BERS EMBEDDING OF THE TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS

YOHEI KOMORI AND TOSHIYUKI SUGAWA

Abstract. In this note, we present a method of computing monodromies of projective structures on a once-punctured torus or a four-times punctured sphere. This leads to an algorithm numerically visualizing the shape of the Bers embedding of a one-dimensional Teichmüller space. As a by-product, the value of the accessary parameter of a fourtimes punctured sphere will be calculated in a numerical way as well as generators of a Fuchsian group uniformizing it. Finally, we observe the relation between the Schwarzian differential equation and Heun’s differential equation in this special case.

1. Introduction

The Teichmüller space was, as its name tells us, invented by O. Teichmüller around 1940. At the early stage, however, quasiconformal mappings had not been developed enough, and therefore, his work was difficult to understand for most people. In the late 1950’s, L. V. Ahlfors and L. Bers established a firm foundation to the theory of quasiconformal mappings, which lead to understanding the Teichmüller space well and further development of it. Among their contributions, the idea of simultaneous uniformization due to Bers enabled us to embed the Teichmüller space of a Riemann surface as a bounded domain in the normed vector space of holomorphic quadratic differentials on it. In the latter half of the last century, many authors have revealed a number of remarkable properties of Teichmüller spaces. Most of them, however, are concerned with the internal geometry of the spaces. On the one hand, in connection with the deformation of Kleinian groups, the boundary of the Bers embedding (so-called the Bers boundary) of a Teichmüller space has been investigated. In spite of much efforts to this direction, the shape of the Bers embedding is still mysterious at present because of its highly transcendental nature. At least, the supporting evidence that the Bers boundary is fairly irregular has been reported by, for instance, [20] and [36]. We also note that the Teichmüller space of finite dimension greater than one is not a symmetric domain. Actually, the analytic automorphism group of it is the homomorphic image of the mapping class group on the surface under the discontinuous action on the Teichmüller space (Royden [32]). As Bers stated in his survey [3], even one-dimensional Teichmüller spaces are exciting one’s curiosity enough.

Date: April 23, 2002.

1991 Mathematics Subject Classification. Primary 30F60; Secondary 30F40, 34A20.

Key words and phrases. Teichmüller space, Bers embedding, monodromy, pleating ray, accessary parameter, bending coordinates, once-punctured torus .

The second named author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 9740056. Part of the present research was carried out during the second named author’s visit to the University of Helsinki under the exchange program of scientists between the Academy of Finland and the JSPS. .

The first attempt to compute a boundary point of a one-dimensional Teichmüllder space has been made by R. M. Porter [30] and L. Keen and et al [12], [11], where they succeeded in drawing several real rays of trace functions ending at cusps in the case of the once-punctured square torus. The present authors are wondering why nobody developed their approach further despite the remarkable progress of computing technology in recent years. We note here that in 1980’s C. McMullen, D. Mumford and D. Wright obtained computer pictures of the Maskit embedding of the Teichmüller space of a once-punctured torus. Since the Maskit embedding had been expected to resemble the Bers embedding in shape, they made several conjectures on the Bers embedding (see, for instance, [23]). Moreover, the Bers boundary of a one-dimensional Teichmüllder space was asserted to be a Jordan curve by C. McMullen around 1990. A proof for the assertion in the literature has been given by Y. Minsky as a corollary of his complete description of the space of discrete faithful representations of once-punctured torus groups [25].

Furthermore, using Minsky’s Pivot Theorem, H. Miyachi [26] proved recently that each cusp of the one-dimensional Teichmüller space is really cusp-shaped, namely, to the endpoint of each rational pleating ray one can attach a subdomain like cardioid in the Bers embedding.

On the other hand, B. Maskit [22] (see also [19]) discovered projective structures such that the monodromies are quasi-Fuchsian groups but the developing maps are not univalent (for details of the terminology here, see the next section). Those exotic projective structures have recently been studied intensively by many authors (see, e.g., [24] , [34]). However, the configuration of components of the interior of exotic projective structures on a surface is still far from being clear.

In this note, we will give a practical method of computing the monodromy of a given projective structure by solving a linear ordinary differential equation of the second order which is associated with the Schwarzian differential equation. Employing the bending coordinates, which were developed by L. Keen and C. Series (see, e.g., [14]) and by C. McMullen [24], we explain how we can generate a picture of the Bers embedding with sufficient precision.

Our method can also be applied to obtain the whole picture of exotic components of the discreteness locus of projective structures if available is an algorithm of determining the discreteness of a given Möbius group generated by two elements with parabolic commutator. This will be a main subject of our forthcoming paper [16] with M. Wada and Y. Yamashita.

This note is organized as follows. Section 2 is devoted to basic definitions of the Teichmüller spaces and related notions. In Section 3, we summarize known facts about commensurable Fuchsian groups with given Fuchsian group Γ uniformizing a once-punctured torus. The explicit description of coverings between corresponding quotient surfaces will be helpful to transfer the computation on the once-punctured torus to that on the commensurable four-times punctured sphere in Section 4. In this way, we can avoid to use elliptic functions in actual computations. This method can also be used to obtain exact values of the Poincaré density of the once-punctured square torus [35]. Section 5 gives an enumeration of free homotopy classes of (non-peripheral, unoriented) simple closed curves on the topological once-punctured torus by using the notion of Farey neighbours. In Section 6 we describe a natural “polar coordinates,” called the bending coordinates,

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 3

of the Teichmüller space of a once-punctured torus by following C. McMullen [24]. Our main theorem states that some sort of radial ray, called a rational pleating ray, can be described as a real locus of the entire function representing the trace of monodromy of a simple closed curve corresponding to the ray. Using results in the preceding sections, we can draw numerically all rational pleating rays in a recursive way. As a by-product, we can compute the value of the accessory parameter and a standard generator pair of a Fuchsian group uniformizing a given once-punctured torus as well. In the final section, we investigate the relation between the Schwarzian differential equation in the four-times punctured sphere and Heun’s differential equation. We will observe that a simple change of indeterminants transforms the former one to the latter. Thus the monodromy can be computed also by solving Heun’s differential equation instead of the Schwarzian differential equation.

Acknowledgements. The authors should like to express deep thanks to Yoshitsugu Takei and Shun Shimomura, who both pointed out the connection with Huen’s differential equation. They also wish to thank Caroline Series and Hideki Miyachi for enjoyable conversations about Theorem 6.3 during the first author’s stay at Warwick.

2. Basic definitions

Let Γ be a Fuchsian group acting on the unit disk 𝔻, in other words, Γ is a discrete subgroup (possibly with torsion elements) of the analytic automorphism group Aut(𝔻) of 𝔻. Note that Aut(𝔻) is Möbius conjugate to Aut(ℍ) ≅ PSL(2, ℝ), where ℍ denotes the upper half plane {z ∈ ℂ; Im z > 0}. We denote by Q(𝔻, Γ) the set of holomorphic quadratic differentials φ(ζ)dζ^{2} for Γ on the unit disk, i.e., φ are holomorphic in 𝔻 and satisfy the functional equation φ ∘ γ · (γ′)^{2} = φ for each γ ∈ Γ. (We often identify an element φ(ζ)dζ^{2} in Q(𝔻, Γ) with the corresponding coefficient φ(ζ).)

