ON THE SPACE OF SCHLICHT PROJECTIVE STRUCTURES

ON COMPACT RIEMANN

SURFACES WITH BOUNDARY

1994/12

Toshiyuki Sugawa

Abstract. In the present paper, we shall establish that IntS(Γ) = T (Γ) for any Fuchsian group uniformizing a compact Riemann surface with nonempty boundary, i.e., for any finitely generated, purely hyperbolic Fuchsian group of the second kind, where S(Γ) denotes the Schwarzian derivative of all the Γ -equivariant schlicht holomorphic functions and T (Γ) is the Teichmüller space of Γ. We also include some results concerning with IntS(Γ) for general Fuchsian groups Γ.

§1. Introduction

Let Γ be an arbitrary Fuchsian group acting on the upper half plane ℍ = {z ∈ ℂ; Imz > 0}. We donote by S(Γ) the set consisting of the Schwarzian derivative S_{f} of all the univalent meromorphic functions f on ℍ with f ∘ γ = χ(γ) ∘ f on ℍ for some group homomorphism χ : Γ → Möb. Then it turns out that S(Γ) is a bounded closed subset of the complex Banach space B_{2}(ℍ, Γ) (see §2 for its precise definition). It is an intersting topic to investigate how (the Bers model of) the Teichmüller space T (Γ) is embedded in S(Γ). Generally, \overline{T (Γ)} ⫋ S(Γ) holds. In fact, first Gehring has shown that \overline{T (1)} ⫋ S(1) in [7], and later the author proved in [14] that \overline{T (Γ)} ⫋ S(Γ) for any Fuchsian group Γ of the second kind. Moreover, recently K. Matsuzaki showed in [9] the existence of certain infinitely generated Fuchsian groups Γ of the first kind such that \overline{T (Γ)} ⫋ S(Γ). But, it is a still now difficult problem to decide whether \overline{T (Γ)} = S(Γ) for a finitely generated Fuchsian group Γ of the first kind. (We remark that this problem is equivalent to the Thurston conjecture: any b-group is a boundary group of the Teichmüller spaces.)

On the other hand, Gehring has shown in [6] that IntS(1) = T (1). Furthermore Žuravlev showed in [17] that T (Γ) is the zero component of IntS(Γ) for an arbitrary Fuchsian group Γ. Thus, it is naturally conjectured that IntS(Γ) = T (Γ) for any Γ. In this direction, Shiga proved in [13] that the above conjecture holds if Γ is finitely generated Fuchsian group of the first kind, equivalently, if B_{2}(ℍ, Γ) is finite dimensional.

The main theorem in this article (Theorem 2.1) is the claim that IntS(Γ) = T (Γ) for any Fuchsian group Γ uniformizing a compact (bordered) Riemann surface with nonempty boundary, in other words, for finitely generated, purely hyperbolic Fuchsian group Γ of the second kind. In order to prove this theorem, we shall

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utilize Gehring’s method in [6] with several localization techniques for overcoming difficulties caused by the group action. Here we remark that our proof does not depend on Žuravlev’s result.

The proof of the main theorem divides into several steps as follows. In §2, we prepare terminologies and notations for later use, and state the main theorem and some lemmas. Let φ ∈ IntS(Γ) and f be a univalent function such that S_{f} = φ. To say that φ ∈ T (Γ), we have to show essentially that D = f(ℍ) is a quasidisk, or equivalently, D is a locally connected John domain (if D is bounded). In §3, we will show that D is locally connected, but “locally” in Ω(G) where G = f^{−1}Γ f, Γ is an arbitrary Fuchsian group (Proposition 3.1). Roughly speaking, in the “island” D, there is no very deep bay. In fact, if such a deep bay exists, one can construct a G-equivariant meromorphic map h on D with small Schwarzian which shuts its inlet (thus, is not univalent) by bending D a little, and this will leads to a contradiction. As a corollary of this result, we see that ∂D = ∂(ℂ̂ \ \overline{D}), in particular, ℂ̂ \ \overline{D} ≠ ∅, for any φ ∈ IntS(Γ).

In §4, we also see that D is a John domain, at least “locally” in Ω(G) (Proposition 4.1). Roughly speaking again, there is no peninsula so much constricted in the island D. In fact, if such a peninsula, one can construct a G-equivariant meromorphic map h on D with small Schwarzian which touches the opposite shore of D by lengthening a narrow part of the peninsula, and this also will lieads to a contradiction. In both steps, we shall accomplish the construction of g as follows: first, we construct a G-equivariant quasi-analytic (in fact, quasiconformal locally, but not necessarily injective) map h with small deformation which satisfies the same properties as g except the holomorphy. By appropriate construction of h, the Beltrami coefficient μ of h^{−1} can be well-defined, so we can choose w^{μ} ∘ h as g, where w^{μ} is a μ-qc map of ℂ̂ (here, for example, μ was extended to 0 in h(D)^{c}). For estimation of the norm of the Schwarzian derivative of w^{μ} ∘ h, we shall utilize the “local norm technique” as in [16].

In §5, for a Fuchsian group uniformizing a compact bordered Riemann surface with nonempty boundary, we prove that the boundary of D/G in Ω(G)/G is a disjoint union of quasi-analytic curves by invoking the annular covering argument. Thus, in particular, the induced conformal map F : ℍ/Γ → D/G by f can be naturally extended to a homeomorphism \overline{ℍ/Γ} → \overline{D/G}. Furthermore, in §6, F turns out to be extended to a quasiconformal map F̃ : Ω(Γ)/Γ → Ω(G)/G which can be lifted to a quasiconformal map f̃: Ω(Γ) → Ω(G). This fact follows essentially from the existence of a G-equivariant quasiconformal reflection with respect to ∂D. Since f̃ may be continued to a quasiconformal self-map of ℂ̂, it has shown that φ = S_{f} ∈ T (Γ).

Finally, the author would like to express his sincere gratitude to Prof. M. Taniguchi for encouragement and worthy advices.

§2. Preliminaries and the main theorem

In this section, we shall fix the terminologies needed below and state related facts and the main theorem. As a general reference, we refer to the textbook [10] by S. Nag.

Projective structures on a Riemann surface. Let R be a hyperbolic Riemann surface and p : ℍ → R be a holomorphic universal covering of R, where ℍ = {z ∈ ℂ; Imz > 0} is the upper half plane. A projective chart on the Riemann surface R is a complex chart on R such that the transition functions are (locally) restrictions of Möbius transformations. Two projective charts on R are equivalent if their union is also a projective chart. Equivalence classes of projective charts on R are called projective structures on R.

Let σ be a projective structure on R represented by a chart {ψ_{α} : U_{α} → V_{α}; α ∈ A}. Set Ũ_{α} := p^{−1}(U_{α}), and write ψ̃_{α} = ψ_{α} ∘ h on Ũ_{α} for α ∈ A. Then (Ũ_{α})_{α∈A} is an open covering of ℍ. Define ψ_{αβ} = ψ_{α} ∘ ψ_{β}^{−1} on ψ_{β}(U_{α} ∩ U_{β}), then ψ_{αβ} is a restriction of Möbius transformation on each component of ψ_{β}(U_{α} ∩ U_{β}), by the very hypothesis. And we have

(2.1)  ψ̃_{α} = ψ_{αβ} ∘ ψ̃_{β} on Ũ_{α} ∩ Ũ_{β}

for any α, β ∈ A.

Here, we recall some of the properties of the Schwarzian derivative. The Schwarzian derivative S_{f} of a non-constant meromorphic function f on a plane domain is defined by

S_{f} = (\frac{f′′}{f′} )′ − \frac{1}{2} (\frac{f′′}{f′} )^{2} .

S_{f} is holomorphic at a point if and only if f is locally schlicht (=univalent) at the point. And, S_{f} = 0 on a domain D ⊂ ℂ̂ if and only if f is a restriction of a Möbius transformation. Further, if f and g are meromorphic functions and if f ∘ g is defined, then the following very important formula (the Cayley identity) holds:

(2.2)  S_{f∘g} = (S_{f}) ∘ g · (g′)^{2} + S_{g}.

By the above properties and (2.1), we obtain that

Sψ̃_{α} = Sψ̃_{β} on Ũ_{α} ∩ Ũ_{β}

for any α, β ∈ A. Thus a holomorphic function φ : ℍ → ℂ is well-defined by

φ = Sψ̃_{α} on Ũ_{α}

for any α. Moreover, by the relation (2.2), we can see that the holomorphic function φ satisfies the following functional equations:

(φ ∘ γ) · (γ′)^{2} = φ

for all γ ∈ Γ, where Γ < Möb is the covering transformation group of p : ℍ → R. The above φ is called a holomorphic quadratic differential for Γ on ℍ. We will denote by Q(ℍ, Γ) the set of all the holomorphic quadratic differentials for Γ on ℍ.

Conversely, let a holomorphic quadratic differential φ for Γ on ℍ be given. Consider the following homogeneous linear ordinary differential equation:

(2.3)  y′′ + \frac{1}{2}φy = 0 on ℍ.

Since ℍ is simply connected, there exists a pair of fundamental solutions (y_{0}, y_{1}) on ℍ uniquely determined by the initial condition

(2.4)  y_{0}(i) = 0, y′_{0}(i) = 1 ; y_{1}(i) = 1, y′_{1}(i) = 0.

Noting that y′_{0}y_{1} − y_{0}y′_{1} ≡ 1, we obtain that f^{φ} := y_{0}/y_{1} satisfies the following conditions:

(2.5)  S_{f^{φ}} = φ on ℍ,

(2.6)  f^{φ}(z) = (z − i) + O(|z − i|^{3}) as z → i = \sqrt{−1}.

It should be remarked that f^{φ} : ℍ → ℂ̂ is uniquely determined by the above conditions (2.5) and (2.6).

Let γ ∈ Γ be represented by γ(z) = \frac{az+b}{cz+d} for some (_{c}^{a}_{d}^{b}) ∈ SL(2, ℂ). Conventionally, we write (γ′)^{−1/2} = cz + d, then ỹ_{j} = (y_{j} ∘ γ)(γ′)^{−1/2} becomes a solution of (2.3) again. (ỹ_{j} may be considered as the analytic continuation of the solution y_{j} along with a path from i to γ(i).) Therefore ỹ_{0} and ỹ_{1} are uniquely represented by linear combinations of y_{0} and y_{1} as

(2.7)  \begin{aligned}ỹ_{0} & = Ay_{0} + By_{1} \\ ỹ_{1} & = Cy_{0} + Dy_{1}, \end{aligned}

where A, B, C and D are constants. Since y′_{0}ỹ_{1} − ỹ_{0}y′_{1} ≡ 1, (_{C}^{A}_{D}^{B}) ∈ SL(2, ℂ).

We denote by χ^{φ}(γ) the Möbius transformation \frac{Az+B}{Cz+D} , which is independent of the choice of signature of (γ′)^{−1/2}. The group homomorphism χ^{φ} : Γ → Möb is called a holonomy homomorphism associated with φ. By (2.7), we have the following transformation formula of f^{φ} :

(2.8)  f^{φ} ∘ γ = χ^{φ}(γ) ∘ f^{φ} for all γ ∈ Γ.

Such f^{φ} as S_{f^{φ}} = φ is called a developing map of φ, and also the pair (f^{φ}, χ^{φ}) is called a deformation of the Fuchsian group Γ.

In this article, we will call φ a schlicht projective structure if its developing map f^{φ} is schlicht(=univalent) in ℍ. Let S(Γ) denote the set of totality of schlicht projective structures for Γ on ℍ. The Nehari-Kraus theorem states that if f^{φ} is schlicht in ℍ then

∥φ∥_{ℍ} = sup_{z∈ℍ} |φ(z)|(2Imz)^{2} ≤ 6.

So it is natural to consider a complex Banach space B_{2}(ℍ, Γ) = {φ ∈ Q(ℍ, Γ); ∥φ∥_{ℍ} < ∞}. Of course S(Γ) ⊂ B_{2}(ℍ, Γ).

By Hurwitz’s theorem, it turns out that S(Γ) is closed in B_{2}(ℍ, Γ).

On the other hand, S(Γ) is closely related to the Teichmüller space T (Γ) of Γ, where (the Bers model of) the Teichmüller space T (Γ) of Γ is defined by

T (Γ) = {φ ∈ Q(ℍ, Γ); f^{φ} can be extended to a Γ -compatible

quasiconformal homeomorphism of ℂ̂},

where we say that f is Γ -compatible if f ∘ γ ∘ f^{−1} ∈ Möb for all γ ∈ Γ.

It is well-known fact that T (Γ) is a bounded connected open set of B_{2}(ℍ, Γ). Clearly T (Γ) ⊂ S(Γ), and it is conjectured that T (Γ) = IntS(Γ), where IntS(Γ) denotes the interior of S(Γ) in the Banach space B_{2}(ℍ, Γ).

Now we state the main theorem, which is a generalization of Gehring’s result in [6].

2.1. Main Theorem. If Γ is a finitely generated, purely hyperbolic Fuchsian group of the second kind, then T (Γ) = IntS(Γ).

2.2. Remark. For a Fuchsian group Γ acting on ℍ, the followings are mutually equivalent:

(i) Γ is finitely generated, purely hyperbolic and of the second kind, (ii) Γ is a Schottky group, (iii) Γ is a uniformizing group of a compact bordered Riemann surface with

nonempty boundary, more presicely, Γ is the covering transformation group of a holomorphic universal covering p : ℍ → R, where R is a compact Riemann surface of genus g(≥ 0) with mutually closed topological disks \overline{D}_{1},…, \overline{D}_{m} removed (m ≥ 1).

In case of (iii), we say that R is of conformal type (g, 0, m), and we should note that such Γ is a free group of rank 2g + m − 1.

In the sequel, we are mainly concerned with the properties of a point in IntS(Γ) for an arbitrary Fuchsian group Γ, or almost equivalently, the shape of the domain D = D^{φ} = f^{φ}(ℍ) for φ ∈ IntS(Γ).

First, for φ ∈ S(Γ), the holonomy homomorphism χ^{φ} : Γ → Möb is injective and G = χ^{φ}(Γ) < Möb acts on D = f^{φ}(ℍ) discontinuously, therefore D ⊂ Ω(G), in particular, G is a Kleinian group. Furthermore, for φ ∈ IntS(Γ), χ^{φ} : Γ → G enjoies the following property.

2.3. Lemma (cf.[15]). For φ ∈ IntS(Γ), the holonomy homomorphism χ^{φ} : Γ → G = χ^{φ}(Γ) is a type-preserving isomorphism.

