ON THE TEICHMÜLLER SPACES OF FUCHSIAN GROUPS OF SCHOTTKY TYPE AND THE SCHWARZIAN DERIVATIVES OF UNIVALENT FUNCTIONS

Toshiyuki Sugawa Department of Mathematics, Faculty of Science, Kyoto University

§1. The main result.

Let Γ be a Fuchsian group acting on the upper half plane ℍ. We denote by B_{2}(Γ) the Banach space of all the holomorphic function φ on ℍ which satisfies the functional equation (φ ∘ γ)(γ′)^{2} = φ for all γ ∈ Γ, with finite norm ∥φ∥ = sup_{z∈ℍ} |φ(z)|(Im z)^{2}. We shall consider the following subsets of B_{2}(Γ) :

S(Γ) = {φ ∈ B_{2}(Γ) : ∃univalent functionf on ℍ with S_{f} = φ},
T (Γ) = {S_{f} ∈ S(Γ) : f extends to a (Γ -compatible) qc-map of Ĉ},

where S_{f} denotes the Schwarzian derivative of f difined as follows: S_{f} = (f′′/f′)′ − \frac{1}{2}(f′′/f′)^{2}.

It is known that S(Γ) is closed and T (Γ) is open in B_{2}(Γ). T (Γ) is called (the Bers model of) the Teichmüller space of Γ. It is an interesting problem how near T (Γ) is to S(Γ). For a cofinite Fuchsian group (i.e., finitely generated Fuchsian group of the first kind) Γ, the statement \overline{T (Γ)} = S(Γ) is equivalent to the Bers conjecture: every B-group is obtained as a boundary group of Teichmüller space. (This conjecture is still now unsolved.)

On the other hand, for any Fuchsian group Γ of the second kind, it is known that \overline{T (Γ)} ⫋ S(Γ) (cf. [G2], [Sug]).

But a weaker statement that T (Γ) = IntS(Γ) is proved for some cases ( [G1: Γ = 1 ], [Shiga: cofinite Γ ]). The main result of this article is the validity of the

Typeset by AMS-TEX 2 TOSHIYUKI SUGAWA

above statement for all Fuchsian groups of Schottky type, where a Fuchsian group Γ is called Schottky type in this article, if Γ is a Schottky group simultaneously, in other words, Γ uniformizes a topologically finite Riemann surface of genus g with m holes, where m ≥ 1. Also, the Schottky type Fuchsian group can be characterized as the finitely generated, purely hyperbolic Fuchsian group of the second kind.

Main Theorem. IntS(Γ) = T (Γ) for any Fuchsian group Γ of Schottky type.

§2. Sketch of proof.

Let Γ be a Fuchsian group of Schottky type. Then, the quotient surface S_{0} = ℍ/Γ is a topologically finite Riemann surface of genus g with m holes and its double S = Ω(Γ)/Γ is a compact Riemann surface of genus N = 2g + m − 1, where Ω(Ω) ⊂ Ĉ denotes the region of discontinuity of Γ. Let φ ∈ IntS(Γ) and F : ℍ → Ĉ be a holomorphic map such that S_{f} = φ. By the Γ -automorphy of φ, G = F Γ F^{−1} is a subgroup of Möb which acts on D = f(ℍ). Since φ is an interior point of S(Γ), it turns out that G is purely loxodromic. Since G(≅ Γ) is a free group of finite rank, Maskit’s characterization theorem tells us that G is also a Schottky group of rank N = 2g + m − 1. So, the quotient surface R = Ω(G)/G is a compact genus N surface. Let p_{0} : Ω(Γ) → S and p : Ω(G) → R be the natural projections. Set R_{0} = p(D) = D/G, which is isomorphic to S_{0} = ℍ/Γ by the conformal map f induced by F : ℍ → D. We shall investigate the way of embedding R_{0} ↪ R. Now, the proof of Main Theorem devides into several steps.

Step 1. ∂R_{0} consists of mutually disjoint m simple closed curves.

This step needs a localization of Gehring’s method [G1]. In this step, essential is the fact that φ is an interior point of S(Γ).