A quadratic differential φ(ζ)dζ^{2} ∈ Q(𝔻, Γ) is called bounded if the norm

∥φ∥_{𝔻} = sup_{ζ∈𝔻}(1 − |ζ|^{2})^{2}|φ(ζ)|

is finite. We denote by B_{2}(𝔻, Γ) the complex Banach space consisting of all bounded holomorphic quadratic differentials for Γ on 𝔻 with the norm defined above.

It is well known that B_{2}(𝔻, Γ) is finite dimensional if and only if the Fuchsian group Γ is cofinite, in other words, finitely generated and of the first kind. If Γ is torsion-free and uniformizes a Riemann surface X of finite analytic type (g, n), where g is the genus of X and n is the number of punctures of X with 2g−2+n > 0, then dim B_{2}(𝔻, Γ) = 3g−3+n. In particular, dim B_{2}(𝔻, Γ) = 1 if and only if the signature (g, n) is equal to (1, 1) or (0, 4). An element of B_{2}(𝔻, Γ) is sometimes called a cusp form for a cofinite Γ. Each cusp form admits at most a simple pole at a puncture or a branch point of the quotient Riemann surface (or orbifold).

We now explain a well-known method enabling us to construct a locally univalent meromorphic function on 𝔻 for which the Schwarzian derivative coincides with the given holomorphic quadratic differential. For a good account including the historical background, we refer the reader to [9, Chap. 10]. Let φ ∈ Q(𝔻, Γ) be given. Then consider the following linear homogeneous ordinary differential equation of the second order:

2η′′ + φη = 0 on 𝔻. (2.1)

Since 𝔻 is simply connected, a unique solution η exists on 𝔻 for the given initial data η(0) = a and η′(0) = b. We denote by η_{1} and η_{2} the special solutions of (2.1) determined by η_{1}(0) = 0, η′_{1}(0) = 1 and η_{2}(0) = 1, η′_{2}(0) = 0. Then the Wronskian η′_{1}η_{2} − η_{1}η′_{2} is identically 1. We note that the ratio f := η_{1}/η_{2} has Taylor expansion in the form

f(ζ) = ζ + a_{3}ζ^{3} + ⋯

near the origin. A simple calculation shows that the Schwarzian derivative S_{f} = (f′′/f′)′− (f′′/f′)^{2}/2 is equal to the given differential φ. Hereafter, we denote by f_{φ} the meromorphic function f = η_{1}/η_{2} just described above and call it the (normalized) developing map of φ when we regard φ as a projective structure on the surface 𝔻/Γ.

Here we briefly recall facts about the analytic automorphism group Aut(𝔻) of the unit disk 𝔻. Let SU(1, 1) be the subgroup of SL(2, ℂ) consisting of all matrices U in the form

(\begin{matrix}u & v \\ v̄ & ū \end{matrix}) with |u|^{2} − |v|^{2} = 1. Then the canonical projection SU(1, 1) → PSU(1, 1) =

SU(1, 1)/{±I} gives the natural homomorphism P : SU(1, 1) → Aut(𝔻), in other words, P (U)(z) = (uz + v)/(v̄z + ū). It is well known that SU(1, 1) is conjugate with SL(2, ℝ) in SL(2, ℂ).

We define the pullback U_{−1/2}^{∗}η of η under U by

U_{−1/2}^{∗}η(z) = (v̄z + ū)η(P (U)(z)).

Here we remark that the function v̄z + ū is a global branch of (P (U)′)^{−1/2}.

A direct computation shows that U_{−1/2}^{∗}η satisfies differential equation (2.1) with P (U)_{2}^{∗}φ = φ ∘ P (U) · (P (U)′)^{2} instead of φ. In particular, when P (U) ∈ Γ, we see that U_{−1/2}^{∗}η_{j} is again a solution of (2.1) for j = 1, 2, and hence, we can write

(2.2)  U_{−1/2}^{∗}η_{1} = aη_{1} + bη_{2}
U_{−1/2}^{∗}η_{2} = cη_{1} + dη_{2},

where a, b, c and d are constants, because η_{1} and η_{2} form a basis of the vector space of solutions of (2.1). (The relations in (2.2) can be regarded as “additive lows” for the solutions.) We note that we can deduce ad − bc = 1 from the fact that the Wronskian of U_{−1/2}^{∗}η_{1}, U_{−1/2}^{∗}η_{2} is identically 1.

Let Γ̃ be the inverse image P^{−1}(Γ) of Γ under the projection P. We now define the map

χ̃_{φ} : Γ̃ → SL(2, ℂ) by χ̃_{φ}(γ̃) = (\begin{matrix}a & b \\ c & d \end{matrix}) , where a, b, c and d are the above constants for

U = γ̃ ∈ Γ̃. It is easy to see that χ̃_{φ} is actually a group homomorphism for each φ. By the holomorphic dependence of solutions on the coefficients, the correspondence φ ↦ χ̃_{φ}(γ̃) is a holomorphic map from Q(𝔻, Γ) to SL(2, ℂ), where the space Q(𝔻, Γ) is endowed with the Fréchet space structure determined by the uniform convergence on compacta. Note that χ_{0} is nothing but the identity. Furthermore, for a γ̃ ∈ Γ̃, we set

Trγ̃(φ) = tr χ̃_{φ}(γ̃) = a + d. (2.3)

The trace function Trγ̃ is also holomorphic in φ and plays an important role in the present investigation.

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 5

Taking the ratio of relations (2.2), we obtain

f_{φ} ∘ γ = χ_{φ}(γ) ∘ f_{φ}

on 𝔻, where γ = P (γ̃) and χ_{φ}(γ) is the Möbius transformation induced by χ̃_{φ}(γ̃), i.e., (az+b)/(cz+d). The element χ_{φ}(γ) is well-defined because (az+b)/(cz+d) is independent of the choice of γ̃ ∈ P^{−1}(γ) ⊂ Γ̃. Thus, the map χ_{φ} : Γ → PSL(2, ℂ) is also a group homomorphism and will be called the monodromy homomorphism of Γ associated with φ ∈ Q(𝔻, Γ).

By virtue of the following theorem, we can see that considering only the monodromy homomorphism would lose nothing in investigation of deformations of Γ.

Theorem A. Let Γ be a cofinite Fuchsian group. For φ, ψ ∈ B_{2}(𝔻, Γ), the coincidence χ_{φ} = χ_{ψ} on Γ implies φ = ψ.

This result traces back to Poincaré for the case when Γ uniformizes a compact Riemann surface. The generalization above is due to I. Kra [17].

The (Bers embedded) Teichmüller space of a Fuchsian group Γ will be denoted by T (Γ) and defined as the set of those holomorphic quadratic differentials φ ∈ Q(𝔻, Γ) for which the developing maps f_{φ} are univalent and admit quasiconformal extensions to ℂ̂ compatible with the action of Γ, where a quasiconformal extension F of f_{φ} is said to be compatible with Γ if for each γ ∈ Γ there exists a Möbius transformation γ′ such that F ∘ γ = γ′ ∘ F on ℂ̂.

Remark 2.1. For simplicity, we have adopted a slightly different definition of the Teichmüller space from the usual one. Our space T (Γ) here means, in the standard definition, the Teichmüller space of the Fuchsian group acting on the exterior of the unit disk, or, more intuitively, the Teichmüller space of the mirror image of the surface 𝔻/Γ. Therefore, the reader should refer with special care to another paper. For example, our standard generator pair θ_{0}, θ_{∞} (see Section 3 below) should be replaced by θ_{0}^{−1}, θ_{∞}.