Proof. First, we remark that the mapping φ ↦ tr^{2}χ^{φ}(γ) is holomorphic on B_{2}(ℍ, Γ), where γ is a fixed element of Γ and tr^{2}g = (a + d)^{2} if the Möbius transformation g is represented by g(z) = \frac{az+b}{cz+d} with ad − bc = 1.

Second, if γ is parabolic or elliptic, i.e., tr^{2}γ = 4 cos^{2} \frac{π}{n} for some natural number n, then tr^{2}χ^{φ}(γ) = 4 cos^{2} \frac{π}{n} for φ ∈ T (Γ) by quasiconformal homogeneity of T (Γ). Since T (Γ) is open in B_{2}(ℍ, Γ), the identity theorem implies that tr^{2}χ^{φ}(γ) = 4 cos^{2} \frac{π}{n} for all φ ∈ B_{2}(ℍ, Γ). Thus χ^{φ} preserves types elliptic and parabolic.

Finally, let γ ∈ Γ be hyperbolic. Since χ^{φ} is injective for φ ∈ S(Γ), χ^{φ}(γ) must not be elliptic. Suppose that χ^{φ}(γ) becomes parabolic, i.e., tr^{2}χ^{φ}(γ) = 4, for some φ_{0}, the identity theorem produces again that tr^{2}χ^{φ}(γ) is a nonconstant map on V, so the image of V under this map is open neighborhood of 4. In particular, for sufficiently large n ∈ ℕ there exists a point φ_{1} in V such that tr^{2}χ^{φ_{1}}(γ) = 4 cos^{2} \frac{π}{n}, which is a contradiction. □

2.4. Remark. The proof of the above lemma in [15] relies upon the λ-lemma. The author has learned the idea in the above proof from H. Shiga.

Hyperbolic sup norm. For later use, we shall fix several notations in the more general situation. Let D be a hyperbolic simply connected domain with the Poincaré metric ρ_{D}(z)|dz| of negative constant curvature −4, and let G be a Kleinian group acting on D (, which is not necessarily a component of Ω(G)). The complex Banach space B_{2}(D, G) is defined as the set

{φ : D → ℂ : holomorphic map; (φ ∘ L)(L′)^{2} = φ for all L ∈ G, ∥φ∥_{D} < ∞},

where ∥φ∥_{D} = sup_{z∈D} |φ(z)|ρ_{D}(z)^{−2}. If G is the trivial group, simply we write B_{2}(D) = B_{2}(D, G). Let f : ℍ → D be a conformal map, then by the conformal invariance of the Poincaré metric, we have

(2.9)  ∥(φ ∘ f)(f′)^{2}∥_{ℍ} = ∥φ∥_{D} for φ ∈ B_{2}(D).

G-Schwarzian domains. Under the above preparations, we shall state a characterization of such a domain D obtained as f^{φ}(ℍ) for some φ ∈ IntS(Γ).

2.5. Lemma. Let Γ be an arbitrary Fuchsian group acting on ℍ. For φ ∈ S(Γ), φ belongs to IntS(Γ) if and only if the domain D = f^{φ}(ℍ) has the following property: There exists a positive constant ε > 0 such that any non-constant meromorphic map g : D → ℂ̂ with S_{g} ∈ B_{2}(D, G)_{ε} must be univalent in ℍ, where G = χ^{φ}(Γ) and B_{2}(D, G)_{ε} = {ψ ∈ B_{2}(D, G); ∥ψ∥_{D} < ε}.

When a Kleinian group G acts on a hyperbolic domain D (not necessarily simply connected), D is called a G-Schwarzian domain with constant ε if the above property holds.

Proof. We write f = f^{φ} for short. By (2.2) and (2.9) we obtain the equality

∥S_{g∘f} − φ∥_{ℍ} = ∥S_{g}∥_{D},

which implies what we need here. □

Next we refer to the local quasiconformal homogeneity of IntS(Γ), which plays an important role in §4. The proof of the following proposition is deeply indebted to a group equivariant version of the λ-lemma.

2.6. Proposition ([15]). Let V be a connected component of IntS(Γ). For any φ_{1}, φ_{2} ∈ V, there exists a quasiconformal self-map F of ℂ̂ with the following properties:

(1) f^{φ_{2}} = F ∘ f^{φ_{1}} on ℍ,

(2) χ^{φ_{2}}(γ) = F ∘ χ^{φ_{2}}(γ) ∘ F^{−1} on ℂ̂ for all γ ∈ Γ.

An estimate of the hyperbolic sup norm. In what follows, it becomes important to estimate the magnitude of the hyperbolic sup norm ∥φ∥_{D}, and so we now give a method to controle the norm by the another (relatively) local data easy to treat. Let A ∈ [1, ∞) be a constant and D a proper subdomain of ℂ. Define D_{A}(D) by the collection of such disks B(z_{0}, r) = {z ∈ ℂ; |z − z_{0}| < r} as B(z_{0}, Ar) ⊂ D.

In this article, an orientation-preserving homeomorphism (or, non-constant continuous map) f : D_{1} → D_{2} shall be called, conventionally, a k-quasiconformal map (or, k-quasi-analytic map, respectively) where k ∈ [0, 1) is a constant if f has locally L^{2}-derivatives such that |∂z̄f| ≤ k|∂_{z}f| almost everywhere in D_{1}. A quasiconformal map is often called a qc map, for short. And, we denote here by μ[f] the Beltrami coefficient ∂z̄f/∂_{z}f of quasiconformal map (or, quasi-analytic map) f. We remark that, since ∂_{z}f ≠ 0 a.e., μ[f] is well-defined. Thus, f is k-qc if and only if ∥μ[f]∥_{∞} ≤ k. We should remark that such a f is ordinarily called K-quasiconformal where K = \frac{1+k}{1−k} ∈ [1, ∞), and this terminology has a adovantageous property that the composition map f_{1} ∘ f_{2} is K_{1} · K_{2}-qc if f_{1} is K_{1}-qc and f_{2} is K_{2}-qc. With these notations, we have the following

2.7. Proposition (cf.[2], [16]). Let D be a simply connected hyperbolic subdomain of ℂ, A ≥ 1 and k ∈ [0, 1) be constants, and f be a non-constant meromorphic function on D. If f|_{Δ} can be extended to a k-qc map of ℂ̂ for any Δ ∈ D_{A}(D), then ∥S_{f}∥_{D} ≤ 96kA^{2}.

Conversely, if ∥S_{f}∥_{D} ≤ 2kA^{2} then f|_{Δ} can be extended to a k-qc map of ℂ̂ for any Δ ∈ D_{A}(D).

Bers projection. The measurable Riemann mapping theorem due to Ahlfors-Bers claims that, for μ ∈ L^{∞}(ℂ) with ∥μ∥_{∞} < 1, there exists a unique quasiconformal homeomorphism of ℂ̂, denoted w^{μ}, such that ∂z̄w^{μ} = μ∂_{z}w^{μ} a.e. and w^{μ}(0) = 0, w^{μ}(1) = 1, w^{μ}(∞) = ∞.

Let G be a Kleinian group acting on an open set D ⊂ ℂ̂. We set E = ℂ \ D. L^{∞}(E, G) and M(E, G) denote the complex Banach space {μ ∈ L^{∞}(ℂ); μ = 0 on D, (μ∘L)·\overline{L′}/L′ = μ a.e. for all L ∈ G\} and its open unit ball, respectively. If G = 1, we shall write L^{∞}(E) = L^{∞}(E, 1) and M(E) = M(E, 1) for simplicity.

For μ ∈ M(E, G), by the automorphy of μ, w^{μ} conjugates G to another Kleinian group, i.e., w^{μ}G(w^{μ})^{−1} < Möb, and w^{μ}|_{D} is conformal since μ = 0 on D. As a result, the Schwarzian derivative of w^{μ}|_{D} is well-defined and turns out to be a (bounded) holomorphic quadratic differential for G on D. Particularly, when D is simply connected domain of hyperbolic type, we denote by Φ_{D}(μ) the Schwarzian derivative of w^{μ}|_{D}, and which is called the (generalized) Bers projection of μ ∈ M(E, G). As is well-known, Φ_{D} : M(E, G) → B_{2}(D, G) is holomorphic and its differential at the origin is represented as an integral operator (cf. [15]):

d_{0}Φ_{D}[ν](z) = −\frac{6}{π} ∫∫_{E} \frac{ν(ζ)}{(ζ − z)^{4}} dξdη (ζ = ξ + iη)

for every ν ∈ L^{∞}(E, G). In the special case that D = ℍ, Φ_{ℍ} is the original Bers projection and its image Φ_{ℍ}(M(ℍ^{c}, G)) is the Teichmüller space T (G) of the Fuchsian group G.

As a corollary of the main theorem, we can verify the following:

2.8. Corollary. Let D be a simply connected subdomain of ℂ̂ of hyperbolic type and E its complement. Suppose that a Schottky group G acts on D. Then the following conditions are equivalent to each other:

(1) d_{0}Φ_{D} : L^{∞}(E, G) → B_{2}(D, G) is surjective,

(2) d_{0}Φ_{D} : L^{∞}(E) → B_{2}(D) is surjective, and

(3) D is a quasidisk.

Proof. As the claim (2) ⇔ (3) is a special case G = 1 of (1) ⇔ (3), it suffices to prove (1) ⇔ (3). The part (3) ⇒ (1) is a direct consequence of the submersivity of the generalized Bers projection (cf. Bers [3], Earle-Nag [5]). Thus we have only to prove that (1) implies (3). First observe that if d_{0}Φ_{D} : L^{∞}(E, G) → B_{2}(D, G) is surjective then Φ_{D}(M(E, G)) is a neighborhood of 0 in B_{2}(D, G) (see, for instance, [1] Proposition 2.5.9), that is, D is a G-Schwarzian domain. Let f : ℍ → D be a Riemann mapping function of D and φ its Schwarzian derivative. Then, the above observation shows that φ ∈ IntS(Γ) where Γ denotes the Fuchsian group f^{−1}Gf. Here we may assume that f = f^{φ}. Since χ^{φ} : Γ → G is a typepreserving isomorphism by Lemma 2.3, Γ is also a Schottky group. (Here note that Schottky groups are characterized as the finitely generated, purely loxodromic free Kleinian groups by Maskit’s theorem [8].) Therefore Theorem 2.1 produces that IntS(Γ) = T (Γ). Thus we have shown that φ ∈ T (Γ), in particular, D = f^{φ}(ℍ) is a quasidisk. □

Quasidisks. Finally, we shall mention a characterization of the quasidisks, where we recall that the quasidisk is defined as an image of the unit disk (or the upper half plane) under a quasiconformal self-map of ℂ̂. Before stating the result, we shall define a distance (“path diameter distance” w.r.t. the Euclidean metric) δ_{D} on any open subset D of ℂ. For given two points z_{1}, z_{2} in D, we set

δ_{D}(z_{1}, z_{2}) = inf_{α⊂D} diamα,

where the infimum is taken over the paths α connecting z_{1} and z_{2} in D and diamα = sup_{w_{1},w_{2}∈α} |w_{1} − w_{2}|. If w_{1} and w_{2} do not belong to the same component of D, we define δ_{D}(w_{1}, w_{2}) = ∞. As is easily seen, δ_{D} satisfies the axiom of distance except that δ_{D} possibly takes the value ∞. In particular, δ_{D} is certainly a distance on D if D is a domain, and δ_{D}(z_{1}, z_{2}) ≥ |z_{1} − z_{2}| by definition.

A bounded simply connected domain D is called linearly connected if δ_{D}(z_{1}, z_{2}) ≤ C|z_{1} − z_{2}| for any z_{1}, z_{2} ∈ D, or equivalently, for an arbitrary disk Δ, any two points in D ∩ Δ can be connected by a path in D ∩ Δ_{A}, where C and A is constants (≥ 1) depending only on D and Δ_{A} denotes {|z − z_{0}| < Ar} if Δ = {|z − z_{0}| < r}. It is worthy to know the fact that a linearly connected, bounded, simply connected domain is always a Jordan domain (see Theorem 3.3 below).

A bounded simply connected domain D is called a John domain if, for an arbitrary disk Δ, any two points in D \ Δ_{A} can be connected by a path in D \ Δ, where A is a constant (≥ 1) depending only on D.

2.9. Theorem (cf. Gehring [6], Pommerenke [12]). A bounded simply connected domain D is a quasidisk if and only if D is a linearly connected John domain.

§3. The first construction of non-univalent meromorphic map with G-invariant small Schwarzian

In this section, we shall proceed in the more general situation. Let G be an arbitrary Kleinian group, D be a G-invariant hyperbolic plane domain and p : Ω(G) → R = Ω(G)/G be the natural projection. Here we should remark that D ⊂ Ω(G), for ℂ̂ \ D is a G-invariant closed set containing at least 3 points, thus ℂ̂ \ D ⊃ Λ(G). In this section, it is our main job to prove the following

3.1. Proposition. Suppose that a Kleinian group G acts on a simply connected plane domain D ⊂ ℂ of hyperbolic type. If D is a G-Schwarzian domain with constant ε > 0, the following is valid for an appropriate constant B > 1 depending only on ε : for an arbitrary Δ ∈ D_{B}(Ω(G)) such that p|_{Δ_{B}} is injective, any two points in Δ ∩ D can be joined by a path in Δ_{B} ∩ D.

Before stepping into the proof of the above proposition, we state a few corollaries.

3.2. Corollary. If a hyperbolic simply connected plane domain D is G-Schwarz-ian for some Kleinian group G acting on D, and if Λ(G) ≠ ∂D then ∂D = ∂D^{∗} where D^{∗} is the exterior of D. In particular, D^{∗} ≠ ∅.

Proof of Corollary 3.2. Since Λ(G) ⊂ ∂D generally, the hypothesis implies that there exists a point z_{0} in ∂D \ Λ(G) = ∂D ∩ Ω(G). The limit set Λ(G) is contained in the closure of the orbit G · z_{0} of z_{0}, on the other hand G · z_{0} is contained in ∂D ∩ Ω(G), thus we obtain that Λ(G) ⊂ \overline{G · z_{0}} ⊂ \overline{D ∩ Ω(G)}. As a consequence, we have \overline{∂D ∩ Ω(G)} = ∂D. Clearly, ∂D^{∗} ⊂ ∂D, so it is sufficient to prove that ∂D ∩Ω(G) ⊂ ∂D^{∗}. If not, since the free regular set ^{∘}Ω(G) = Ω(G) \ {elliptic fixed points of G} is dense in Ω(G), there exists a point w_{0} ∈ ∂D ∩^{∘}Ω(G) \ ∂D^{∗}. Pick and fix another point w_{1} in ∂D. Then, since w_{0} ∈ Int\overline{D} ∩^{∘}Ω(G), there exists an injective disk Δ with center w_{0} and radius r > 0 such that Δ ⊂ \overline{D} and w_{1} ∉ Δ. Because ∂Δ_{1/2} ∩ D is dense in ∂Δ_{1/2}, we can take finitely many points a_{1}, a_{2},…, a_{n} ∈ ∂Δ_{1/2} ∩ D such that |a_{j} − a_{j+1}| < \frac{r}{4B} for j = 1, 2,…, n where B is the constant in Proposition 3.1 and a_{n+1} = a_{1}.

