separates Λ(G), we have r_{1} − r_{0} ≤ 2, hence r_{1}/r_{0} ≤ 1 + 2/r_{0}. By construction, the closed disk E = {z; |z − a| ≤ r_{0}} intersects Λ_{n} = B_{n} ∩ Λ(G) or Λ′_{n} = B′_{n} ∩ Λ(G) for some n ∈ ℤ. Conjugating by h^{n}, we may assume that E intersects Λ_{0} or Λ′_{0}. Assume that E ∩ Λ_{0} ≠ ∅. (The other case can be treated similarly, so we omit it.) We further divide the case into three parts.

1. In the case E ∩Λ(G)\Λ_{0} ≠ ∅, we have 2r_{0} ≥ 2−2r^{2}, thus r_{1}/r_{0} ≤ 1+2/(1−r^{2}) < 5.

2. In the case E ∩ Λ(G) = Λ_{0}, we have 2r_{0} ≥ diamΛ_{0} ≥ r^{2}, thus r_{1}/r_{0} ≤ 1 + 2/r^{2}.

3. In the case E∩Λ(G) ⊊ Λ_{0}, the annulus A′ separates Λ_{0}, therefore r_{1}−r_{0} ≤ diamΛ_{0} ≤ 2r^{2}. On the other hand, by assumption, we can take points z ∈ ω ∩ A′ and w ∈ E∩Λ(G). Then r_{1}+r_{0} ≥ |z−w| > dist(∂B, B∩Λ(G)) ≥ 1−r^{2}. By these inequalities, it follows that r_{1}/r_{0} ≤ (1 + r^{2})/(1 − 3r^{2}) ≤ 5.

In any case, e^{m(A)} − 2 log 3 ≤ r_{1}/r_{0} ≤ 1 + 2/r^{2} holds provided that r ≤ 1/4. Hence we have M_{Ω}^{∘} ≤ log(1 + 2/r^{2}) + 2 log 3, which implies L(Ω) ≥ c_{1} = c_{1}(r) > 0.

In particular, by Theorem A, δ(z)ρ_{Ω}(z) ≥ C_{1} holds for z ∈ Ω, where δ(z) = dist(z, ∂Ω) and C_{1} is a positive constant depending only on r. Using this fact, we next show λ^{∗}(G) ≥ c_{2} > 0. First we note the following estimate:

λ^{∗}(G) = inf_{f∈G\(G_{e}∪G_{p})} inf_{z∈ω} d_{Ω}(z, f(z)) ≥ inf_{f∈G\⟨h⟩} inf_{z∈ω} d_{Ω}(z, f(z)).

For f = g^{±1}, we can use Theorem 6.2 to show

d_{Ω}(z, f(z)) ≥ l_{g,Ω} ≥ C_{1} log |λ|,

where λ is the multiplier of g with |λ| ≥ 1. By the relation λ + λ^{−1} = 4t^{2}/αr^{2} − 2, we have |λ| ≥ 4t^{2}/r^{2} − 3 > 1.

Now suppose that f ∈ G \ ⟨h⟩ is not g^{±1}. Let z_{0} be a point in ω. If f(z_{0}) ∈ B then f(ω̂) ⊂ B, equivalently, f^{−1}(ℂ̂ \ B) ⊂ ℂ̂ \ ω̂, which implies f^{−1}(z_{0}) ∈ ℂ \ ω̂. When f(z_{0}) ∈ B′, the same result holds. Since f ∉ ⟨h⟩, we actually see f^{−1}(z_{0}) ∈ ∪_{n∈ℤ\{0}}(B_{n} ∪ B′_{n}) in these cases. Thus, without loss of generality, we may assume that z_{0} belongs to B_{n} or B′_{n} for some non-zero integer n because d_{Ω}(z_{0}, f(z_{0})) = d_{Ω}(z_{0}, f^{−1}(z_{0})). Let α be a geodesic arc joining z_{0} and f(z_{0}) in Ω such that ∫_{α} ρ_{Ω} = d_{Ω}(z_{0}, f(z_{0})). Since ω_{n} := h^{n}(ω) separates f(z_{0}) from z_{0}, a component α_{0} of α ∩ ω_{n} connects a bounded boundary component of ω_{n} with an unbounded one. We now estimate the hyperbolic length of α_{0} in Ω from below. By translation invariance under h and symmetry of Ω, we may assume that α_{0} connects ∂B with {z; Imz = 1}.

Letting ζ_{0} and ζ_{1} be endpoints of α_{0} with |ζ_{0}| = r and Imζ_{1} = 1, we can estimate as follows:

∫_{α} ρ_{Ω}(z)|dz| ≥ ∫_{α_{0}} ρ_{Ω}(z)|dz| ≥ C_{1} ∫_{α_{0}} \frac{|dz|}{δ(z)} ≥ C_{1} ∫_{α_{0}} \frac{|dz|}{|z|} ≥ C_{1} log \frac{1}{r}.

Here, we used the fact δ(z) ≤ |z|. Consequently, we obtain λ^{∗}(G) ≥ c_{2}, where c_{2} = C_{1} min{log(4t^{2}/r^{2} − 3), log 1/r}. Whence, the inequality (7.1) is now shown.

Finally, we show the inequality (7.2). Let β : [−1, 1] → Ω be the curve given by β(s) = 1 + si. Then the image of β in X(G) is a non-trivial closed curve which is not homotopic to any puncture. Therefore, by the method same as in Example 7.2, we have

L^{∗}(X(G)) ≤ ℓ_{Ω}(β) ≤ \frac{2}{t − r} ≤ \frac{K}{t} ,

where K = 2/(1 − r).

Remark. Further, in the same fashion as in Example 7.2, we can construct an infinitely generated free Kleinian group G with the property that min{L(Ω(G)), λ^{∗}(G)} > 0 while L^{∗}(X(G)) = 0.

Acknowledgement. The author wishes to express his gratitude to Professor Katsuhiko Matsuzaki for telling him the main idea of Proposition 6.1.

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Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

  E-mail address: sugawa@kusm.kyoto-u.ac.jp