When Γ is the trivial group 1, T (1) is called the universal Teichmüller space. By definition, we see that T (Γ_{1}) ⊂ T (Γ_{2}) if Γ_{1} ⊃ Γ_{2}, in particular, the universal Teichmüller space contains all Teichmüller spaces of Fuchsian groups.

The Kraus-Nehari theorem says that ∥φ∥_{𝔻} ≤ 6 if f_{φ} is univalent. Hence T (Γ) is a bounded subset of B_{2}(𝔻, Γ). We also note that the Ahlfors-Weill theorem implies {φ ∈ B_{2}(𝔻, Γ); ∥φ∥_{𝔻} < 2} ⊂ T (Γ). It is well known also that T (Γ) is a connected open subset of B_{2}(𝔻, Γ). For details, see, for instance, [10]. By the existence of a conformally natural extension operator from quasisymmetric homeomorphisms of S^{1} to quasiconformal ones of 𝔻 (see Tukia [37] or Douady-Earle [4]), we can see that

T (Γ) = T (1) ∩ B_{2}(𝔻, Γ) (2.4)

and that T (Γ) is contractible.

More generally, we consider the set K(Γ) of those quadratic differentials φ ∈ B_{2}(𝔻, Γ) for which the monodromy images χ_{φ}(Γ) are discrete in PSL(2, ℂ). The following result is due to H. Shiga [33].

Theorem B. The Bers embedded Teichmüller space T (Γ) coincides with the connected component of Int K(Γ) which contains the origin.

Moreover, each element of Int K(Γ) is known to correspond to an isomorphism onto a quasi-Fuchsian group (Shiga and Tanigawa [34]). Thus, the trace function Trγ̃ for a hyperbolic element γ̃ ∈ Γ̃ satisfies

Trγ̃(Int K(Γ)) ⊂ ℂ \ [−2, 2].

By virtue of Theorem B, we would see the shape of the Bers embedding if we could mark the grids contained in K(Γ) for sufficiently fine mesh in the space B_{2}(𝔻, Γ). This approach is out of our scope here, however, it will be a basis of our forthcoming paper [16] in the case when 𝔻/Γ is a once-punctured torus. It also has the advantage that one can draw a picture (up to translation) even if one does not know about the value of the accessary parameter (see Section 5). To do that, we need an algorithm determining the discreteness of a given Möbius group. Such an algorithm, however, is available only for the special case when the group is generated by two elements with parabolic commutator at present.

Our method here will rely on the internal geometry of the Teichmüller space, and hence, it might be applicable to even higher dimensional cases.

3. Commensurability relations

All one-dimensional Teichmüller spaces are mutually conformally equivalent because they are all simply connected. In particular, the Teichmüller space of a once-punctured torus is biholomorphic to that of a four-times punctured sphere. There is, however, an intrinsic relation between them as is described below.

Let Γ be a Fuchsian group uniformizing a once-punctured torus X. It is a basic fact that Γ has free generators θ_{0} and θ_{∞} with the properties: (i) both are hyperbolic; (ii) the commutator [θ_{0}, θ_{∞}] is parabolic; and (iii) the (signed) intersection number θ_{0} · θ_{∞} equals 1. We call such (θ_{0}, θ_{∞}) a standard generator pair of Γ.

If a standard generator pair is given, then there exists a unique complex number τ with Im τ > 0 satisfying the following property: The quotient surface (ℂ \ {m + nτ; m, n ∈ ℤ})/L_{τ} = (ℂ \ L_{τ})/L_{τ}, where L_{τ} is the lattice group generated by 1 and τ over ℤ, is conformally equivalent to X in such a way that the curves t + (1 + τ)/2 and tτ + (1 + τ)/2 in ℂ parametrized by t ∈ [0, 1] correspond to θ_{0} and θ_{∞}, respectively.

We call τ the Teichmüller parameter of the (marked) Fuchsian group Γ with standard generator (θ_{0}, θ_{∞}). This parameter τ is known to give a global analytic coordinate of the Teichmüller space of Γ onto the upper half plane ℍ (see [10]).

Now consider the four-times punctured torus Z given as the quotient (ℂ \ {(m + nτ)/2; m, n ∈ ℤ\})/L_{τ} = (ℂ \ \frac{1}{2}L_{τ})/L_{τ}. Then the mapping z ↦ 2z induces a four-sheeted (unbranched) covering Z → X. Note that the covering group of Z → X is isomorphic to ℤ_{2} ⊕ ℤ_{2}.

On the other hand, the conformal involution z ↦ −z induces a 2-sheeted (unbranched) covering Z → Y, where Y is the four-times punctured sphere described as ℂ̂\{e_{0}, e_{1}, e_{2}, e_{3}}, where e_{0} = ℘(0) = ∞, e_{1} = ℘(1/2), e_{2} = ℘(τ/2), e_{3} = ℘((1 + τ)/2), in terms of the Weierstrass ℘-function with period lattice L_{τ}. We note that e_{1} + e_{2} + e_{3} = 0.

Since the covering Z → X is Galois, we obtain a natural homomorphism from Γ to Aut(Z). We denote by Γ_{0} the kernel of this homomorphism. Then Γ_{0} is a Fuchsian group uniformizing Z. By the above remark, we note that Γ/Γ_{0} ≅ ℤ_{2} ⊕ ℤ_{2}.

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 7

The composition of the universal covering 𝔻 → 𝔻/Γ_{0} ≅ Z and the above projection Z → Y is a universal covering of the four-times punctured sphere Y, whose covering group will be denoted by Γ′. Then Γ′ is an extension of Γ_{0} such that Γ′/Γ_{0} ≅ ℤ_{2}.

Similarly, by the involution z ↦ −z of ℂ \ {m + nτ; m, n ∈ ℤ}, we obtain a 2-sheeted covering X → W, where W is a Riemann orbifold of signature (0; 2, 2, 2, ∞), in fact, W is realized as the complex plane with ramification of order 2 at the three points e_{1}, e_{2} and e_{3} via the covering map [z] ↦ ℘(z). This orbifold appears also as the quotient space of Y by the action of the group G(e_{0}, e_{1}, e_{2}, e_{3}) described below.

Let a_{0}, a_{1}, a_{2} and a_{3} be distinct points in the Riemann sphere ℂ̂ and set Ω = ℂ̂ \ {a_{0}, a_{1}, a_{2}, a_{3}}. Unless these points are in a special position, we cannot permute these in an arbitrary way by a Möbius transformation. The domain Ω, however, has a special kind of symmetry always.

For example, there exists a unique Möbius transformation A_{1} swapping a_{0} and a_{1} and swapping a_{2} and a_{3}. In fact, we take the Möbius transformation A sending a_{0}, a_{1}, a_{2} to a_{1}, a_{0}, a_{3}, respectively. Since A swaps a_{0} and a_{1}, the transformation A must be an involution, i.e., A^{2} = id. Therefore we conclude A(a_{3}) = A^{−1}(a_{3}) = a_{2}.

Similarly, we can take Möbius transformations A_{2} and A_{3} such that A_{2}(a_{0}, a_{1}, a_{2}, a_{3}) = (a_{2}, a_{3}, a_{0}, a_{1}), A_{3}(a_{0}, a_{1}, a_{2}, a_{3}) = (a_{3}, a_{2}, a_{1}, a_{0}).