Then the disk B(a_{j}, r/4) is included in the injective disk Δ, so Proposition 3.1 guarantees the existence of a path γ_{j} ⊂ B(a_{j}, r/4) ∩ D connecting a_{j} and a_{j+1} (j = 1,…, n). Therefore γ = ∪_{j=1}^{n}γ_{j} is a closed path in D separating w_{0} from w_{1}, which contradicts the connectedness of ∂D. □

By the next characterization of Jordan domains, we obtain a further information about G-Schwarzian simply connected domains.

3.3. Theorem (Newman [11] Chap.VI, Theorem 14.1 and Theorem 16.2). A hyperbolic simply connected domain D ⊂ ℂ̂ is a Jordan domain if and only if D is uniformly locally connected, more precisely, for any positive number ε there exists a positive η such that, for all pairs of points x, y ∈ D, d(x, y) < η implies that δ_{D}(x, y) < ε, where δ_{D} denotes the “path diameter distance” with respect to the spherical metric of ℂ̂.

3.4. Corollary. Let D be a hyperbolic, G-Schwarzian, simply connected plane domain for some Kleinian group G and D′ a Jordan domain such that ∂(D ∩ D′) ⊂ ^{∘}Ω(G), where ^{∘}Ω(G) denotes the free regular set of G, i.e., ^{∘}Ω(G) = Ω(G) \ { elliptic fixed points of G}. Then, each component of D ∩ D′ is a Jordan domain.

Proof. Since D ∩ D′ is simply connected, it suffices to show that each component of D ∩ D′ is uniformly locally connected, by Theorem 3.3. By Corollary 3.2, we can assume that D and D′ are both bounded domains. Let 0 < ε < diam∂D ∩ ∂D′. Since some compact neighborhood of ∂(D ∩ D′) is contained in free regular set of G, Proposition 3.1 yields that there exists a positive η_{1} such that δ_{D}(x, y) < ε/2 for any x, y ∈ D ∩ D′ with |x − y| < η_{1}.

On the other hand, Theorem 3.3 implies that there exists a positive η_{2} such that δ_{D′}(x, y) < ε/2 for any x, y ∈ D′ with |x − y| < η_{2}.

Let D_{1} be a component of D ∩D′ and x, y ∈ D_{1} with |x−y| < η_{0} = min{η_{1}, η_{2}}. Then, there exist paths γ_{1}, γ_{2} from x to y such that γ_{1} ⊂ D, γ_{2} ⊂ D′ and that diamγ_{j} < ε/2. And, since D_{1} is connected, there exists a path γ_{0} in D_{1} from x to y. Since D and D′ are both simply connected, γ_{1} is homotopic to γ_{0} in D and γ_{2} is homotopic to γ_{0} in D′, and so γ_{1} is homotopic to γ_{2} in D ∪ D′. Let γ denote the closed curve γ_{1} · γ_{2}^{−1}, then γ is null homotopic in (∂D ∩ ∂D′)^{c}. Since diamγ < ε < diam∂D∩∂D′, γ bounds no points of ∂D∩∂D′. And therefore γ is null homotopic in Δ \ (∂D ∩ ∂D′) = (Δ \ ∂D) ∪ (Δ \ ∂D′), where Δ = {z ∈ ℂ; |z − x| < ε/2}. (Remark that γ_{j} ⊂ Δ.)

Therefore by Alexander’s lemma (to be stated below) x and y are known to be connected by a path γ_{3} in (Δ \ ∂D) ∩ (Δ \ ∂D′) = Δ \ (∂D ∪ ∂D′). Since γ_{3} ⊂ D_{1} and diamγ_{3} < diamΔ = ε, it is proved that D_{1} is uniformly locally connected. □

3.5. Alexander’s lemma (cf. Newman [11]). Let O_{1} and O_{2} be open sets in ℂ̂. Suppose that x, y ∈ O_{1} ∩ O_{2} are connected by paths γ_{i} in O_{i} (i = 1, 2). Then, if γ_{1} ∪γ_{2} is null homotopic in O_{1} ∪O_{2}, x and y are connected by a path in O_{1} ∩O_{2}.

As the final corollary, we state a rather technical lemma which will be used in §5.

3.6. Lemma. Under the same hypothesis of Proposition 3.1, let w_{0} be a point of ∂D ∩^{∘}Ω(G) and Δ ⊂ ^{∘}Ω(G) an injective disk centered at w_{0}. Then there exists a connected open neighborhood V of w_{0} in Δ such that V ∩ D is connected.

Proof. Consider the disk Δ_{1/B} ∈ D_{B}(Ω(G)), where B is the constant which appeared in Proposition 3.1. Let W be a connected component of Δ ∩ D which includes a point in Δ_{1/B} ∩ D. Then, by Proposition 3.1, Δ_{1/B} ∩ D ⊂ W. Therefore, we can adopt Δ_{1/B} ∪ W as a neighborhood V. □

Proof of Proposition 3.1. We choose B ≥ C > 1 so that B ≥ 6C and C ≥ 9 + 2^{11} · 3^{3}/ε. For some disk Δ = B(z_{0}, r) ∈ D_{B}(Ω(G)), suppose that p|_{Δ_{B}} is injective and that two points z_{1}, z_{2} ∈ Δ ∩ D cannot be connected by any path in Δ_{C} ∩ D (see Fig. 3.1). Let D_{j} be the connected component of Δ_{C} ∩ D containing z_{j} (j = 1, 2). Noting that D_{1} ∩ D_{2} = ∅ by the hypothesis, we can choose a component of ∂Δ_{C} \ \overline{D_{1}}, say J, containing a point of D_{2}. Then the closed interval I = ∂Δ_{C} \ J has the following properties:

(1) \overline{D}_{1} ∩ ∂Δ_{C} ⊂ I,

(2) D_{2} ∩ I = ∅, and

(3) ∂I ⊂ ∂D.

Furthermore, interchanging D_{1} and D_{2} if necessary, we may assume that

(4) |I| ≤ \frac{1}{2}|Δ_{C}| where | · | denotes the arc length.

Fig. 3.1.

We assign the anti-clockwise orientation to I with initial point w_{0} and terminal point w_{1}. For a time, we shall consider the things by moving w_{0}, w_{1} to 0, ∞ respectively through the Möbius transformation Q(z) = \frac{z−w_{0}}{z−w_{1}} . First, we remark that the quantity λ = Q(z_{2})/Q(z_{1}) is near to 1. Precisely, for the principal value of σ of log λ, we have the next

3.7. Lemma.

|σ| ≤ \frac{4|w_{0} − w_{1}|}{(C − 1)^{2}r} (≤ \frac{8C}{(C − 1)^{2}} ) , and |σ| ≤ \frac{8}{C − 1}.

Proof of Lemma 3.7. If we set u = \frac{z_{2}−z_{1}}{z_{1}−w_{0}} and v = \frac{z_{2}−z_{1}}{z_{1}−w_{1}} , then we have |u|, |v| ≤ \frac{2}{C−1}(< \frac{1}{2}), |u − v| ≤ \frac{2|w_{0}−w_{1}|}{(C−1)^{2}r} and

|σ| = |log \frac{1 + u}{1 − u}| = |∫_{u}^{v} \frac{dζ}{1 + ζ} | ≤ 2 ∫_{[u,v]} |dζ| = 2|u − v| ≤ \frac{4|w_{0} − w_{1}|}{(C − 1)^{2}r}.

Similarly, we have |σ| ≤ 2(|u| + |v|) ≤ \frac{8}{C−1}. □

Let ω = Imσ = argλ (|ω| ≤ σ < π/4) and θ_{0} be an angle of the ray Q(I) with the positive real line, i.e., Q(I) = {re^{iθ_{0}}; 0 ≤ r ≤ ∞}. In order to construct a tame deformation such that its images of z_{1} and z_{2} coincide, we first define a map T̃ : ℂ̂ → ℂ̂ by

T̃(z) = \{ \begin{matrix}z exp(\frac{π/4+θ}{π/4−ω} σ) & (−π/4 ≤ θ < −ω) \\ λz & (−ω ≤ θ < π) \\ z exp(\frac{5π/4−θ}{π/4} σ) & (π ≤ θ < 5π/4) \\ z & (5π/4 ≤ θ < 7π/4) \end{matrix}

where z = re^{i(θ+θ_{0})}, T̃(0) = 0, and T̃(∞) = ∞, and further set T = Q^{−1} ∘ T̃ ∘ Q.

3.8. Lemma. T̃ and T are \frac{8}{C−9}-qc homeomorphism of ℂ̂.

Proof. If we set

t = t(re^{i(θ+θ_{0})}) = \{ \begin{matrix}\frac{σ}{π/4−ω} & (−π/4 < θ < −ω) \\ −4σ/π & (π < θ < 5π/4) \\ 0 & ( otherwise ), \end{matrix}

then s := ir \frac{∂T̃}{∂r}/\frac{∂T̃}{∂θ} = \frac{i}{t+i} = \frac{1}{1−it} a.e. and we have an estimate |t| ≤ \frac{|σ|}{π/4−|σ|} ≤ 2|σ| since |σ| ≤ \frac{8}{C−1} ≤ \frac{1}{4} < π/4 − \frac{1}{2}. Therefore,

|μ[T̃]| = |\frac{s − 1}{s + 1}| = |\frac{it}{2 − it}| ≤ \frac{|t|}{2 − |t|} ≤ \frac{|σ|}{1 − |σ|} ≤ \frac{8}{C − 9},

i.e., ∥μ[T]∥_{∞} = ∥μ[T̃]∥_{∞} ≤ \frac{8}{C−9}. □

If we set

Ẽ_{1} = {z ∈ ℂ^{∗}; θ_{0} − π/4 < argz ≤ θ_{0}},
Ẽ_{2} = {z ∈ ℂ^{∗}; θ_{0} + π ≤ argz < θ_{0} + π + ω},

and E_{j} = Q^{−1}(Ẽ_{j}) (j = 1, 2), then we can obtain the following.

3.9. Lemma. T (Δ_{C} ∪ E_{1}) ⊂ Δ_{C} ∪ E_{1} ∪ E_{2} ⊂ Δ_{4C}(⊂ Δ_{B}).

Proof. The first inclusion is clear by construction of T. Now we prove the second inclusion. With a suitable normalization, we may assume that Δ_{C} = B(0, 1) and w_{1} = \overline{w}_{0} = e^{iφ_{0}} (0 < φ_{0} < π). The condition (4) implies that 0 < φ_{0} ≤ π/2. Thus we observe that E_{1} is largest if φ_{0} = π/2, in that case

(3.1)  E_{1} ⊂ B(1, \sqrt{2}) ⊂ B(0, 1 + \sqrt{2}) ⊂ B(0, 4) = Δ_{4C}.

Next, we shall consider E_{2}. We may assume that ω > 0, for otherwise E_{2} = ∅. We need to calculate the radius ρ and the center c < 0 of the circle Q^{−1}{argz = θ_{0} + ω modπ} (see Fig. 3.2). The elementary geometry tells us that

ρ sin(φ_{0} − ω) = sin φ_{0},
ρ cos(φ_{0} − ω) = cos φ_{0} − c,

and the latter implies that −c ≤ ρ. By Lemma 3.2,

ω ≤ \frac{4|w_{1} − w_{0}|}{(C − 1)^{2}} = \frac{8 sin φ_{0}}{(C − 1)^{2}} ≤ \frac{φ_{0}}{2} ,

since C ≥ 5. So, φ_{0}−ω ≥ φ_{0}/2, and therefore ρ = sin φ_{0}/ sin(φ_{0}−ω) ≤ sin φ_{0}/ sin(φ_{0}/2) = 2 cos(φ_{0}/2) ≤ 2. The above estimate enables us to deduce that E_{2} ⊂ B(c, ρ) ⊂ B(0, ρ − c) ⊂ B(0, 2ρ) ⊂ B(0, 4) = Δ_{4C} ⊂ Δ_{B}. The proof is now comleted. □

Fig. 3.2

Now, we shall go to the next step. We construct a locally injective quasi-analytic map h : D → ℂ̂ as follows. Let ψ(z) = ξ(z)+i(η(z)+θ_{0}) be a branch of log Q(z) in D such that η = 0 on I ∩D_{1}, where we note that log Q(z) has the same imaginary part on I ∩ D_{1} by the property (1). Then, clearly −2π < η < π, η(z_{1}) > 0, η(z_{2}) < −π, and |η(z_{1}) − η(z_{2}) − 2π| = |Reσ| ≤ \frac{8}{C−1}.

At first, we define h on Δ_{B} ∩ D by

h(z) = \{ \begin{matrix}z & if η(z) ≤ −π/4, \\ T (z) & if − π/4 ≤ η(z). \end{matrix}

Observe that h(Δ_{B} ∩ D) ⊂ Δ_{B} and h = id on ∂Δ_{B} ∩ D by Lemma 3.9. Since ∪_{L∈G}L(Δ_{B}) is a disjoint union, we can extend h to a continuous map on D (denoted the same letter h) as follows:

h(z) = \{ \begin{matrix}L ∘ h ∘ L^{−1}(z) & if z ∈ L(Δ_{B} ∩ D) for some L ∈ G, \\ z & otherwise. \end{matrix}

By construction, h satisfies the following.

(a) h is \frac{8}{C−9}-quasi-analytic,

(b) h ∘ L = L ∘ h for all L ∈ G,

(c) h(z_{1}) = h(z_{2}),

(d) h(Δ_{B} ∩ D) ⊂ Δ_{B}, and h(D \ Δ_{B}) ∩ Δ_{B} = ∅. And, a crucial point of the above construction is the validity of the following

3.10. Lemma. The Beltrami coefficient μ[h^{−1}] on h(D) of the (local) inverse of h is well-defined, i.e., it is independent of a particular branch of h^{−1}.

Proof of Lemma 3.10. By equivariance of the definition of h and the property (d), it is sufficient to prove only on Δ_{B}. Let

E_{3} = {z ∈ Δ_{B} ∩ D; #h^{−1}(h(z)) > 1}, and
E_{4} = {z ∈ Δ_{B} ∩ D; ∂z̄h(z) ≠ 0}.