Step 2. There exists a self-homeomorphism h of R with the following properties: (i) h ∘ h = id_{R}, (ii) h|_{∂R_{0}} = id_{∂R_{0}}, (iii) h(R_{0}) ∩ R_{0} = ∅,

ON THE TEICHMÜLLER SPACES OF FUCHSIAN GROUPS OF SCHOTTKY TYPE AND THE SCHWARZIAN DERIVATIVES OF UNIVALENT FUNCTIONS3

(iv) there exits a homeomorphism H : Ω(G) → Ω(G) such that p ∘ H = h ∘ p on Ω(G) and that H = id_{∂D∩Ω(G)}.

This step is covered by rathar algebraic arguments. For example, the following lemma is utilized.

Lemma (general property of the normal coverings).

Suppose that p : (Ω, z_{0}) → (R, a_{0}) is a normal covering between (connected) pointed manifolds. Let R_{0} be a subdomain of R such that a_{0} ∈ R_{0} and ι : R_{0} → R denote the inclusion map. Then ι naturally induces the homomorphism ι_{∗} : π_{1}(R_{0}, a_{0}) → π_{1}(R, a_{0}). Let λ : π(R, a_{0}) → G be the lifting 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}) iff the final point of the lift α̃ of α with initial point z_{0} coincides with g(z_{0}). Then, the followings hold.

(i) Each component of p^{−1}(R_{0}) is simply connected ⟺ λ ∘ ι_{∗} is injective.

(ii) p^{−1}(R_{0}) is connected ⟺ λ ∘ ι_{∗} is surjective.

In particular, if p^{−1}(R_{0}) is a simply connected domain, then 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).

First of all, we can naturally extend f to a homeomorphism f : \overline{S_{0}} → \overline{R_{0}} by Step

1. Further, by use of Step 2, we can extend f to a homeomorphism f̃: S → R in the following way.

f̃ = \{ \begin{matrix}f on \overline{S_{0}}, \\ h ∘ f ∘ j on S \ \overline{S_{0}}, \end{matrix}

where j denotes the involution map S → S induced by conjugation J(z) = z̄. By construction, f̃ can be lifted, that is, there exists a homeomorphism F̃ : Ω(Γ) → Ω(G) such that p ∘ F̃ = f̃∘ p_{0}. By purely topological arguments, it turns out that F̃ can be naturally extended to a homeomorphism F̃ : Ĉ → Ĉ. In particular, it is known that D is an image of ℍ under the self-homeomorphism F̃ of Ĉ, so D is a Jordan domain.

Step 3. ∂R_{0} is a disjoint union of quasi-analytic curves.

4 TOSHIYUKI SUGAWA

Here, the “quasi-analytic curve” means the quasiconformal image of a circle. For the proof of Step 3, we need just more delicate arguments than in Step 1. By the way, one can prove the following Proposition. Let S and R be compact Riemann surfaces and S_{0} ⊂ S, R_{0} ⊂ R be subdomains with quasi-analytic boundaries. Suppose that f̃ : S → R is an orientation-preserving homeomorphism such that f̃(S_{0}) = R_{0} and the restriction map f̃|_{S_{0}} : S_{0} → R_{0} is quasiconformal. Then, there exists a quasiconformal map f̃_{1} : S → R which is homotopic to f̃ and f̃_{1} = f̃ on R_{0}.

By virture of this proposition, we can choose a quasiconformal f̃ : S → R as the extension of f. Then, a topological extension F̃ : Ĉ → Ĉ of a lift of f̃ is quasiconformal on Ω(Γ), so F̃ : Ĉ → Ĉ is a quasiconformal self-homeomorphism since Λ(Γ) = Ĉ \ Ω(Γ) ⊂ ℝ̂ is a quasiconformally removable set. Therefore D = F̃(ℍ) is a quasi-disk, the proof is completed.

References

[G1].F.W.Gehring, Univalent functions and the universal Teichmüller space, Comment. Math.

Helv. 52 (1977), 561–572. [G2].F.W.Gehring, Spirals and the universal Teichmüller space, Acta Math. 141 (1978), 99–113. [Shiga]. H.Shiga, Characterization of quasi-disks and Teichmüller spaces, Tôhoku Math. J. 37 (1985),

541–552. [Sug]. T.Sugawa, On the Bers conjecture for Fuchsian groups of the second kind, J. Math. Kyoto

Univ. 32 (1992), 45–52.