We now have the group, denoted by G(a_{0}, a_{1}, a_{2}, a_{3}), formed by the elements id, A_{1}, A_{2} and A_{3}, which acts on Ω as analytic automorphisms and is isomorphic to ℤ_{2} ⊕ ℤ_{2}. The quotient space Ω/G(a_{0}, a_{1}, a_{2}, a_{3}) is a Riemann orbifold of signature (0; 2, 2, 2, ∞). We remark that the fixed points of A_{1} can be expressed by

\frac{a_{0}a_{1} − a_{2}a_{3} ± \sqrt{(a_{0} − a_{2})(a_{1} − a_{2})(a_{0} − a_{3})(a_{1} − a_{3})}}{a_{0} + a_{1} − a_{2} − a_{3}}.

We now return to our case. The covering map R : Y → Y/G(e_{0}, e_{1}, e_{2}, e_{3}) = W can be expressed explicitly by

R(z) = \frac{z^{4} − 2(e_{1}e_{2} + e_{2}e_{3} + e_{3}e_{1})z^{2} + 8e_{1}e_{2}e_{3}z + (e_{1}e_{2} + e_{2}e_{3} + e_{3}e_{1})^{2}}{4(z − e_{1})(z − e_{2})(z − e_{3})}

= \frac{z^{4} + g_{2}z^{2}/2 + 2g_{3}z + (g_{2}/4)^{2}}{4z^{3} − g_{2}z − g_{3}} ,

where g_{2} and g_{3} are well-known constants determined by the lattice L_{τ}. This rational function comes from the additive low in the theory of elliptic functions, i.e., ℘(2z) = R(℘(z)), and was used to construct an example of complex dynamics such that the Julia set equals the whole sphere ℂ̂ by Lattès in 1918 (see §4.3 of [2]).

Let Γ_{1} be the covering group of the universal cover 𝔻 → X → W of W. We note that Γ_{0} ◁ Γ ◁ Γ_{1}, Γ_{0} ◁ Γ′ ◁ Γ_{1}, Γ_{0} = Γ ∩ Γ′, and that Γ_{1} = ⟨Γ, Γ′⟩.

In particular, B_{2}(𝔻, Γ_{1}) ⊂ B_{2}(𝔻, Γ) and B_{2}(𝔻, Γ_{1}) ⊂ B_{2}(𝔻, Γ′), and these are all onedimensional vector spaces, hence all equal. Now relation (2.4) leads to the following result.

Lemma 3.1. The Bers embedded one-dimensional Teichmüller spaces of the commensurable Fuchsian groups Γ, Γ′ and Γ_{1} all coincide:

T (Γ) = T (Γ_{1}) = T (Γ′).

Therefore, to consider the Bers embedding of the Teichmüller space of a once-punctured torus is equivalent to do that of the corresponding four-times punctured sphere. The latter sometimes has an advantage making things simple in the calculation as we shall see in the next section.

4. The monodromy on a four-times punctured sphere

Let Γ be the Fuchsian group uniformizing a once-punctured torus and Γ′ be the Fuchsian group commensurable with Γ uniformizing the corresponding four-times punctured sphere. Although the space B_{2}(𝔻, Γ) = B_{2}(𝔻, Γ′) is complex one-dimensional, it seems quite hard to calculate numerically the value of functions in this space. On the other hand, an element of B_{2}(𝔻, Γ) can be regarded as a holomorphic quadratic differential on the quotient space 𝔻/Γ, which is more tractable than its lift to 𝔻. Indeed, R. M. Porter [30] and L. Keen [11] employed this idea and succeeded in computing the real rays of trace functions for several words of special type. In authors’ opinion, appearance of elliptic functions in their methods has made the computation somewhat difficult.

So, the authors would like to propose an idea using quadratic differentials on the fourtimes punctured sphere Y = 𝔻/Γ′ instead of those on the once-punctured torus X = 𝔻/Γ. Our formulation will need only rational functions, so the involved computation will be much simpler than the other methods. In this section, we will provide a method of computing the monodromy homomorphism χ_{φ} : Γ → PSL(2, ℂ) up to conjugation by PSL(2, ℂ), where the conjugation does not depend on φ. In fact, our construction gives global holomorphic mappings μ_{0}, μ_{∞} : B_{2}(𝔻, Γ) → SL(2, ℂ) such that the homomorphism χ̃′_{φ} : Γ̃ → SL(2, ℂ) determined by χ̃′_{φ}(θ̃_{0}) = μ_{0}(φ) and χ̃′_{φ}(θ̃_{∞}) = μ_{∞}(φ) induces the same homomorphism Γ → PSL(2, ℂ) as χ_{φ} up to PSL(2, ℂ)-conjugacy (Theorem 4.3). Here θ̃_{r} is a representative of θ_{r} in SU(1, 1) for r = 0, ∞. We remark also that the argument developed below can also be applied to the case of n-times punctured spheres, n > 4.

Recall the relation 𝔻/Γ′ ≅ ℂ\{e_{1}, e_{2}, e_{3}}. For simplicity, by the map z ↦ (z −e_{2})/(e_{1} − e_{2}), we transform the domain to the other one ℂ\{0, 1, λ}, denoted still by the same letter Y as before, where λ = (e_{3} − e_{2})/(e_{1} − e_{2}) = λ(τ) is known to be a holomorphic universal covering of the domain ℂ \ {0, 1} and usually called an elliptic modular function (cf. [1, Chapter 7]).

We denote by p the universal covering 𝔻 → Y ≅ 𝔻/Γ′ constructed above and set z_{0} = p(0). Then we can associate the isomorphism, denoted by δ, from Γ′ onto π_{1}(Y, z_{0}) with p by assigning the homotopy class of the image p([0, γ(0)]) to a given γ ∈ Γ′.

Let B_{2}(Y) denote the Banach space of bounded holomorphic quadratic differentials on Y, where a holomorphic quadratic differential ψ(z)dz^{2} on Y is said to be bounded if the norm

∥ψ∥_{Y} := sup_{z∈Y} |ψ(z)|ρ_{Y} (z)^{−2}

is finite, where ρ_{Y} (z)|dz| denotes the complete hyperbolic metric on Y of the constant negative curvature −4, i.e., 1/(1 − |ζ|^{2}) = ρ_{Y} (p(ζ))|p′(ζ)|. We remark that a holomorphic quadratic differential ψ satisfies ∥ψ∥_{Y} < ∞ if and only if ψ has at most simple pole at each puncture of Y.

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 9

By definition, the spaces B_{2}(𝔻, Γ′) and B_{2}(Y) are isometrically isomorphic via the pull-back p_{2}^{∗} : B_{2}(Y) → B_{2}(𝔻, Γ′) defined by p_{2}^{∗}ψ = ψ ∘ p · (p′)^{2}. In particular, B_{2}(Y) is one-dimensional, too.

On the other hand, the rational function

ψ_{0}(z) = \frac{1}{z(z − 1)(z − λ)} (4.1)

belongs to B_{2}(Y), thus, ψ_{0} forms a basis of the vector space B_{2}(Y).