First, h(E_{3}) ⊂ T ({z ∈ Δ_{B} ∩ D; −π/4 < η(z) < π}) ∩ {z ∈ Δ_{B} ∩ D; −2π < η(z) < −π/4} ⊂ Q^{−1}({−2π < argz < −π}), by definition of h. On the other hand, h(E_{4}) ⊂ T ({−π/4 ≤ η(z) ≤ −ω}) ⊂ Q^{−1}({−π/4 ≤ argz ≤ 0}), and so it follows that h(E_{3}) ∩ h(E_{4}) = ∅, this forces E_{3} ∩ E_{4} = ∅, i.e., h is holomorphic on E_{3}. The proof is finished. □

The above lemma says that the next definition:

μ = \{ \begin{matrix}μ[h^{−1}] & on h(D) \\ 0 & on ℂ \ h(D) \end{matrix}

is well-defined and the property (a) and (b) imply that μ ∈ M(ℂ, G) and ∥μ∥_{∞} ≤ \frac{8}{C−9}(< 1). Now, we define a quasi-analytic map g : D → ℂ̂ by g = w^{μ} ∘ h, then by the chain rule for quasi-analytic mappings, we can see that μ[g] = 0 a.e. on D. By virtue of Weyl’s lemma for quasi-analytic mappings, g : D → ℂ̂ is known to be meromorphic, moreover property (c) of h implies that g(z_{1}) = g(z_{2}), that is, g is not univalent. By the fact that μ ∈ M(ℂ, G), w^{μ} transforms the Kleinian group G to another one by conjugation, thus g does so, in other words, S_{g}(z)dz^{2} is G-invariant.

Finally, we shall give an estimate of the hyperbolic-sup norm of S_{g} which will lead to a contradiction.

Before into the final step, we prepare some lemmas. The proof of the first is quite elementary (see [16] Proof of Proposition 2.4).

3.11. Lemma. For any constant A ≥ 1, the following is valid. If Δ′ ∈ D_{A}(D) and if L ∈ Möb satisfies that A(D) ⊂ ℂ then L(Δ′) ∈ D_{A′}(D) where A′ = \frac{A+A^{−1}}{2} .

3.12. Lemma. Let E = {z ∈ D; −π/4 < η(z) < −ω} and A ≥ 3 be a constant. Then, for Δ′ ∈ D_{A}(D) such that Δ′ ∩ E ≠ ∅, h coincides with T on Δ′.

Proof of Lemma 3.12. We may assume that Δ_{C} = B(0, 1). Suppose that Δ′ = B(c, ρ) ∈ D_{A}(D) satisfies Δ′ ∩ E ≠ ∅. Then clearly −5π/4 < η < π − ω on Δ′. And we note that

(3.2)  Aρ ≤ |c − w_{i}| (i = 0, 1).

from the assumtion and the fact that w_{i} ∈ ∂D. Pick a point ζ from Δ′ ∩ E, then |c − ζ| < ρ, |ζ − w_{1}| ≤ diamE ≤ diamE_{1} ≤ 2\sqrt{2} and |ζ| < 1 + \sqrt{2} by (3.1). Hence, Aρ ≤ |c − w_{1}| ≤ |c − ζ| + |ζ − w_{1}| < ρ + 2\sqrt{2}, thus we have ρ ≤ \frac{2\sqrt{2}}{A−1} ≤ \sqrt{2}. Since |c| ≤ |c−ζ|+|ζ| < ρ+(1+\sqrt{2}), we have that Δ′ ⊂ B(0, |c|+ρ) ⊂ B(0, 1+\sqrt{2}+2ρ) ⊂ B(0, 6) ⊂ Δ_{B}. Thus we need only to prove that Δ′ ∩ E′ = ∅ where E′ = Q^{(}−1{z ∈ ℂ^{∗}; −5π/4 < argz < −3π/4}) = Q^{−1}({z ∈ ℂ^{∗}; 3π/4 < argz < 5π/4}).

Suppose that Δ′ ∩ E′ ≠ ∅. We may assume that |w_{0} − c| ≤ |w_{1} − c|. Let C_{1} = {z; |z − a_{1}| = r_{1}} and C_{2} = {z; |z − a_{2}| = r_{2}} denote the circles including circular arcs γ_{1} = Q^{−1}({ζ; argζ − θ_{0} = −π/4}) and γ_{2} = Q^{−1}({ζ; argζ − θ_{0} = −3π/4}), respectively (where C_{2} is possibly a line). Remark that, by assumption, γ_{1} and γ_{2} perpendicularly intersects at the two points w_{0} and w_{1} and that γ_{j} ∩ ∂Δ′ ≠ ∅ for j = 1, 2.

Let ζ_{j} denote the intersection point of C_{j} and the ray starting from a_{j} and passing through c for j = 1, 2. If ζ_{j} ∈ C_{j} \ γ_{j}, then Aρ ≤ |c − w_{0}| = dist(c, γ_{2}) < ρ, this is impossible. Thus we conclude that ζ_{j} ∈ γ_{j}, and hence we have

(3.3)  |c − ζ_{j}| = dist(c, γ_{j}) < ρ for j = 1, 2.

Fig.3.3 : Case 1

Now let c_{j} be the orthogonal projection of the point c to the line through w_{0} and a_{j} (j = 1, 2).

Case 1: W = Q^{−1}({ζ; −3π/4 < argζ −θ_{0} < −π/4}) is unbounded (see Fig. 3.3). In this case, we have |c_{j} −w_{0}| ≤ |c−ζ_{j}| < ρ for j = 1, 2. Thus we get an estimate

that

|c − w_{0}| = \sqrt{|c_{1} − w_{0}|^{2} + |c_{2} − w_{0}|^{2}} < \sqrt{2}ρ.

By (3.2), we have Aρ < \sqrt{2}ρ, which is a contradiction.

Case 2: W is bounded.

We may assume that c is in the inside of C_{1} and in the outside of C_{2} (see Fig. 3.4).

Fig.3.4 : Case 2

Then, as in Case 1, we know that

(3.4)  |c_{2} − w_{0}| < ρ.

Next, because |c − a_{1}| ≤ |c − c_{1}| + |c_{1} − a_{1}| = |c_{2} − w_{0}| + (r_{1} − |c_{1} − w_{0}|), we obtain that

(3.5)  |c_{1} − w_{0}| ≤ |c_{2} − w_{0}| + r_{1} − |c − a_{1}| = |c_{2} − w_{0}| + |c − ζ_{1}| < 2ρ

by (3.3) and (3.4). So, by (3.4) and (3.5), we have

|c − w_{0}| = \sqrt{|c_{1} − w_{0}|^{2} + |c_{2} − w_{0}|^{2}} < \sqrt{5}ρ.

Combined with (3.2), we can deduce that Aρ < \sqrt{5}ρ, which contradicts the hypothesis that A ≥ 3. □

Proof of Propositon 3.1 (continued). At first, we shall examine the local qc extensibility of h. Let A ≥ 6 and Δ ∈ D_{A}(D). If Δ ∩ L(E) ≠ ∅ for some L ∈ G, then L^{−1}(Δ) ∩ E ≠ ∅ and, by Lemma 3.11, L^{−1}(Δ) ∈ D_{A′}(D) where A′ = \frac{A+A^{−1}}{2} ≥ 3. So, Lemma 3.12 implies that h = T on L^{−1}(Δ), i.e., h = L∘h∘L^{−1} = L∘T ∘L^{−1} on Δ. From Lemma 3.8, we know that h|_{Δ} can be extended to a k-qc homeomorphism L ∘ T ∘ L^{−1} of ℂ̂, where k = \frac{8}{C−9}.

Otherwise, Δ∩(∪_{L∈G}L(E)) = ∅, so h is a restriction of a Möbius transformation on Δ. Thus, in any case, h|_{Δ} can be extended to a k-qc homeomorphism of ℂ̂. On the other hand, w^{μ} is originally a global k-qc map, and hence g|_{Δ} = w^{μ} ∘ h|_{Δ} can be extended to a k′-qc homeomorphism of ℂ̂, where \frac{1+k′}{1−k′} = (\frac{1+k}{1−k})^{2}. By quite easy calculations, we have k′ = \frac{2k}{1+k^{2}} ≤ \frac{16(C−9)}{(C−9)^{2}+64} ≤ \frac{16}{C−9}, therefore combined with Lemma 2.7, we obtain an estimate

∥S_{g}∥_{D} ≤ 96A^{2}k′ ≤ 96 · 6^{2} · \frac{16}{C − 9} < ε.

From the first hypothesis, g must be univalent on D, which contradicts the fact that g(z_{1}) = g(z_{2}). Thus, z_{1} and z_{2} ∈ Δ ∩ D must be connected by a path in Δ_{C} ∩ D, therefore, in Δ_{B} ∩ D. □

§4. The second construction of non-univalent meromorphic map with G-invariant small Schwarzian

In this section, we shall make another construction of non-univalent meromorphic map, which is, in a sense, a dual of the one in §3. At first, we prove a rather technical proposition, which holds for general Fuchsian groups of the second kind.

4.1. Proposition. Suppose that a Kleinian group G acts on a simply connected plane domain D ⊂ ℂ of hyperbolic type and let p : Ω(G) → Ω(G)/G be the natural projection. If D is a G-Schwarzian domain with constant ε > 0, the following is valid for an appropriate constant C > 1 depending only on ε : for each disk Δ ∈ D_{C}(Ω(G)) such that \overline{E} ⊂ Ω(G) and that p|\overline{E} is injective where E = D \ D_{0} and D_{0} is some connected component of D \ \overline{Δ} (that is the “main” component in a sense), any two points z_{0}, z_{1} ∈ D \ \overline{Δ}_{C} can be joined by a path in D \ \overline{Δ}.

Proof. Let C > 5 + 2^{1}0 · 3 · 5^{2}/ε and Δ ∈ D_{C}(Ω(G)), D_{0} and E as in the statement above. Suppose that some two points z_{1}, z_{2} ∈ D \ \overline{Δ}_{C} cannot be joined by any path in D \ \overline{Δ}. We denote by D_{k} the connected component of D \ \overline{Δ} which contains z_{k} for k = 1, 2. By the assumption, D_{1} ≠ D_{2}. Let I be the connected component of ∂Δ \ \overline{D}_{1} containing points of ∂Δ ∩ \overline{D}_{2}. Obviously ∂Δ ∩ \overline{D}_{2} ⊂ I, and ∂Δ ∩ \overline{D}_{0} ⊂ I or ∂Δ ∩ \overline{D}_{0} ⊂ ∂Δ \ \overline{I}. Replacing I by ∂Δ \ \overline{I} and interchanging D_{1} and D_{2} if necessary, we may assume that

(4.1)  ∂Δ ∩ \overline{D}_{1} ⊂ ∂Δ \ \overline{I}, ∂Δ ∩ \overline{D}_{2} ⊂ I, and ∂Δ ∩ \overline{D}_{0} ⊂ I.

We assign the anti-clockwise orientation to I, and let w_{0}, w_{1} be the initial and terminal point of I, respectively. Note here that w_{i} ∈ ∂D by construction. Now we introduce a Möbius transformation Q(z) = \frac{z−w_{0}}{z−w_{1}} . Let σ be the principal value of log λ, where λ = Q(z_{2})/Q(z_{1}). By Lemma 3.2, we know that |σ| ≤ \frac{8}{C−1}. We define a quasiconformal map T̃_{t} : ℂ̂ → ℂ̂ for 0 ≤ t ≤ 1 by

T̃_{t}(z) = \{ \begin{matrix}z exp (\frac{θ−π/3}{π/3}tσ) & (π/3 ≤ θ < 2π/3) \\ z exp (tσ) & (2π/3 ≤ θ < 4π/3) \\ z exp (\frac{5π/3−θ}{π/3} tσ) & (4π/3 ≤ θ < 5π/3) \\ z & (0 ≤ θ < π/3 or 5π/3 ≤ θ < 2π) \end{matrix}

where z = re^{i(θ+θ_{0})} and θ_{0} is an angle of the ray Q(I) with the positive real line. Further, let T_{t} = Q^{−1} ∘ T̃_{t} ∘ Q for 0 ≤ t ≤ 1. By the same way as in the proof of Lemma 3.8, we have the following

4.2. Lemma. T̃_{t} and T_{t} are \frac{4t}{C−5}-qc homeomorphism of ℂ̂ for 0 ≤ t ≤ 1.

Let ψ(z) = ξ(z) + i(η(z) + θ_{0}) be a branch of log Q(z) in D such that η = 0 on ∂D_{0} ∩ D (⊂ I). Then it is clear that −π < η < 2π on D, η < 0 on D_{0} ∪ D_{1} and that η > π on D_{2}. First, define h_{t}(z) for z ∈ E = D \ D_{0} by the rule

h_{t}(z) = \{ \begin{matrix}z & if η(z) < 0, \\ T_{t}(z) & if 0 ≤ η(z) < π, \\ Q^{−1}(e^{tσ}Q(z)) & if π ≤ η(z). \end{matrix}

Noting that ∪_{L∈G}L(E) is a disjoint union by assumption, next we extend h_{t} to a mapping on ∪_{L∈G}L(E) (still written by the same notation h_{t}) as follows: h_{t} = L ∘ h_{t} ∘ L^{−1} on L(E). Since h(z) = z for any z ∈ ∂D_{0} ∩ D by the choice of η, we can continuously extend h_{t} by difining as the identity map on D \ ∪_{L∈G}L(E). By (4.1), these quasi-analytic mappings h_{t} : D → ℂ̂ (0 ≤ t ≤ 1) satisfy the following conditions.

(a) h_{t} is \frac{4t}{C−5}-quasi-analytic,

(b) h_{t} ∘ L = L ∘ h_{t} for any L ∈ G,

(c) h_{1}(z_{1}) = h_{1}(z_{2}) = z_{1}, and

(d) h_{t}(z) continuously depends on (t, z) ∈ [0, 1] × D. Now we let J = {t ∈ [0, 1]; ∃L ∈ G \ {1} s.t. h_{t}(E^{∘}) ∩ L(h_{t}(E^{∘})) ≠ ∅}, where E^{∘} = IntE = D \ \overline{D}_{0}.

