ON THE SCHLICHT PROJECTIVE STRUCTURES ON

COMPACT BORDERED RIEMANN SURFACES WITH

Toshiyuki Sugawa

Abstract. In the present session board, the main result in [Sug] is explained, that is, IntS(Γ) = T (Γ) for any Fuchsian group uniformizing a compact bordered 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 derivatives of all the Γ -equivariant schlicht holomorphic functions and T (Γ) is the (Bers model of) Teichmüller space of Γ. The paper [Sug] of the author also contains 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} and Q(ℍ, Γ) be the space consisting all the holomorphic quadratic differntial φ on ℍ for Γ, i.e., φ is holomorphic function on ℍ satisfying a condtion φ ∘ γ˙(γ′)^{2} = φ for γ ∈ Γ. 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, by the Kraus-Nehari theorem, it turns out that S(Γ) is a bounded closed subset of the complex Banach space B_{2}(ℍ, Γ) = {φ ∈ Q(ℍ, Γ)|∥φ∥_{ℍ} < ∞, where ∥φ∥_{ℍ} = sup_{z∈ℍ}(2Imz)^{2}|φ(z)|.

Furthermore, we define a subset T (Γ) of S(Γ) by {φ = S_{f} ∈ S(Γ)|f(ℍ) is a quasidisk }. This subset T (Γ) is called (the Bers model of) the Teichmüller space of the Fuchsian group Γ, and is known to be a bounded connected open subset of B_{2}(ℍ, Γ).

It is an intersting topic to investigate how 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 still a difficult problem to decide whether \overline{T (Γ)} = S(Γ) for a finitely generated Fuchsian group Γ of the first kind.

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 result in this article is the following

Typeset by AMS-TEX

Theorem 1.

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

Remark.

For a Fuchsian group Γ acting on ℍ, the following conditions 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 Γ is a free group of rank 2g + m − 1.

§2. Proof of the main theorem

In the beginning, we consider a general Fuchsian group Γ of the second kind. Let φ = S_{f} ∈ IntS(Γ) and set D = f(ℍ). We denote χ : Γ → Möb the monodromy homorphism for f, that is, f ∘ γ = χ(γ) ∘ f for γ ∈ Γ. First, it is easy to see that χ is type preserving monomorphism (cf. [16] or [Sug]) and G = χ(Γ) is a Kleinian group acting on D.

Essentially, it is sufficient to prove that D is a quasidisk. Now, we recall a characterization of the quasidisk due to Gehring [G1]: a bounded simply connected domain D is a quasidisk if and only if there exists a constant A > 1 such that

(1) for an arbitrary disk Δ, any two points in D ∩ Δ can be connected by a path in D ∩ Δ_{A}, and

(2) for an arbitrary disk Δ, any two points in D \ Δ_{A} can be connected by a path in D \ Δ, where Δ_{A} denotes {|z − z_{0}| < Ar} if Δ = {|z − z_{0}| < r}.

First, in the line along the above characterization, we can show the following propositions.

Propostion 1.

Suppose that a Kleinian group G acts on a simply connected plane domain D ⊂ ℂ of hyperbolic type. And suppose that 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.

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