Now we consider the developing map f_{φ} : 𝔻 → ℂ̂, where φ is the pull-back of a given ψ ∈ B_{2}(Y) under the projection p. Then the (local) branch g of the function f_{φ} ∘ p^{−1} near the basepoint z_{0} = p(0) with g(z_{0}) = 0 satisfies the relation

p_{2}^{∗}ψ (= φ = S_{f_{φ}}) = p_{2}^{∗}(S_{g}) + S_{p}. (4.2)

The function g has the expansion

g(z) = \frac{1}{p′(0)}(z − z_{0}) − \frac{p′′(0)}{2p′(0)^{3}} (z − z_{0})^{2} + ⋯ (4.3)

around the point z_{0}. We should remember the fact that the first two coefficients do not depend on the particular choice of φ = p_{2}^{∗}ψ. Here the relation S_{p} = −p_{2}^{∗}(S_{p^{−1}}) holds and S_{p^{−1}} depends only on the domain Y, in fact, this is independent of the choice of the branch and the covering map p because of the invariance property of the Schwarzian derivative: S_{A∘f} = S_{f} for any Möbius transformation A. This holomorphic quadratic differential S_{p^{−1}}(z)dz^{2} will be written by ν_{Y} (z)dz^{2} in this article and called the uniformizing connection of Y (see [18]).

By the behaviour of the universal covering near the puncture, the quadratic differential ν_{Y} (z)dz^{2} is known to have a pole of the second order at every puncture of Y with residue 1/2. By this observation, we have the following.

Lemma 4.1 ([8, Theorem 3.1], [18, (2.3.1)]). The uniformizing connection ν_{Y} (z)dz^{2} of Y has the form

ν_{Y} (z) = \frac{1}{2z^{2}(z − 1)^{2}} + \frac{1}{2(z − λ)^{2}} + \frac{c(λ)}{z(z − 1)(z − λ)},

where c(λ) is a constant determined by λ.

The constant c(λ) was called the accessory parameter by Poincaré and is known to be difficult to compute in general. I. Kra [18] showed that c(λ) is real analytic but not complex analytic in λ.

If a Möbius transformation A leaves Y invariant, we have ν_{Y} = A_{2}^{∗}ν_{Y} , in particular, ν_{Y} is invariant under the action of the group G(0, 1, ∞, λ). By this invariance, when Y has a good symmetry, the value c(λ) can be computed explicitly. For example, we easily see that c(λ) is real for a real λ and that c(λ) is pure imaginary if Re λ = 1/2. In particular, we see that c(1/2) = 0. The following examples can be found in [8].

Example 4.1. c(−1) = 1. Note that λ(−1 + i) = −1.

Example 4.2. c((1 + \sqrt{3}i)/2) = −i/\sqrt{3}. Note that λ((1 + \sqrt{3}i)/2) = (1 + \sqrt{3}i)/2.

For the moment, we proceed ahead as if we know the value of the accessary parameter c(λ). We will return to this problem in the final part of Section 5.

Since S_{p} = −p_{2}^{∗}(ν_{Y}), we can see from (4.2) that S_{g} is a globally defined quadratic differential in Y :

S_{g} = ψ + ν_{Y} . (4.4)

Now we write ψ in the form ψ = tψ_{0}, where t is a complex number and ψ_{0} is given by (4.1), and consider the linear ordinary differential equation on Y of the following form:

2y′′ + \{\frac{1}{2z^{2}(z − 1)^{2}} + \frac{1}{2(z − λ)^{2}} + \frac{t + c(λ)}{z(z − 1)(z − λ)}\} y = 0. (4.5)

We remark that I. Laine and T. Solvali [21] investigated the general differential equation y′′ + C y = 0 with coefficient C meromorphic in a symply connected domain and obtained a nice condition under which the (local) monodromy of the equation becomes trivial (see Corollary 4.2 therein).

Introducing the auxiliary unknown function x, we can transform this equation to the system of linear ordinary differential equations of the first order:

y′ = yA, A = (\begin{matrix}0 & 1 \\ −\frac{1}{2}(ψ + ν_{Y}) & 0 \end{matrix}) , (4.6)

where y = (x y) .

For a given point z_{1} ∈ Y, let y_{1} = (x_{1} y_{1}) and y_{2} = (x_{2} y_{2}) be the (local) solutions of (4.6) near the point z_{1} satisfying y_{1}(z_{1}) = (1 0) and y_{2}(z_{1}) = (0 1) . Then the matrix

F_{z_{1}} = (\begin{matrix}y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix}y′_{1} & y_{1} \\ y′_{2} & y_{2} \end{matrix})

will be called the fundamental matrix of differential equation (4.6) at z_{1}. Note that det F_{z_{1}} ≡ 1. In particular, the matrix F_{z_{1}} is a holomorphic map from a neighbourhood of z_{1} in Y to the complex Lie group SL(2, ℂ) with F_{z_{1}}(z_{1}) = I, where I denotes the identity matrix. The matrix F_{z_{1}} itself can be regarded as an SL(2, ℂ)-valued (local) solution of the differential equation (4.6). An arbitrary (local) solution y of (4.5) near the point z_{1} with the initial conditions y′(z_{1}) = a and y(z_{1}) = b can be expressed by

(y′ y) = (a b) F_{z_{1}},

and vice versa.

The holomorphic map F_{z_{1}} can be analytically continued along any path α from z_{1} to z_{2} in Y. The resulting germ at z_{2} depends only on the homotopy class [α] of α in Y, and hence, will be denoted by H_{[α]}. Since the germ H_{[α]} is an SL(2, ℂ)-valued local solution of (4.6) near the point z_{2}, there exists a unique constant matrix B ∈ SL(2, ℂ) such that H_{[α]} = BF_{z_{2}}. The matrix B is called the transition matrix along [α] and will be denoted by L[α] or, more precisely, L_{t}[α]. In what follows, we will denote by Y [z_{1}, z_{2}] the set of homotopy classes of paths connecting z_{1} and z_{2} in Y.

By definition, we can see the following fundamental properties of the transition matrix:

(i) H_{[α]} = L[α]F_{z_{2}} for [α] ∈ Y [z_{1}, z_{2}], (ii) L[α] = H_{[α]}(z_{2}) for [α] ∈ Y [z_{1}, z_{2}], and (iii) L[α]L[β] = L[α · β] for [α] ∈ Y [z_{1}, z_{2}] and [β] ∈ Y [z_{2}, z_{3}].

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 11

Property (ii) means that the transition matrix L[α] can be computed by (numerically) solving differential equation (4.6) along the path α. (In practical computations, a polygonal line should be taken as α.)

We set l_{t}[α] = L_{t}[α] for [α] ∈ π_{1}(Y, z_{0}) = Y [z_{0}, z_{0}]. Then, by property (iii), the map l_{t} : π_{1}(Y, z_{0}) → SL(2, ℂ) is a group homomorphism and will be called the monodromy homomorphism associated with ψ = tψ_{0}.

Now we consider the ratio y_{1}/y_{2}, which will be denoted by h_{z_{1}}, of y_{1} and y_{2} appearing in the fundamental matrix F_{z_{1}} at z_{1}. In other words, h_{z_{1}} = Π ∘ F_{z_{1}}, where Π : SL(2, ℂ) →

ℂ P^{1} = ℂ̂ is defined by Π : (\begin{matrix}a & b \\ c & d \end{matrix}) ↦ b/d. Then h_{z_{1}} satisfies the Schwarzian differential

equation (4.4) and has expansion in the form h(z) = (z − z_{1}) + b_{3}(z − z_{1})^{3} + ⋯ near the point z_{1}. In particular, when we take z_{0} as the reference point, comparing the above expansion with (4.3), we obtain

(4.7) g = k ∘ h, where k(z) = \frac{p′(0)^{2}z}{p′(0) − p′′(0)z/2} and h = h_{z_{0}}.