4.3. Lemma. For any t ∈ [0, 1] \ J, the following Beltrami coefficient μ_{t} is welldefined:

μ_{t} = \{ \begin{matrix}μ[h_{t}^{−1}] & on h_{t}(D), \\ 0 & elsewhere. \end{matrix}

Proof of Lemma 4.3. Set E_{1} = {z ∈ D; π/3 ≤ η(z) ≤ 2π/3} (⊂ Δ). Then, by definition, h_{t} is holomorphic off ∪_{L∈G}L(E_{1}). Therefore, it is sufficient to show that

(4.2)  h_{t}(E_{1}) ∩ h_{t}(D \ E_{1}) = ∅.

Since h_{t} = T_{t} on Δ ∩ E, h_{t} is injective in Δ ∩ E, so we have

(4.3)  h_{t}(E_{1}) ∩ h_{t}(Δ ∩ E \ E_{1}) = ∅.

Noting here that |arg(T̃_{t}(z)/z)| ≤ |arg(e^{tσ})| = t|Imσ| ≤ \frac{8t}{C−1} < π/6, we have |arg(Q(h_{t}(z))/Q(z)| < π/6 for all z ∈ E. So that, we obtain

(4.4)  Q(h_{t}(E_{1})) ⊂ {ζ; π/6 < argζ − θ_{0} < 5π/6}, and

(4.5)  Q(h_{t}(E \ Δ)) ⊂ {ζ; 5π/6 < argζ − θ_{0} < 13π/6}.

In particular, we have

(4.6)  h_{t}(E_{1}) ∩ h_{t}(E \ Δ) = ∅.

By (4.3) and (4.6), we can see that

(4.7)  h_{t}(E_{1}) ∩ h_{t}(E \ E_{1}) = ∅.

Further (4.4) implies that

(4.8)  h_{t}(E_{1}) ∩ D_{0} = ∅

since Q(D_{0}) ⊂ {ζ; −π < argζ < 0}. Noting that h_{t} = id on D′_{0} := D_{0}\∪_{L∈G\{1}}L(E), we have the following equality

h_{t}(D \ E_{1}) = h_{t}(E \ E_{1}) ∪ h_{t}(D_{0}) = h_{t}(E \ E_{1}) ∪ D′_{0} ∪ (∪_{L∈G\{1}}L(h_{t}(E))) .

Combining h_{t}(E_{1}) ∩ (∪_{L∈G\{1}}L(h_{t}(E))) = ∅ with (4.7) and (4.8), we obtain (4.2), thus we finish the proof. □

Let t ∈ [0, 1] \ J. By property (a) and (b), one can see that μ_{t} ∈ M(ℂ, G) with ∥μ_{t}∥_{∞} ≤ \frac{4t}{C−5}. We define a quasi-analytic map g_{t} : D → ℂ̂ by g_{t} = w^{μ_{t}} ∘ h_{t}. Then, it follows that g_{t} is meromorphic, for μ[g_{t}] = 0 a.e. Since g_{t} ∘ L = χ_{t}(L) ∘ g_{t}, where χ_{t}(L) = w^{μ_{t}} ∘ L ∘ (w^{μ_{t}})^{−1} ∈ Möb for all L ∈ G, the Schwarzian derivative of g_{t} is G-automorphic. Now we shall estimate ∥S_{g_{t}}∥_{D}.

4.4. Lemma. Let A ≥ 5, then h_{t} = T_{t} on Δ′ for any Δ′ ∈ D_{A}(D) with Δ′ ∩ E_{1} ≠ ∅.

Proof of Lemma 4.4. Let Δ′ = B(c, r) ∈ D_{A}(D) such that Δ′ ∩ E_{1} ≠ ∅. Then it suffices to show that Δ′ ⊂ Δ. Suppose that Δ′ ⊈ Δ, then one and only one of the following happens:

(1) I ∩ ∂Δ′ ≠ ∅ and Q^{−1}({ζ : argζ − θ_{0} = π/3}) ∩ ∂Δ′ ≠ ∅.

(2) (∂Δ \ \overline{I}) ∩ ∂Δ′ ≠ ∅ and Q^{−1}(\{ζ : argζ − θ_{0} = 2π/3\}) ∩ ∂Δ′ ≠ ∅.

In both cases, there are two circular arcs γ_{1} and γ_{2} with the following properties.

(a) γ_{j} is a subarc of a circle centered at a_{j} with endpoints w_{0} and w_{1},

(b) γ_{j} ∩ ∂Δ′ ≠ ∅ for j = 1, 2, and

(c) γ_{1} intersects γ_{2} at w_{i} with angle π/3 for i = 0, 1.

Without loss of generarity, we may assume that |w_{0} − c| ≤ |w_{1} − c|. Since Δ′ ∈ D_{A}(D) and w_{0} ∈ ∂D we have

(4.9)  |w_{0} − c| ≥ Aρ.

Now let ζ_{j} be the intersection point of the circle C_{j} and the ray starting from a_{j} and passing through c, where we should note that c ≠ a_{j} by (4.9). Moreover by (b) and (4.9), we can see that ζ_{j} ∈ γ_{j} and |c − ζ_{j}| = dist(c, γ_{j}) < ρ, so we have that

(4.10)  |ζ_{j} − w_{0}| ≥ |w_{0} − c| − |ζ_{j} − c| > (A − 1)ρ.

Let m_{j} be the midpoint of γ_{j}, that is, m_{j} ∈ γ_{j} such that |m_{j} − w_{0}| = |m_{j} − w_{1}|. By the property (c), the elementary geometry tells us that ∠m_{1}w_{0}m_{2} = π/6. Set θ = ∠ζ_{1}w_{0}ζ_{2} ∈ (0, π), then we can verify that

(4.11)  θ ≤ π/6.

In fact, this is evident if the region W bounded by γ_{1} ∪ γ_{2} is covex. Next, we consider the case that W is not convex. We may assume that m_{2} is contained in the inner domain of the circle C_{1} (see Fig. 4.1).

Fig. 4.1

Let ψ_{j} = ∠m_{j}a_{j}ζ_{j} ∈ (0, π) for j = 1, 2. Then ψ_{1} = ∠m_{1}a_{1}c ≤ ∠m_{1}a_{2}c = ψ_{2}. Noting that ∠m_{j}w_{0}ζ_{j} = ψ_{j}/2, thus we have that θ = π/6 + (ψ_{1} − ψ_{2})/2 ≥ π/6.

Now the cosine formula says that

|ζ_{1} − ζ_{2}|^{2} = |ζ_{1} − w_{0}|^{2} + |ζ_{2} − w_{0}|^{2} − 2|ζ_{1} − w_{0}||ζ_{2} − w_{0}| cos θ

≥ |ζ_{1} − w_{0}|^{2} + |ζ_{2} − w_{0}|^{2} − 2|ζ_{1} − w_{0}||ζ_{2} − w_{0}| cos π/6 (by (4.11))

≥ (A − 1)^{2}ρ^{2}(2 − \sqrt{3}) (by (4.10)) .

On the other hand, |ζ_{1} − ζ_{2}| ≤ |ζ_{1} − c| + |ζ_{2} − c| < 2ρ, so we obtain that \sqrt{2 − \sqrt{3}}(A − 1)ρ < 2ρ, that is, A < 1 + 2/\sqrt{2 − \sqrt{3}} < 5. This contradicts the assumption that A ≥ 5. □

Proof of Proposition 4.1 (continued). Take A ≥ 1 such that A′ = \frac{A+A^{−1}}{2} ≥ 5. Let Δ′ ∈ D_{A}(D). If Δ′ ∩ (∪_{L∈G}L(E_{1})) = ∅, then h_{t}|′_{Δ} is a restriction of a Möbius transformation by construction. If Δ′ ∩ (∪_{L∈G}L(E_{1})) ≠ ∅, then

Δ′ ∩ L^{−1}(E_{1}) ≠ ∅ for some L ∈ G.

Since L(Δ′) ∩ E_{1} ≠ ∅, Lemma 4.4 yields that h_{t} = T_{t} on L(Δ′), hence h_{t} = L^{−1} ∘ h_{t} ∘ L = L^{−1} ∘ T_{t} ∘ L on Δ′. Consequently, h_{t}|_{Δ′} can be extended to a \frac{4t}{C−5}-qc map by Lemma 4.2. In any cases, h_{t}|_{Δ′} can be extended to a global k-qc map, where k = \frac{4t}{C−5}. Because w^{μ_{t}} is originally a k-qc map, g|_{Δ′} can be extended to a k′-qc map, where \frac{1+k′}{1−k′} = (\frac{1+k}{1−k})^{2}. Since k′ = \frac{2k}{1+k^{2}} ≤ 2k = \frac{8t}{C−5}, by Proposition 2.7, we obtain an estimate

∥S_{g_{t}}∥_{D} ≤ 96A^{2}k′ ≤ 96A^{2} · \frac{8t}{C − 5}.

By assumption on C, we have ∥S_{g_{t}}∥_{D} < ε if we take A = 10, for exmaple, in the above. Let f : ℍ → D be a Riemann mapping of D, then the above implies that S_{g_{t}∘f} belongs to the ball centered at S_{f} of radius ε in B_{2}(ℍ, Γ) where Γ = f^{−1}Gf. Remarking here the ball above is contained in S(Γ) by the assumption on D, we can deduce from Proposition 2.6 that there exists a global qc extension g̃_{t} : ℂ̂ → ℂ̂ of g_{t} such that g̃_{t} ∘L = χ_{t}(L)∘g̃_{t} on ℂ̂ for all L ∈ G. (Or, by utilizing a group equivariant version of the ultimate λ-lemma due to Slodkowski (cf. Earle-Kra-Krushkal [4]), we have directly this result.) Therefore h_{t} also has a global qc extension (w^{μ_{t}})^{−1} ∘ g̃_{t}, in particular, the next lemma follows.

4.5. Lemma. For any t ∈ [0, 1] \ J, h_{t} : D → ℂ̂ can be extended to a homeomorphism h̃_{t} of ℂ̂ commuting with G.

Now suppose that J = ∅, then the above lemma implies that h_{1} : D → ℂ̂ is injective, which contradicts the property (c). So, J must be nonempty. Let t_{0} be the infimum of J. Since J is open in (0, 1], we remark that t_{0} ∈ [0, 1]\J, in particular Lemma 4.5 is applicable to t_{0}. Let t_{n} (n = 1, 2,…) be a sequence in J converging to t_{0}. As t_{n} ∈ J,

(4.12)  h_{t_{n}}(E^{∘}) ∩ h_{t_{n}}(L_{n}(E^{∘})) ≠ ∅ for some L_{n} ∈ G \ {1}.

If there exists an L ∈ G\{1} such that L = L_{n} for infinitely many n’s. Then, by the property (d) and the fact that \overline{E} ⊂ Ω(G), (4.12) forces that h̃_{t_{0}}(\overline{E})∩h̃_{t_{0}}(L(\overline{E})) ≠ ∅, which is impossible because \overline{E} ∩ L(\overline{E}) = ∅ and h̃_{t_{0}} : ℂ̂ → ℂ̂ is injective. Thus, we may assume that L_{n} is a distinct sequence in G \ {1}. In this case, as is easily seen from (4.12), h̃_{t_{0}}(\overline{E}) ∩ Λ(G) ≠ ∅, which is a contradictory to the fact that \overline{E} ∩ Λ(G) = ∅ and that h̃_{t_{0}}(Λ(G)) = Λ(G).

Anyway, contradictions are deduced, which is caused by the falsity of the assumption that z_{1}, z_{2} ∈ D \ \overline{Δ}_{C} cannot be joined by any path in D \ \overline{Δ}. □

§5. The boundary of R_{0} = D/G is a disjoint union of quasi-analytic curves

Let Γ be an arbitrary Fuchsian group and φ ∈ IntS(Γ). We denote by R the (possibly disconnected) Riemann surface Ω(G)/G where G = χ^{φ}(Γ).

In this sectoin, we shall study the relative boundary ∂R_{0} of the subdomain R_{0} = D/G = f^{φ}(ℍ)/G in R. Our main aim here is to prove that ∂R_{0} is a disjoint union of quasi-analytic curves under the suitable hypothesis. To this end, we shall introduce a notion of the annular covering. Let α be a homotopically nontrivial simple closed curve in a hyperbolic Riemann surface R. Let π : ℍ → R be a holomorphic universal covering of R, and γ an element of the covering transformation group Γ_{0} < Möb of π which covers α, i.e., the terminal point of a lifting curve α̃ of α with respect to π equals to γ(z_{0}) where z_{0} is the initial point of α̃.

As is easily seen, the quotient Riemann surface ℍ/⟨γ⟩ is conformally equivalent to an annulus A = {z ∈ ℂ; c < |z| < 1}, where 0 ≤ c < 1 satisfies the relation cosh(π^{2}/ log c) = |trγ|/2, which is not so significant below.

Let π_{1} : ℍ → A be a holomorphic covering with the covering transformation group ⟨γ⟩. The induced holomorphic covering map q = q_{α} : A → R such that π = q ∘π_{1} is called an annular covering with respect to α. By construction, α̂ = π_{1}(α̃) is the unique closed lift of α, in other words, any other lift of α than α̂ is not closed. Using the tool above, we shall prove the following result.

5.1. Theorem. Let Γ be an arbitrary Fuchsian group of the second kind acting on the upper half plane ℍ. For φ ∈ IntS(Γ), let G = χ^{φ}(Γ), D = f^{φ}(ℍ), R = Ω(G)/G, and R_{0} = D/G. If a connected component α of ∂R_{0} is compact and contains no branch points of the natural projection p : Ω(G) → R, then α is a quasi-analytic curve and a one-sided boundary component.

Where, the quasi-analytic curve means the quasiconformal homeomorphic image of the circle.

5.2. Corollary. In particular, when Γ is a finitely generated, purely hyperbolic Fuchsian group of the second kind, the relative boundary ∂R_{0} of R_{0} is a disjoint union of finitely many quasi-analytic curves, and thus the conformal map F^{φ} : S_{0} → R_{0} induced by f^{φ} : ℍ → D, φ ∈ IntS(Γ), naturally extends to a homeomorphism F^{φ} : \overline{S_{0}} → \overline{R_{0}}, where S_{0} = ℍ/Γ and \overline{S_{0}} is its closure in S = Ω(Γ)/Γ, in other words, f^{φ} : ℍ → D naturally extends to a homeomorphism f^{φ} : \overline{ℍ} \ Λ(Γ) → \overline{D} \ Λ(G).

Proof of Corollary 5.2. By the hypothesis, R is compact, thus so ∂R_{0} is. Therefore the former part of the above assertion directly follows from Theorem 5.1. In particular, ∂S_{0} and ∂R_{0} consist of mutually disjoint simple closed curves, therefore the latter part can be deduced from the general version of the famous Carathéodory theorem.