Note that the Möbius transformation k depends only on the universal covering p of Y and that the function h can be analytically continued to the function l̄_{t}[α] ∘ h along a curve α, where l̄_{t}[α] is the Möbius transformation induced by l_{t}[α].

Let us look at the relation between χ_{tφ_{0}} and l_{t} for t ∈ ℂ, where φ_{0} = p_{2}^{∗}(ψ_{0}) ∈ B_{2}(𝔻, Γ′). First note that the monodromy homomorphism χ_{tφ_{0}} is defined over the Fuchsian group Γ_{1} = ⟨Γ, Γ′⟩ because B_{2}(𝔻, Γ′) = B_{2}(𝔻, Γ_{1}). We also recall that δ : Γ′ → π_{1}(Y, z_{0}) is the isomorphism mapping γ to p([0, γ(0)]). Now we are ready to state the following theorem.

Theorem 4.2. The monodromies χ_{tφ_{0}} and l̄_{t} are essentially same. More precisely,

χ_{tφ_{0}}(γ) = k ∘ l̄_{t}(δ(γ)) ∘ k^{−1} (4.8)

for each t ∈ ℂ and γ ∈ Γ′, where φ_{0} = p_{2}^{∗}ψ_{0} and k is the Möbius transformation given in (4.7).

Proof. Set g = k ∘ h, where h = h_{z_{0}}. Then, as was explained, f = g ∘ p near the origin. By the analytic continuation along the image curve α = δ(γ) of the oriented hyperbolic geodesic segment joining 0 and γ(0) in the unit disk 𝔻, the function h changes to l̄_{t}(α)∘h. Hence, analytically continuing both sides of f = k ∘ h ∘ p along this segment, we see f = A ∘ k ∘ h ∘ p = A ∘ g ∘ p near the point γ(0), here we write A = k ∘ l̄_{t}(δ(γ)) ∘ k^{−1} for short. In particular, f ∘ γ = A ∘ g ∘ p ∘ γ = A ∘ g ∘ p = A ∘ f near the origin. This implies χ_{φ}(γ) = A.

Remark 4.1. The monodromy homomorphism χ_{φ} : Γ → PSL(2, ℂ) has no natural homomorphic lifts Γ → SL(2, ℂ) unless we specify a homomorphic section s : Γ → SU(1, 1) ⊂ SL(2, ℂ) of Γ. In contrast, l_{t} : π_{1}(Y, z_{0}) → SL(2, ℂ) is intrinsically homomorphic. Particularly, the composed map l_{0} ∘ δ : Γ′ → SU(1, 1) gives a homomorphic lift of the Fuchsian group Γ′. As we can see from the standard local theory of Fuchsian differential equations around regular singular points, tr (l_{t}[α]) = −2 holds for every simple loop [α] rounding about a puncture of Y. By this constraint, the lift of Γ′ is uniquely determined (note that Γ′ is a free group generated by three elements corresponding to punctures). Moreover, Y. Okumura [28] observes under this constraint that tr (l_{0}[α]) ≤ −2 holds for each dividing simple loop α in Y (even in more general context). In particular, we have tr (l_{0}[β_{0}]) < −2 and tr (l_{0}[β_{∞}]) < −2, where β_{0} and β_{∞} are the (dividing) simple loops which will be defined below. This implies that the homomorphism l_{0} ∘ δ has no homomorphic extensions to Γ_{1} = ⟨Γ, Γ′⟩. In fact, if such an extension, say χ : Γ_{1} → SL(2, ℝ), exists then it must satisfy tr (χ(γ^{2})) = tr (χ(γ)^{2}) = tr (χ(γ))^{2} − 2 ≥ 2 for each element γ ∈ Γ. Letting γ = θ_{0}, we would have tr (χ(γ^{2})) = tr (l_{0}[β_{0}]) < −2, which is a contradiction.

As is well known, for determination of a homomorphism χ : Γ → PSL(2, ℂ), it is enough to know the values of tr^{2}(χ(γ)) for γ = θ_{0}, θ_{∞} and θ_{1} = θ_{0}θ_{∞}, where (θ_{0}, θ_{1}) is a standard generator pair of Γ. Using Theorem 4.2, we can compute tr^{2}(χ_{φ}(θ_{r})) for r = 0, 1, ∞. In fact, since δ(θ_{r}^{2}) is a dividing loop in Y, we see tr^{2}(χ_{φ}(θ_{r})) = ±tr (χ_{φ}(θ_{r}^{2})) + 2 = −tr (l_{t}(δ(θ_{r}^{2}))) + 2 by the above remark. However, sometimes it is more convenient to have the representation χ_{φ} itself rather than just the traces. In order to directly compute χ_{φ}(γ) for each γ ∈ Γ (up to Möbius conjugation) we use the symmetry of the domain Y which was explained in the previous section.

Given a once-punctured torus X = 𝔻/Γ represented by the Teichmüller parameter τ, it suffices to compute the monodromy images of the standard generator pair (θ_{0}, θ_{∞}) of Γ. The fact that φ = p_{2}^{∗}ψ ∈ B_{2}(𝔻, Γ_{1}) for each ψ ∈ B_{2}(Y) implies that ψ(z)dz^{2} is invariant under the action of Γ_{1}/Γ′ ≅ G(0, 1, ∞, λ) < Aut(Y), where λ = λ(τ). Since ν_{Y} (z)dz^{2} is also invariant, A_{−1/2}^{∗}y becomes a local solution of (4.5) for each local solution y of (4.5) and for A ∈ G(0, 1, ∞, λ) by the same reasoning given in Section 2, where A_{−1/2}^{∗}y is

defined by y ∘ A · (A′)^{−1/2} and (A′)^{−1/2}(z) = rz + s for a representative (\begin{matrix}p & q \\ r & s \end{matrix}) ∈ SL(2, ℂ)

of A. We set for F_{A(z_{1})} = (\begin{matrix}y′_{1} & y_{1} \\ y′_{2} & y_{2} \end{matrix})

A^{∗}(F_{A(z_{1})}) := (\begin{matrix}(A_{−1/2}^{∗}y_{1})′ & A_{−1/2}^{∗}y_{1} \\ (A_{−1/2}^{∗}y_{2})′ & A_{−1/2}^{∗}y_{2} \end{matrix}) = M_{z_{1}}(A)F_{z_{1}}, (4.9)

where the matrix M_{z_{1}}(A) is defined by

M_{z_{1}}(A) = (\begin{matrix}1/(rz_{1} + s) & 0 \\ r & rz_{1} + s \end{matrix}) .

We consider the simple closed curves β̂_{0} and β̂_{∞} in Z induced by the curves s+(1+τ)/4 and sτ + (1 + τ)/4 parametrized by s ∈ [0, 1] in ℂ \ \frac{1}{2}L_{τ} and we take the reference point z_{0} in Y so that the intersection point of β̂_{0} and β̂_{∞} projects to z_{0} under the projection Z → Y. Then the image curves β_{0} and β_{∞} in Y represent the homotopy classes δ(θ_{0}^{2}) and δ(θ_{∞}^{2}), respectively.