In order to prove Theorem 5.1, first we need the next

5.3. Lemma. In addition to the hypothesis in Theorem 5.1, further assume that Γ has infinitely many elements. Then, there exists a certain exhausting sequence (R_{n})_{n=1}^{∞} of R_{0} with the following properties:

(0) each R_{n} is a subdomain of R_{0},

(i) R_{1} ⊂ R_{2} ⊂ …, ∪_{n=1}^{∞}R_{n} = R_{0}, (ii) each α_{n} = ∂R_{n} ∩ R_{0} is a homotopically non-trivial smooth simple closed

curve which is freely homotopic to any other α_{m}, and moreover if R is not biholomorphic to the punctured plane ℂ^{∗} then α_{n} is not homotopic to any puncture of R, (iii) every limit point of (α_{n})_{n=1}^{∞} is contained in α, and (iv) R_{0} \ R_{1} contains no branch points of p.

Proof of Lemma 5.3. First we shall show that α is open in ∂R_{0}, in other words, V ∩ ∂R_{0} = α for some open neighbourhood V of α in R. In fact, let T be a compact neighborhood of α with smooth boundary such that \overline{T} ⊂ ^{∘}Ω(G) and that ∂T ∩ ∂R_{0} = ∅, then it suffices to show that T_{0} := T ∩ R_{0} consists of finitely many components each of which has at most finitely many boundary components, i.e., T_{0} is a finite union of surfaces of finite topological types. Here, we should note that a Riemann surface X is of infinite topological type if and only if there is an infinite family of mutually disjoint, homotopically independent simple closed curves γ_{j} (j = 1, 2,…) in X. (We call γ_{1}, γ_{2},… is homotopically independent when each γ_{j} is not freely homotopic to any γ_{k} for k ≠ j nor null-homotopic.) Let C = {γ_{1},…, γ_{l}} be a finite family of mutually disjoint, homotopically independent simple closed curves in T_{0}. Then each γ_{j} is not null-homotopic in T, too. Otherwise, γ_{j} bounds a topological disk Δ in T, on the other hand, β = Δ ∩ ∂R_{0} ≠ ∅ for γ_{j} is not null-homotopic in T_{0}. Since Δ is simply connected domain and since Δ ⊂ ^{∘}R, p|Δ̃ : Δ̃ → Δ is biholomorphic, where Δ̃ is a component of p^{−1}(Δ). Thus, ∂Δ̃ ⊂ R_{0} separates β̃:= (p|Δ̃)^{−1}(β) ⊂ ∂D from ∂D \ β̃, this contradicts the connectedness of ∂D. So we have seen that γ_{j} is not null-homotopic in T.

Next suppose that γ_{j} is freely homotopic to γ_{k} for some k ≠ j in T. Then γ_{j} and γ_{k} bound a topological annulus (ring domain) A in T. Since γ_{j} is not freely homotopic to γ_{k} in T_{0}, β := A ∩ ∂R_{0} ≠ ∅. Moreover, β separates γ_{j} from γ_{k} in A. Indeed, if not, there exists an arc δ in A \ β connecting γ_{j} and γ_{k}, then A \ δ is simply connected and contains a nonempty subset β of ∂R_{0}, which leads to a contradiction in the same way as in the above.

Further suppose that γ_{j} is freely homotopic to γ_{k′} for k′ ≠ j, k in T. Then similarly γ_{j} and γ_{k′} bound a topological annulus A′ in T, and β′ := A′ ∩ ∂R_{0} separates γ_{j} from γ_{k′} in A′. Because ∂A ∩ ∂A′ = γ_{j}, we have A ⊂ A′, A′ ⊂ A or A ∩ A′ = ∅. In any cases, β ∪ β′ devides R_{0} into two pieces, which contradicts the connectedness of R_{0}.

Thus we conclude that each γ_{j} is freely homotopic in T to at most one other curve. So we can renumber C = {γ_{1},…, γ_{l}} so that γ_{j} is freely homotopic to γ_{s+t+j} in T for j = 1,…, s and that γ_{s+j} is not freely homotopic in T to any γ_{k} for k ≠ s + j, where integers s, t ≥ 0 satisfy 2s + t = l. In particular, {γ_{1},…, γ_{s+t}} is a family of mutually disjoint homotopically independent simple closed curves in T.

On the other hand, as is well-known, a Riemann surface X of finite topological type (g, 0, m) has at most 3g −3+2m mutually disjoint homotopically independent simple closed curves if (g, m) ≠ (0, 0), (0, 1), (1, 0). Therefore, l = 2s+t ≤ 2(s+t) ≤ 2(3g − 3 + 2m) if T is of topological type (g, 0, m), hence l is bounded. This means that T_{0} must be a finite union of surfaces of finite topological types. Thus, we have proved that compact component α of ∂R_{0} is always isolated in ∂R_{0}.

Let p_{0} : Ω(Γ) → S = Ω(Γ)/Γ be the natural projection, and F : S_{0} = p_{0}(ℍ) → R_{0} the conformal mapping induced by f = f^{φ} : ℍ → D. From the above observation, we can take a curve β̃ : [0, 1] → Ω(G) such that β̃([0, 1)) ⊂ D and β̃(1) ∈ p^{−1}(α). Let β̃_{0} := f^{−1} ∘ β̃: [0, 1) → ℍ, then we know that β̃_{0}(t) has a limit, say β̃_{0}(1), as t → 1 (see, for example, Pommerenke [12] Proposition 2.14).

Clearly β̃_{0}(1) ∈ ℝ̂ ∩ Ω(Γ), and let I be the connected component of ℝ̂ ∩ Ω(Γ) containing β̃_{0}(1). Now we shall show that the stabilizer H of I in Γ is generated by a hyperbolic or parabolic element. (Here we remark that the latter case happens only when Γ is generated by a single parabolic element.) Otherwise, H must be trivial, and so p_{0} is injective on a neighborhood V of I. Thus we can select a sequence (I_{n})_{n=1}^{∞} of simple arcs in V ∩ ℍ with the same end points as I such that E_{n} ⊃ E_{n+1} (n = 1, 2,…) and ∩_{n=1}^{∞}\overline{E}_{n} = \overline{I}, where E_{n} is the region bounded by I_{n} ∪ \overline{I}. First, we remark that the cluster set C = ∩_{n=1}^{∞}\overline{F (p_{0}(E_{n}))} is contained in ∂R_{0}. Since α is a compact isolated component of ∂R_{0}, we can choose a compact neighborhood W of α such that ∂W ∩ ∂R_{0} = ∅. Here, F (p_{0}(I_{n})) is not relatively compact in R and ∅ ≠ β̃_{0}([0, 1)) ∩ I_{n} ⊂ p_{0}^{−1}(F^{−1}(W ∩ R_{0})) for sufficiently large n, therefore we may pick a point Q_{n} ∈ F (p_{0}(I_{n})) ∩ ∂W for large n. Because ∂W is compact, we may assume that Q_{n} converges to a point Q ∈ ∂W. On the other hand, Q ∈ C ⊂ ∂R_{0}, which contradicts the fact ∂W ∩ ∂R_{0} = ∅. Thus, we have proved that H = Stab_{Γ} (I) is generated by a hyperbolic or parabolic element γ_{0}. Let (I_{n})_{n=1}^{∞} be a sequence of circular arcs (or horocycles) in ℍ with the same end points as I such that E_{n} ⊃ E_{n+1} (n = 1, 2,…), and ∩_{n=1}^{∞}E_{n} = ∅, where E_{n} is the region bounded by I_{n} ∪ \overline{I}. Then R_{n} := R_{0} \ F (p_{0}(E_{n} ∪ I_{n})) is a desired exhaustion of R_{0} for sufficiently large n. □

Proof of Theorem 5.1. We shall prove in the case that Γ is an infinite group. (When Γ is finite, R = ℂ̂/Γ = ℂ̂ by the Riemann-Hurwitz formula, thus the proof is much easier.) Moreover, we may assume that R is hyperbolic. In fact, even if not, taking a sufficiently small closed disk E ⊂ R such that E ∩ \overline{R_{0}} = ∅ (see Corollary 3.2), we have only to replace R by the hyperbolic surface R′ = R \ E. Therefore we assume that R is hyperbolic and Γ is an infinite group in the sequel.

Let R_{n}, α_{n} (n = 1, 2,…) be as in Lemma 5.2 and π : A = {c < |z| < 1} → R be an annular covering with respect to α_{n}. (Remark that for freely homotopic curves, we can take the same annular covering.)

Denote by α̂_{n} the unique closed lift of α_{n} via π and let W_{n} ⊂ U = {z; |z| < 1} be a Jordan domain bounded by α̂_{n}. Compositing the map z ↦ c/z to π if necessary, we may assume that W_{n} ⊂ W_{n+1} for all n ≥ 1, here we should note that c > 0 by Lemma 5.3 (ii). Set W = ∪_{n=1}^{∞}W_{n} and α̂ = A ∩ ∂W. Clearly, π(α̂) ⊂ α. We shall show that the restricted map π|α̂ : α̂ → α is a homeomorphism.

Let w_{0} be an arbitrary point of α. By Lemma 3.6, there exists a connected open neighborhood V of w_{0} such that V ∩R_{0} is connected, V ∩R_{1} = ∅, V ∩(∂R_{0} \α) = ∅ and that V is contained in a topological disk Ṽ in R. Now we claim that there exists a (unique) component V_{0} of π^{−1}(V) with the following conditions:

(1) V_{0} ∩ α̂ ≠ ∅,

(2) (π^{−1}(V) \ V_{0}) ∩ α̂ = ∅,

(3) V_{0} ∩ π^{−1}(α) = V_{0} ∩ α̂.

To prove the above claim, we first remark that π_{1} := π|_{W \\overline{W}_{1}} : W \ \overline{W}_{1} → R_{0} \ \overline{R}_{1} is biholomorphic. Let V_{0} be a component of π^{−1}(V) containing a point of π_{1}^{−1}(V ∩ R_{0}). Then V_{0} ∩ (W \ \overline{W}_{1}) = π_{1}^{−1}(V ∩ R_{0}) because V ∩ R_{0} is connected.

In particular, (π^{−1}(V) \ V_{0}) ∩ (W \ \overline{W}_{1}) = ∅, thus the condition (2) follows. Since π_{0} := π|_{V_{0}} : V_{0} → V is biholomorphic, we have

π^{−1}(α) ∩ V_{0} = π_{0}^{−1}(α ∩ V) = π_{0}^{−1}(∂R_{0} ∩ V) = ∂π_{0}^{−1}(R_{0} ∩ V) ∩ V_{0}
= ∂π_{1}^{−1}(R_{0} ∩ V) ∩ V_{0} = ∂(V_{0} ∩ (W \ \overline{W}_{1})) ∩ V_{0}
= ∂W ∩ V_{0} = α̂ ∩ V_{0},

where we use the fact that V_{0} ∩ W_{1} = ∅ thus (3) is proved. By the condition (3), ζ_{0} := π_{0}^{−1}(w_{0}) ∈ V_{0} ∩ π^{−1}(α) ⊂ α̂, which implies (1), and as a by-product we obtain that α = π(α̂). Moreover, condition (3) yields the injectivity of π|α̂ : α̂ → α, therefore we have proved that π|α̂ : α̂ → α is a homeomorphism.

We shall continue the proof of Theorem 5.1. Since π is a conformal map in a neighborhood of α̂, it is sufficient to prove that α̂ is a quasi-circle.

Here we mention a lemma which is a direct conclusion of the Koebe distorsion therem: \frac{a}{(1+a)^{2}} ≤ |f(z)| ≤ \frac{a}{(1−a)^{2}} for |z| = a < 1 if f is univalent in the unit disk and f(0) = f′(0) − 1 = 0.

5.4. Lemma. Let f be a conformal mapping from a disk Δ into ℂ. For K > 1, let a ∈ (0, 1) satisfy the equation K = \frac{(1−a)^{2}}{4a} . Then there exists a disk Δ̃ such that f(Δ_{a}) ⊂ Δ̃ and Δ̃_{K} ⊂ f(Δ).

First, we shall show the following lemma.

5.5. Lemma. The domain W constructed above is linearly connected.

Proof of Lemma 5.5. Since α̂ = ∂W ⊂ A is compact, there exists a positive constant δ < diamW with the following property: if a disk Δ with diamΔ ≤ δ has a nonempty intersection of α̂, then Δ ⊂ A \ (π^{−1}(∂R_{0}) \ α̂) and π is injective in Δ.

Take 0 < a < 1 satisfing the equation B = \frac{(1−a)^{2}}{4a} where B is the constant which appeared in Propostion 3.1. Let Δ be an arbitrary disk. In order to prove linear connectedness, we shall consider several cases.

Case 1. Δ ∩ α̂ = ∅.

In this case Δ ⊂ W or Δ ∩ W = ∅, so any two points in Δ ∩ W can be always joined by a path in Δ ∩ W.

Case 2. Δ ∩ α̂ ≠ ∅ and diamΔ < aδ.

Then Δ_{1/a} ⊂ A, Δ_{1/a} ∩ (π^{−1}(∂R_{0}) \ α̂) = ∅, and π is injective in Δ_{1/a} by the choice of δ.

Let V be a connected component of p^{−1}(π(Δ_{1/a})). Remark that p is injective in V because p is a covering map and π(Δ_{1/a}) is simply connected. Now we apply Lemma 5.4 to the conformal mapping f = (p|_{V})^{−1} ∘ π : Δ_{1/a} → V. Thus we know that f(Δ) ⊂ Δ̃ and Δ̃_{B} ⊂ f(Δ_{1/a}) = V ⊂ Ω(G) for some disk Δ̃. Then, any two points in f(Δ) ∩ D can be joined by a path in Δ̃_{B} ∩ D, so in V ∩ D by Proposition 3.1. Since f(Δ_{1/a} ∩ W) = f(Δ_{1/a}) ∩ D = V ∩ D by the condition Δ_{1/a} ∩ (π^{−1}(∂R_{0}) \ α̂) = ∅, any two points in Δ ∩ W = f^{−1}(f(Δ) ∩ D) can be joined by a path in Δ_{1/a} ∩ W.

Case 3. Δ ∩ α̂ ≠ ∅ and diamΔ ≥ aδ. If we set M = 1 + \frac{2diamW}{aδ} (> 1/a), then Δ_{M} ⊃ W. Thus any two points in Δ ∩ W can be joined by a path in Δ_{M} ∩ W = W.