Note that β_{0} separates 0 and λ from 1 and ∞. Hence we see that A_{0}(z_{0}) = p(θ_{0}(0)), where A_{0}(z) = (z−λ)/(z−1) ∈ G(0, 1, ∞, λ) is determined by A_{0}(0, λ, 1, ∞) = (λ, 0, ∞, 1). Similarly, noting that β_{∞} separates 0 and ∞ from 1 and λ, we have A_{∞}(z_{0}) = p(θ_{∞}(0)), where A_{∞}(z) = λ/z ∈ G(0, 1, ∞, λ) is determined by A_{∞}(0, ∞, 1, λ) = (∞, 0, λ, 1). For r = 0, ∞, let α_{r} be an oriented curve in Y joining z_{0} and A_{r}(z_{0}) which is homotopic to p_{∗}([0, θ_{r}(0)]), in other words, homotopic to the curve coming from the one s/2 + (1 + τ)/4 for r = 0 or sτ/2 + (1 + τ)/4 for r = ∞ in Z parametrized by s ∈ [0, 1].

Now we are in a position to state one of our main results.

Theorem 4.3. For suitable representatives θ̃_{0} of θ_{0} and θ̃_{∞} of θ_{∞} in SU(1, 1),

χ̃_{tφ_{0}}(θ̃_{0}) = L_{t}[α_{0}]M_{z_{0}}(A_{0}), and χ̃_{tφ_{0}}(θ̃_{∞}) = L_{t}[α_{∞}]M_{z_{0}}(A_{∞})

hold for all t ∈ ℂ up to an SL(2, ℂ)-conjugacy independent of t, where φ_{0} = p_{2}^{∗}(ψ_{0}) and ψ_{0} is given by (4.1).

Remark 4.2. If the point z_{0} and the paths α_{0} and α_{∞} are replaced by other z_{1} ∈ Y, α′_{0} ∈ Y [z_{1}, A_{0}(z_{1})] and α′_{∞} ∈ Y [z_{1}, A_{∞}(z_{1})], the above result remains true as far as α′_{0}·(A_{0})_{∗}(α′_{0}) and α′_{∞} · (A_{∞})_{∗}(α′_{∞}) are freely homotopic to β_{0} and β_{∞} in Y, respectively.

Proof. We set φ = tφ_{0} for a fixed t ∈ ℂ and let f = f_{φ} : 𝔻 → ℂ̂ be the normalized developing map for φ. Then f = k ∘ h_{z_{0}} ∘ p near the origin of 𝔻, where k is the Möbius transformation given by (4.7). Let r represent 0 or ∞. Analytically continuing both sides along the segment [0, θ_{r}(0)] in 𝔻, we obtain

f = k ∘ L_{t}[α_{r}]^{−} ∘ h_{z′_{0}} ∘ p

near the point θ_{r}(0), where we set z′_{0} = A_{r}(z_{0}) and denote by M^{−} the Möbius transformation induced by a matrix M ∈ GL(2, ℂ). Noting h_{z′_{0}} ∘ A_{r} = M_{z_{0}}(A_{r})^{−} ∘ h_{z_{0}} by (4.9) and p ∘ θ_{r} = A_{r} ∘ p, we have

f ∘ θ_{r} = k ∘ L_{t}[α_{r}]^{−} ∘ h_{z′_{0}} ∘ p ∘ θ_{r} = k ∘ L_{t}[α_{r}]^{−} ∘ h_{z′_{0}} ∘ A_{r} ∘ p
= k ∘ L_{t}[α_{r}]^{−} ∘ M_{z_{0}}(A_{r})^{−} ∘ h_{z_{0}} ∘ p = k ∘ L_{t}[α_{r}]^{−} ∘ M_{z_{0}}(A_{r})^{−} ∘ k^{−1} ∘ f

near the origin. This implies χ_{φ}(θ_{r}) = k ∘ L_{t}[α_{r}]^{−} ∘ M_{z_{0}}(A_{r})^{−} ∘ k^{−1}. Hence, χ̃_{φ}(θ̃_{r}) = ±k̃L_{t}[α_{r}]M_{z_{0}}(A_{r})k̃^{−1}, where k̃ is a matrix representing k. Since the quantities χ̃_{tφ_{0}}(θ̃_{r}) and L_{t}[α_{r}]M_{z_{0}}(A_{r}) are globally holomorphic in t, we easily see that the above signature is constant. (Note also that the conjugacy map k does not depend on t.) Therefore, we obtain the required equalities by choosing a suitable signature of θ̃_{r} for each r.

Remark 4.3. The following relation might be useful: For a point z_{1} in Y and A ∈ G(0, 1, ∞, λ), we set z_{2} = A(z_{1}). Then we have L[α]M_{z_{1}} = M_{z_{2}}L[A_{∗}(α)] for any path α connecting z_{1} and z_{2} in Y.

We note also that we can use the orbifold W instead of Y in order to compute χ_{φ}(θ_{r}) directly. The present method, however, has the merit that we can “save” the polygonal path used in solving the differential equation numerically. If we try to solve the differential equation along a closed polygonal path, then we would need more segments.

5. Enumeration of simple closed curves

In the preceding section, we have presented a method of computing the monodromy for a given projective structure and a γ ∈ Γ. We explain here how to compute the values of the trace function corresponding to an arbitrary simple closed geodesic in a recursive and algebraic way using the particular values Trθ̃_{0}, Trθ̃_{∞} and Trθ̃_{1}, where θ̃_{r} is a fixed representative of θ_{r} in SU(1, 1) for each r. The materials here are, more or less, known although rigorous proofs for those are sometimes difficult to locate. The reader is referred to, e.g., [14], [25] and [38] for further discussions. A good introduction to the Teichmüller space of a torus can also be found in [10].

Since we need only the topological structure of a once-punctured torus for a while, we treat first the topological once-punctured torus described as Σ = (ℝ^{2} \ℤ^{2})/ℤ^{2}. We denote by ⟨z⟩ the projection of a point z ∈ ℝ^{2}\ℤ^{2} to Σ. Let [α] and [β] be the homotopy classes in Σ with base point x = ⟨(1/2, 1/2)⟩ which are represented by the curves α(t) = x + ⟨(t, 0)⟩ and β(t) = x + ⟨(0, t)⟩, 0 ≤ t ≤ 1, respectively. Let A and B be the homology classes corresponding to [α] and [β]. It is well known that the fundamental group π_{1}(Σ, x) is a free group generated by [α] and [β] and that the homology group H_{1}(Σ) is a free abelian group generated by A and B. Taking A and B as a basis, we may regard H_{1}(Σ) as the lattice group ℤ ⊕ ℤ.

The fundamental group and the homology group coincide for the unpunctured torus \overline{Σ} = ℝ^{2}/ℤ^{2} and both are isomorphic to ℤ^{2} = ℤ ⊕ ℤ. In particular, the homomorphism π_{1}(Σ, x) → π_{1}(\overline{Σ}, x) induced by the natural inclusion Σ → \overline{Σ} is canonically isomorphic to the abelianization π_{1}(Σ, x) → H_{1}(Σ) = H_{1}(\overline{Σ}).

A simple closed curve γ in Σ is said to be non-peripheral if γ can be continuously shrunken to neither a point nor a puncture in Σ. A non-peripheral simple closed curve γ corresponds to a homology class of the form pA + qB, where p and q are relatively prime integers, that is, {mp + nq; m, n ∈ ℤ} = ℤ. We can easily understand this as follows; cutting Σ along γ, we have a once-punctured annulus, which implies that there is another non-peripheral simple closed curve γ′ such that homology classes of γ and γ′ generate H_{1}(Σ), hence the homology class of γ must be of the above form.