Hence, in any cases, arbitrary two points in Δ ∩ W can be joined by a path in Δ_{M} ∩ W, thus W is linearly connected. □

By Lemma 5.5, in particular, W is a bounded Jordan domain. We should remark that we have proved Lemma 5.5 without results in §4. As for Jordan domains, we mention the next elementary fact, which follows from the uniform continuity of a homeomorphic parametrization S^{1} → ∂W and its inverse map.

5.6. Lemma. Let W be a bounded Jordan domain in ℂ and η any positive number. Then there exists a positive constant δ with the property that, for any cross cut γ of W with diamγ ≤ δ, it holds that

min{diamW_{1}, diamW_{2}} ≤ η,

where W_{1} and W_{2} are two components of W \ γ.

By the help of Lemma 5.6, secondly we shall show the following lemma.

5.7. Lemma. W is a John domain.

Proof of Lemma 5.7. Let 0 < a < 1 such that C = \frac{(1−a)^{2}}{4a} , where C is the constant in the statement of Proposition 4.1. Since α ⊂ A is compact, we can choose a positive η > 0 so small that any disk Δ with \overline{Δ} ∩ α̂ ≠ ∅ and diamΔ ≤ 4η should satisfy Δ ⊂ A \ \overline{W}_{1}, π|_{Δ} is injective, and π(Δ) ⊂ p(^{∘}Ω(G)).

By Lemma 5.6, for sufficiently small δ (0 < δ ≤ η) the following holds: if a disk Δ with diameter< δ has nonempty intersection with α̂ = ∂W, then diam(W \W_{0}) < η where W_{0} is the connected component of W \ \overline{Δ} containing W_{1}.

Now let Δ be an arbitrary disk centered at z_{0}.

Case 1. Δ ∩ α̂ = ∅.

In this case, arbitrary two points in W \ \overline{Δ} can be joined by a path in W \ \overline{Δ}. Case 2. Δ ∩ α̂ ≠ ∅ and diamΔ < aδ. Since \overline{Δ}_{1/a}∩α̂ ≠ ∅ and diamΔ_{1/a} < δ(≤ η), Δ_{1/a}∩W_{1} = ∅ and diam(W \W_{0}) < η

where W_{0} is the component of W \\overline{Δ} containing W_{1}. Set Δ′ = \{z ∈ ℂ; |z−z_{0}| < 2η\}, then clearly \overline{W} \ W_{0} ⊂ Δ′ and \overline{Δ}_{1/a} ⊂ Δ′. Since diamΔ′ = 4η, Δ′ must satisfy that Δ′ ⊂ A \ \overline{W}_{1}, π|_{Δ′} is injective, and that π(Δ′) ⊂ p(^{∘}Ω(G)).

Let V be a connected component of p^{−1}(π(Δ′)), then p|_{V} : V → π(Δ′) is a biholomorphic map for π(Δ′) is simply connected and π(Δ′) ⊂ p(^{∘}Ω(G)). Since π is injective in Δ′, f := (p|_{V})^{−1} ∘ π : Δ′ → V is a conformal homeomorphism. Now we apply Lemma 5.4 for f|_{Δ_{1/a}}, then we obtain that there exists a disk Δ̃ such that f(Δ) ⊂ Δ̃ and Δ̃_{C} ⊂ f(Δ_{1/a}).

Fix a point z_{1} ∈ W_{0} ∩ (Δ′ \ \overline{Δ_{1/a}}). Suppose that there exists a point z_{2} in (W \ W_{0}) \ \overline{Δ_{1/a}}(⊂ Δ′).

We denote by T the component of W \ \overline{Δ} containing z_{2}, so clearly T ∩ W_{0} = ∅ and \overline{T} ⊂ Δ′. Since \overline{T} ⊂ Δ′, ∂f(T) ⊂ V ∩ ∂f(Δ′ ∩ W \ \overline{Δ}) = V ∩ ∂(D \ f(\overline{Δ})), so we obtain that ∂f(T) ⊂ ∂(D \ f(\overline{Δ})), which implies that f(T) is a connected component of D \ f(\overline{Δ}).

On the other hand, w_{j} = f(z_{j}) ∈ f(Δ′ ∩ W \ \overline{Δ_{1/a}}) = V ∩ D \ f(\overline{Δ_{1/a}}) ⊂

D \ \overline{Δ̃_{C}} (j = 1, 2), thus Proposition 4.1 guarantees that w_{1} and w_{2} are connected by a path in D \ \overline{Δ̃} ⊂ D \ f(\overline{Δ}). Since f(T) is a comonent of D \ f(\overline{D}) and w_{2} ∈ f(T), w_{1} must be in f(T) too, i.e., z_{1} ∈ T, which is a contradiction. Therefore we conclude that (W \ W_{0}) \ \overline{Δ_{1/a}} = ∅, i.e., W \ \overline{Δ_{1/a}} ⊂ W_{0}, which implies that any two points in W \ \overline{Δ_{1/a}} are joined by a path in W_{0} ⊂ D \ \overline{Δ}.

Case 3. Δ ∩ α̂ ≠ ∅ and diamΔ ≥ aδ.

Let M = 1 + \frac{2diamW}{aδ} (> 1/a), then Δ_{M} ⊃ W. Thus, trivially it holds that any two points in W \ \overline{Δ}_{M} can be joined by a path in W \ \overline{Δ}.

In any cases, we have proved that any two points in W \ \overline{Δ_{M}} can be joined by a path in W \ \overline{Δ}. Now the proof is completed. □

Combining Lemma 5.5 and Lemma 5.7 with Theorem 2.9, we can immediately obtain Thereom 5.1.

§6. Existence of a topological involution of R w.r.t. ∂R_{0}

In this section, chiefly we shall be concernd with the following result, which is a crucial part of the proof of our main theorem.

6.1. Theorem. Let G be a Schottky group of rank N(≥ 0) and p : Ω(G) → R := Ω(G)/G the natural projection. Suppose that R_{0} is a proper subdomain of R such that D = p^{−1}(R_{0}) is a simply connected domain and that ∂R_{0} consists of mutually disjoint simple closed curves. Let f : ℍ → D be a Riemann mapping of D, Γ the Fuchsian group defined by Γ = f^{−1}Gf and χ : Γ → G the isomorphism defined by χ(γ) ∘ f = f ∘ γ for all γ ∈ Γ.

Then f can be extended to a homeomorphism f̃: ℂ̂ → ℂ̂ satisfying that χ(γ) ∘ f̃ = f̃∘ γ. In particular, D = f̃(ℍ) is a Jordan domain.

To prove Thorem 6.1, the following proposition comprises the key step.

6.2. Proposition. Under the same hypothesis of Theorem 6.1, there exists a topological involution J of R with respect to ∂R_{0}, which can be lifted. More precisely, J : R → R is an orientation-reversing homeomorphism such that J(R_{0}) ∩ R_{0} = ∅, J|_{∂R_{0}} = id_{∂R_{0}}, J ∘J = id_{R} and there exists a homeomorphism j : Ω(G) → Ω(G) which satisfies that p ∘ j = J ∘ p, j|_{∂D\Λ(G)} = id_{∂D\Λ(G)}, j ∘ j = id_{Ω(G)} and that j ∘ L = L ∘ j for all L ∈ G.

6.3. Remark. By Proposition 6.4 below, the above lift j : Ω(G) → Ω(G) naturally extends to a self-homeomorphism of ℂ̂, and then j|_{∂D} = id_{∂D}.

Proof of Theorem 6.1. First remark that the conformal map f : ℍ → D naturally extends to a homeomorphism f : \overline{ℍ} \ Λ(Γ) → \overline{D} \ Λ(G) by Corollary 5.2.

Let j : Ω(G) → Ω(G) be the involution satisfying the statement in Proposition 6.2, then we define f̃: Ω(Γ) → Ω(G) by the rule

f̃ = \{ \begin{matrix}f & on \overline{ℍ} \ Λ(Γ) \\ j ∘ f ∘ j_{0} & on ℂ̂ \ \overline{ℍ}, \end{matrix}

where j_{0} denotes the conjugation map z ↦ z̄.

Since the limit sets of Schottky groups are totally disconnected, it suffices to prove the following purely topological proposition, which essentially follows from the fact that for any plane domain Ω, the Kerékjártó-Stoïlow compactification of Ω is homeomorphic to the quotient space \overline{Ω}/ ∼ obtained by collapsing each boundary component of Ω in ℂ̂ to one point (with the quotient topology). □

6.4. Proposition. Let E_{1}, E_{2} be totally disconnected compact subsets of ℂ̂, and set Ω_{i} = ℂ̂ \ E_{i} for i = 1, 2. Then, any homeomorphism f : Ω_{1} → Ω_{2} (if exists) uniquely extends to a homeomorphism f̃: ℂ̂ → ℂ̂.

Sketch of the proof of Proposition 6.4. Let z ∈ E_{1}. By the Zoretti theorem (cf [N:p.109]), we can take a nesting sequence α_{n}, n = 1, 2,… of Jordan curves in Ω_{1} shrinking to the one point z. Then {f(α_{n})} is a nesting sequence of the Jordan curves shrinking to the exactly one point, say, w. Thus we can assign f̃(z) as the limit point w of f(α_{n}) for z ∈ E_{1}. Defining f̃ = f on Ω_{1} = ℂ̂ \ E_{1} we have a homeomorphic extension f̃: ℂ̂ → ℂ̂ of f. □

Now, our only task is to prove Proposition 6.2! As a preparation, we now state general results about relations between geometric properties of covering spaces and algebraic ones of the fundamental groups. The proof of these results is straightforward, so we shall omit it. Suppose that p : Ω → R is a normal (=Galois) covering between Riemann surfaces (or, more generally, manifolds). Let R_{0} be a subdomain of R and ι : R_{0} → R denote the inclusion map. Pick a point a_{0} from R_{0} and z_{0} from Ω with p(z_{0}) = a_{0}. The inclusion map ι : R_{0} → R naturally induces a homomorphism ι_{∗} : π_{1}(R_{0}, a_{0}) → π_{1}(R, a_{0}). Let λ : π_{1}(R, a_{0}) → G be the monodromy homomorphism with respect to z_{0}, where G is a covering transformation group of p : Ω → R. Namely, g = λ([α]) for g ∈ G and [α] ∈ π_{1}(R, a_{0}) if and only if the final point of the lift α̃ of α with initial point z_{0} coincides with g(z_{0}).

6.5. Proposition.

(1) Any one of the following implies the others.

(1a) Each component of p^{−1}(R_{0}) is simply connected, (1b) λ ∘ ι_{∗} is injective, (1c) ι_{∗} is injective and ι_{∗}(π_{1}(R_{0}, a_{0})) ∩ ker λ = 1.

(2) Any one of the following implies the others.

(2a) p^{−1}(R_{0}) is connected

(2b) λ ∘ ι_{∗} is surjective, (2c) π_{1}(R, a_{0}) = ker λ · ι_{∗}(π_{1}(R_{0}, a_{0})).

6.6. Corollary. The following conditions are equivalent to each other.

(a) p^{−1}(R_{0}) is a simply connected domain

(b) λ ∘ ι_{∗} : π_{1}(R_{0}, a_{0}) → G is an isomorphism,

(c) ι_{∗} : π_{1}(R_{0}, a_{0}) ↪ π_{1}(R, a_{0}) is an embedding and π_{1}(R, a_{0}) = ker λ ⋊ π_{1}(R_{0}, a_{0}) (semi-direct product).

We now return to the case we have considered, i.e., Ω = Ω(G) and p : Ω → R = Ω(G)/G is a Schottky covering. Since p^{−1}(R_{0}) = D is a simply connected domain, by the Corollary 6.6, it turns out that the homomorphism ι_{∗} : π_{1}(R_{0}, a_{0}) → π_{1}(R, a_{0}) has a cross-section s : π_{1}(R, a_{0}) → π_{1}(R_{0}, a_{0}), e.g., s = (λ ∘ ι_{∗})^{−1} ∘ λ. Through the natural homomorphisms h : π_{1}(R_{0}, a_{0}) → H_{1}(R, ℤ) and h_{0} : π_{1}(R_{0}, a_{0}) → H_{1}(R_{0}, ℤ) (Hurewicz homomorphisms), homomorphisms ι_{#} : H_{1}(R_{0}, ℤ) → H_{1}(R, ℤ), s_{#} : H_{1}(R, ℤ) → H_{1}(R_{0}, ℤ) of the first homology groups are induced (see the diagram (6.1)). Here we should remark that the kernel of the Hurewicz homomorphism is the commutator subgroup of the fundamental group.

(6.1)  \begin{matrix}π_{1}(R_{0}, a_{0}) & ⟶^{ι_{∗}} & π_{1}(R, a_{0}) & ⟶^{s} & π_{1}(R_{0}, a_{0}) \\ ^{h_{0}}↓ & & ^{h}↓ & & ^{h_{0}}↓ \\ H_{1}(R_{0}, ℤ) & ⟶^{ι_{#}} & H_{1}(R, ℤ) & ⟶^{s_{#}} & H_{1}(R_{0}, ℤ) \end{matrix}

Since s_{#} ∘ ι_{#} = (s ∘ ι_{∗})_{#} = id_{H_{1}(R_{0},ℤ)}, we obtain the following

6.7. Proposition. The homomorphism ι_{#} : H_{1}(R_{0}, ℤ) → H_{1}(R, ℤ) induced by the inclusion map ι : R_{0} → R is injective.

To prove Proposition 6.2, we must show the next lemma.

6.8. Lemma. The exterior R_{1} = R \ \overline{R_{0}} of R_{0} is homeomorphic to R_{0}.

Proof. First, we recall that the Schottky group G is a free group of finite rank N. Next, let (g, 0, m) be the topological type of R_{0}, i.e., R_{0} is a genus g compact surface (without punctures) with m mutually disjoint closed topological disks removed. As is well-known, the fundamental group π_{1}(R_{0}, ∗) is a free group of rank 2g + m − 1. Now, Corollary 6.6 yields that π_{1}(R_{0}, ∗) is isomorphic to G, thus N = 2g + m − 1.

For a while, suppose that R_{1} is connected. Let (g′, 0, m′) be the topological type of R_{1}, then clearly m = m′ and g + g′ + m − 1 = N since R is of genus N compact surface and R = \overline{R_{0}} ∪ \overline{R_{1}}, thus g′ = g which asserts that R_{0} and R_{1} are of same topological type (g, 0, m). So, we have only to prove the connectedness of R_{0}.

Let c_{1},…, c_{m} be boundary loops of R_{0} which are consistently oriented. Denote by C_{1},…, C_{m} the homology class of c_{1},…, c_{m} in R_{0}, respectively. Here, notice that H_{1}(R_{0}, ℤ) has a generators C_{1},…, C_{m} over ℤ with the sole relation C_{1}+⋯+C_{m} =

0. Suppose that R_{1} was disconnected. Let R_{1}′ be a connected component of R_{1}, then R_{1}′ had some boundary components, say, c_{1},…, c_{l} with 1 ≤ l < m. Let C_{i}′ = ι_{#}(C_{i}) be the homology class of c_{i} in R, then C_{1}′ + ⋯ + C_{l}′ = 0 since C_{1}′ + ⋯ + C_{l}′ = [c_{1} + ⋯ + c_{l}] = [−∂R_{1}′] = 0 in H_{1}(R, ℤ). By Proposition 6.4, ι_{#} : H_{1}(R_{0}, ℤ) → H_{1}(R, ℤ) is injective, so we obtain an extra relation C_{1} + ⋯ + C_{l} = 0 which is a contradiction. □

6.9. Proposition. Under the same hypothesis of Theorem 6.1, we have a system of disjoint simple closed curves {ℓ_{1},…, ℓ_{N}} on R, which satisfies the following conditions:

(a) p : Ω(G) → R is the highest covering which lifts ℓ_{i} to a closed loop for i = 1,…, N,

(b) each ℓ_{i} transversely intersects ∂R_{0} at exactly two points, and

(c) R_{0}′ = R_{0}\∪_{i=1}^{N}ℓ_{i} and R_{1}′ = R_{1}\∪_{i=1}^{N}ℓ_{i} are both simply connected domains, where R_{1} = R \ \overline{R_{0}}.

Proof. Let (g, 0, m) be the topological type of R_{0}, then N = 2g + m − 1 as we have seen before. Let c_{1},…, c_{m} be boundary comopnents of R_{0}, or equivalently, of R_{1}. Then there exist mutually disjoint simple closed arcs u_{1},…, u_{N} on \overline{R_{1}} as follows (see Fig. 6.1):

(i) whole u_{i} is contained in R_{1} except its endpoints, (ii) for i = 1,…, m − 1, u_{i} connects c_{m} with c_{i}, (iii) for i = m,…, m + 2g − 1 = N, u_{i} starts from c_{m} and returns to c_{m}, and (iv) R_{1}′ := R_{1} \ ∪_{i=1}^{N}u_{i} is connected.

Fig. 6.1

In order to advance the proof, we require several lemmas as the following.

6.10 Lemma. R_{1}′ is simply connected.

Proof of Lemma 6.10. Let Σ_{i} denote the compact sufrace (with boundary) which is obtained by cutting \overline{R_{1}} along u_{1} ∪ ⋯ ∪ u_{i}. (We set Σ_{0} = \overline{R_{1}}.) Let χ(Σ) be the Euler characteristic of a compact surface Σ with boundary, that is

χ(Σ) = #{vertices} − #{edges} + #{faces}

for an arbitrary triangulation of Σ. Further remark that χ(Σ_{g,m}) = 2 − 2g − m, where Σ_{g,m} represents a compact orientable surface of genus g with m boundaries. Because Σ_{i} is obtained by cutting Σ_{i−1} along u_{i}, we have χ(Σ_{i}) = χ(Σ_{i−1}) + 1 for i = 1,…, N. Summarizing these equalities, we obtain that

χ(Σ_{N}) = χ(Σ_{0}) + N = 2 − 2g − m + N = 1.

Therefore R_{1}′ must be homeomorphic to the unit disk. □

6.11. Lemma. p^{−1}(R_{1}) is simply connected. In particular, the restriction of p to any component of p^{−1}(R_{1}) is a universal covering of R_{1}.

Proof of Lemma 6.11. p^{−1}(R_{1}) is the comlement of the connected set \overline{D}, therefore the above statement is clear. □

Let R̃_{1}′ be connected component of p^{−1}(R_{1}′). By Lemmas 6.10 and 6.11, the restriction map p|R̃_{1}′ : R̃_{1}′ → R_{1}′ is bijective. Let Σ̃ denote the closure of R̃_{1}′.

By the way, we can take a closed curve w_{i} : [0, 1] → R_{1} which starts from one side of u_{i} and ends to another side of u_{i}, and which satisfies that w_{i}((0, 1)) ⊂ R_{1}′. Here, we should remark that for i = 1,…, m − 1, w_{i} is freely homotopic to the boundary curve c_{i}. Let w̃_{i} : [0, 1] → Ω(G) be the unique lift in Σ̃ of w_{i} and L_{i} be the unique element of G with w̃_{i}(1) = L_{i}(w̃_{i}(0)). Clearly w_{i} is a homotopically nontrivial loop in R_{1}, so that L_{i} ≠ 1 by virture of Proposition 6.5.(1) and Lemma 6.11. Let u_{i}^{+} be the unique lift of u_{i} which passes through w̃_{i}(0), and set u_{i}^{−} = L_{i}(u_{i}^{+}). Then, p^{−1}(u_{i}) ∩ Σ̃ = u_{i}^{+} ∪ u_{i}^{−} and u_{i}^{+} ∩ u_{i}^{−} = ∅ (see Fig. 6.2).

Fig. 6.2

Let, for i = 1,…, N, v̂_{i}^{+} (resp. v̂_{i}^{−}) be the geodesic curve in ℍ which connects images a_{i}^{+}, b_{i}^{+} (resp. a_{i}^{−}, b_{i}^{−}) of endpoints of u_{i}^{+} (resp. u_{i}^{−}) under f^{−1}; that is, v̂_{i}^{±} is the semi-circle in ℍ perpendicularly intersecting ∂ℍ = ℝ̂ at a_{i}^{±} and b_{i}^{±}. Set v_{i}^{±} = f(v̂_{i}^{±}) and ℓ_{i}^{±} = u_{i}^{±}∪v_{i}^{±} for i = 1,…, N. Then it is obvious that ℓ_{i}^{±} are Jordan curves in Ω(G) and L_{i}(ℓ_{i}^{+}) = ℓ_{i}^{−} by construction. Furthermore, ℓ_{1}^{+}, ℓ_{1}^{−},…, ℓ_{N}^{−} are mutually disjoint. Indeed, if some ℓ_{i}^{±} intersects other ℓ_{k}^{±} (possibly with different signature), since v_{i}^{±} transversally intersects v_{k}^{±} at most one point while u_{i}^{±}∩u_{k}^{±} = ∅, the closed curve ℓ_{i}^{±} separates f(a_{k}^{±}) from f(b_{k}^{±}), this is a contradiction.

We shall denote by Extℓ_{i}^{±} the component of ℂ̂ \ ℓ_{i}^{±} which contains R_{1}′. By definition, L_{i}(Extℓ_{i}^{+}) ∩ Extℓ_{i}^{−} = ∅ for i = 1,…, N, and so the subgroup G_{0} :=< L_{1},…, L_{N} > of G generated by L_{1},…, L_{N} is a Schottky group of the same rank N and W := ∩_{i=1}^{N}(Extℓ_{i}^{+} ∩ Extℓ_{i}^{−}) is a fundamental domain of G_{0}. Here we should note that W ∩ ∂D = ∂Σ \ (u_{1}^{+} ∪ u_{1}^{−} ∪ ⋯ ∪ u_{N}^{−}) consists of finite number of lifts of some part of boundary curves c_{1},…, c_{m}, particularly, W ∩ ∂D ⊂ Ω(G). As clearly W ∩ D ⊂ Ω(G), we have Ω(G_{0}) ⊂ Ω(G). On the other hand, trivially Ω(G_{0}) ⊃ Ω(G), thus we conclude that Ω(G_{0}) = Ω(G). Since Ω(G_{0})/G_{0} and R = Ω(G)/G are of the same genus N, the Riemann-Hurwitz theorem implies that the induced covering map Ω(G_{0})/G_{0} = Ω(G)/G_{0} → Ω(G)/G must be univalent, in other words, G = G_{0}. Therefore ℓ_{i} := p(ℓ_{i}^{±}) for i = 1,…, N have all the disired properties, by the construction above. For example, R_{0}′ is known to be simply connected domain by the proof of Lemma 6.10. The proof of Proposition 6.9 is now completed. □

Proof of Proposition 6.2. Let ℓ_{1},…, ℓ_{N} be a system of mutually disjoint simple closed curves on R as in Proposition 6.9. Set R_{0}′ = R_{0}\∪_{i=1}^{N}ℓ_{i} and R_{1}′ = R_{1}\∪_{i=1}^{N}ℓ_{i} are both simply connected domains, where R_{1} = R \ \overline{R_{0}}. Let W be a component of p^{−1}(R \ ∪_{i=1}^{N}ℓ_{i}), then W is a 2N-ply connected domain with boundary curves ℓ_{1}^{+}, ℓ_{1}^{−},…, ℓ_{N}^{−}, where ℓ_{i}^{±} is a closed lift of ℓ_{i}. We denote by L_{i} the unique element of G which maps ℓ_{i}^{+} to ℓ_{i}^{−}. Let R̃_{k}′ be the connected component of p^{−1}(R_{k}′) which is contained in W and let Σ_{k} be the closure of R̃_{k}′ for k = 0, 1. Then, Σ_{k} seems as a 4N-gon with 2N sides ℓ_{i}^{±} ∩Σ_{k} (i = 1,…, N) and 2N sides which are lifts of some part of ∂R_{0}. Since the order of ℓ_{i}^{±}∩Σ_{0} in ∂Σ_{0} well corresponds to the one of ℓ_{i}^{±}∩Σ_{1} in ∂Σ_{1}, we can obtain an orientation-reversing homeomorphism J̃_{0} : ∂Σ_{0} → ∂Σ_{1} with the following two properties:

(1) J̃_{0} = id on Σ_{0} ∩ ∂D = Σ_{1} ∩ ∂D,

(2) J̃_{0} ∘ L_{i} = L_{i} ∘ J̃_{0} on Σ_{0} ∩ ℓ_{i}^{+} for i = 1,…, N.

As Σ_{0} and Σ_{1} are Jordan domains, we can extend J̃_{0} to a homeomorphism J̃_{1} : Σ_{0} → Σ_{1} with J̃_{1}|_{∂Σ_{0}} = J̃_{0}. By Property (1), we can further extend J̃_{1} to an orientation-reversing homeomorphism J̃_{2} : \overline{W} → \overline{W} by the rule:

J̃_{2} = \{ \begin{matrix}J̃_{1} & on Σ_{0} \\ J̃_{1}^{−1} & on Σ_{1}. \end{matrix}

Noting that J̃_{2} ∘ L_{i} = L_{i} ∘ J̃_{2} on ℓ_{i}^{+} for i = 1,…, N, we extend J̃_{2} to a homeomorphism J̃: Ω(G) → Ω(G) as the following:

J̃ = L ∘ J̃_{2} ∘ L^{−1} on L(\overline{W}) for all L ∈ G.

By construction, J̃ satisfies the following conditions:

(a) J̃(D) ∩ D = ∅ and J̃ = id on ∂D ∩ Ω(G),

(b) J̃∘ J̃ = id_{Ω(G)},

(c) J̃∘ L = L ∘ J̃ for any L ∈ G.

Because of (c), J̃ descends to a homeomorphism J : R → R with J ∘ p = p ∘ J̃, which has the desired properties. □

In order to complete the proof of Theorem 2.1, we have only to show the following

6.12. Theorem. Let G be a Schottky group of rank N(≥ 0) and p : Ω(G) → R := Ω(G)/G the natural projection. Suppose that R_{0} is a proper subdomain of R such that D = p^{−1}(R_{0}) is a simply connected domain and that ∂R_{0} consists of mutually disjoint quasi-analytic curves. Let f : ℍ → D be a Riemann mapping of D, Γ the Fuchsian group defined by Γ = f^{−1}Gf and χ : Γ → G the isomorphism defined by χ(γ) ∘ f = f ∘ γ for all γ ∈ Γ.

Then f can be extended to a quasiconformal homeomorphism f̃: ℂ̂ → ℂ̂ satisfying that χ(γ) ∘ f̃ = f̃∘ γ. In particular, D = f̃(ℍ) is a quasidisk.

Proof. By Theorem 6.1, f can be extended to a homeomorphism f̃_{0} : ℂ̂ → ℂ̂ satisfying that χ(γ) ∘ f̃_{0} = f̃_{0} ∘ γ. Let F_{0} denote a homeomorphism from S = ℍ/Γ onto R induced by f̃_{0} : Ω(G) → Ω(G). Since F_{0}|_{S_{0}} : S_{0} → R_{0} is conformal and ∂R_{0} consists of mutually disjoint quasi-analytic curves, F_{0}|_{S_{0}} can be extended to a quasiconformal mapping F_{1} on a neighborhood U of \overline{S}_{0} such that F_{1} is diffeomorphic in U \ \overline{S}_{0}. Here, we used the well-known fact that a quasiconformal map from the unit disk Δ onto a quasidisk can be extended to a quasiconformal self-map of the whole plane ℂ̂ whose restriction to ℂ̂ \ \overline{Δ} is real-analytic (for example, by the Ahlfors-Weill extension). Then, it is easily seen that there exists a diffeomorphism F : S \ \overline{S}_{0} → R \ \overline{R}_{0} which coincides with F_{1} on some neighborhood of ∂S_{0} and which is homotopic to F_{0}|_{S\\overline{S}_{0}} by a homotopy that fixes ∂S_{0} pointwise.

We extend F to a homeomorphism from S onto R by defining F = F_{0} on \overline{S}_{0}, then F becomes quasiconformal and F ≃ F_{0} in S. Since F_{0} can be lifted to f̃_{0}, F also can be lifted to a homeomorphism f̃: Ω(Γ) → Ω(G) such that χ(γ) ∘ f̃ = f̃∘ γ for all γ ∈ Γ. By Proposition 6.4, f̃ naturally extends to a homeomorphism of ℂ̂ (denoted f̃ also). Since F : S → R is quasiconformal, f̃ is also quasiconformal on Ω(Γ) ⊂ ℂ̂ \ ℝ̂. On the other hand, ℝ̂ is a quasiconformally removable set, thus f̃ must be quasiconformal on the whole plane. The proof is finished. □

Proof of Theorem 2.1. Let φ ∈ IntS(Γ). Then G = χ^{φ}(Γ) is a Schottky group by Lemma 2.3 and Maskit’s characterization theorem, and R_{0} := f^{φ}(ℍ)/G is a proper subdomain of R := Ω(G)/G with quasi-analytic boundary by Corollary 5.2. Now Theorem 6.12 implies that f^{φ} can be extended to a Γ -compatible quasiconformal homeomorphism f̃ of ℂ̂, which means that φ ∈ T (Γ). Thus we have proved now that IntS(Γ) ⊂ T (Γ). □

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