The ratio p/q will be called the slope of γ. Note that the slope does not depend on the orientation of the curve.

As might be seen from the linear action of SL(2, ℤ) on ℝ^{2} leaving ℤ^{2} invariant, the mapping class group of Σ is isomorphic to SL(2, ℤ). This is a classical result. However, this could be seen more easily with a stronger assertion about the relation between simple closed curves and their slope.

Note here that the mapping class group M(Σ) of Σ is defined as Homeo^{+}(Σ)/Homeo_{0}(Σ), where Homeo^{+}(Σ) is the group of orientation-preserving self-homeomorphisms of Σ and Homeo_{0}(Σ) is the subgroup consisting of all homeomorphisms which are homotopic to the identity in Σ. We also remark that the mapping class group of Σ is isomorphic to the quotient of the automorphism group of π_{1}(Σ, x) over the inner automorphism group of π_{1}(Σ, x).

Proposition 5.1. The mapping class group of Σ is canonically isomorphic to SL(2, ℤ). For each rational r ∈ ℚ̂ there exists a non-peripheral simple closed curve in Σ with slope r. If two non-peripheral simple closed curves in Σ have the same slope, they are freely homotopic in Σ up to orientation.

The latter assertion can be found in [14].

Proof. Let S be the homeomorphism of Σ defined by S(⟨(s, t)⟩) = ⟨(s, t)⟩ for 0 ≤ t ≤ 1/2 and by S(⟨(s, t)⟩) = ⟨(s + 2t − 1, t)⟩ for 1/2 ≤ t ≤ 1, which realizes a Dehn twist around the curve α. Also let T be the rotation ⟨(s, t)⟩ ↦ ⟨(t, −s)⟩ by π/2. Then the induced homomorphisms S_{∗} and T_{∗} on π_{1}(Σ, x) satisfy S_{∗}(α, β) = (α, αβ) and T_{∗}(α, β) = (β, α^{−1}), and hence, they act on H_{1}(Σ) as matrices (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) , respectively,

under the above identification with ℤ ⊕ ℤ. It is well known that the above two matrices generate SL(2, ℤ), and thus, the natural homomorphism M(Σ) → Aut(H_{1}(Σ)) ≅ SL(2, ℤ) is surjective.

A rational r ∈ ℚ̂ can be written in the form r = p/q, where p and q are relatively prime integers. Then there exist integers p′ and q′ such that pq′ − p′q = 1. We now take an

element f_{∗} in M(Σ) which acts on H_{1}(Σ) as the matrix (\begin{matrix}p & p′ \\ q & q′ \end{matrix}) . Then f_{∗}(α) has slope

r as desired.

We next show the uniqueness part. By means of the action of the mapping class group, it is enough to show that a simple closed curve α^{∗} in π_{1}(Σ, x) with slope 0 is necessarily homotopic to α in Σ. Since Σ\α^{∗} is a punctured annulus, we can connect both boundary components by a simple arc with endpoints at x. That arc can be regarded as a simple closed curve, say β^{∗}, in Σ so that α^{∗} and β^{∗} form standard generators of π_{1}(Σ, x). Furthermore, by repeated application of the Dehn twist along α^{∗}, we can choose β^{∗} so that it is homotopic to β in \overline{Σ}.

If we take an orientation-preserving self-homeomorphism f of Σ fixing the point x such that f_{∗}(α) = α^{∗} and f_{∗}(β) = β^{∗}, then f_{∗} acts on π_{1}(\overline{Σ}, x) trivially. Therefore, f is homotopic to the identity in the unpunctured torus \overline{Σ}. Let f_{t} be an isotopy in \overline{Σ} connecting the identity and f with f_{0} = id and with f_{1} = f. Since \overline{Σ} = ℝ^{2}/ℤ^{2} is an abelian group, f̂_{t} = f_{t} − f_{t}(0) is an isotopy in \overline{Σ} fixing the origin 0 = ⟨(0, 0)⟩. Hence f̂_{t}|_{Σ} is an isotopy in Σ connecting the identity and f, which implies that α^{∗} is freely homotopic to α.

Finally, we show the first assertion in the proposition. It suffices to prove the injectivity of the natural homomorphism M(Σ) → Aut(H_{1}(Σ)) = SL(2, ℤ). Let f_{∗} be in the kernel of the above homomorphism. We may assume that f fixes x. Then f_{∗}(α) and f_{∗}(β) have slope 0 and ∞, respectively. The last argument yields also that f is homotopic to the identity in Σ. Hence f_{∗} is trivial on π_{1}(Σ, x).

We can extract from the proposition the following corollary, which will be utilized effectively below.

Corollary 5.2. Let (γ_{1}, γ_{2}) be a standard generator pair of the fundamental group π_{1}(Σ, x) of the topological once-punctured torus Σ. Then (γ_{1}, γ_{1}γ_{2}), (γ_{1}γ_{2}, γ_{2}) and (γ_{2}, γ_{1}^{−1}) are all standard generator pairs of π_{1}(Σ, x).

We next recall fundamental facts about Farey triangles (cf. [25]). The reader also finds a good account for Farey sequences in [7] as well as an interesting historical remark.

For three points a, b, c in ℝ̂, we denote by Δ(a, b, c) the hyperbolic triangle formed by three hyperbolic geodesics in the upper half plane ℍ connecting two of the three points a, b and c. Let Δ = Δ(0, 1, ∞). Then ℍ is tessellated by Δ and its conjugates by the modular group PSL(2, ℤ). Note that the stabilizer of Δ in PSL(2, ℤ) consists of three elements and permutes the vertices of Δ cyclically. Each triangle which is conjugate with Δ by the action of PSL(2, ℤ) is called a Farey triangle. The initial Farey triangle Δ and its reflection Δ′ = Δ(0, −1, ∞) in the imaginary axis form a fundamental domain of the modular group G_{2} = {±C ∈ PSL(2, ℤ); C ≡ I mod 2} of level 2. We will say that both Δ and Δ′ are of level 0. A Farey triangle which shares a side with that of level 0 will be called of level 1 unless it is of level 0. Similarly, a Farey triangle which shares a side with that of level n will be called of level n + 1 unless it is of level ≤ n. It is important to note that the corresponding graph to the above tessellation is a tree, namely, there is no closed circuit.

It is well known that the orbit of 0 under the action of PSL(2, ℤ) coincides with ℚ̂. We denote by F̃(n) the set of rationals which appear as vertices of Farey triangles of level ≤ n. Set F(n) = F̃(n) \ F̃(n − 1) for n = 0, 1, …. For instance, F(0) = {−1, 0, 1, ∞}, F(1) = {−2, −1/2, 1/2, 2}, F(2) = {−3, −3/2, −2/3, −1/3, 1/3, 2/3, 3/2, 3} and so on. We note that #F(n) = 2^{n+1} for n ≥ 1. An element r in F(n) will be called of level n and designated by level(r) = n. Two rationals r_{1} and r_{2} in ℚ̂ are said to be Farey neighbours if r_{1} and r_{2} are distinct vertices of a common Farey triangle. For convenience, we adopt the notation r_{1} ∼ r_{2} to mean that r_{1} and r_{2} are Farey neighbours. It should be remembered that this is not an equivalence relation. Note that if r_{1} ∼ r_{2} then |level(r_{1}) − level(r_{2})| ≥ 1 unless level(r_{1}) = level(r_{2}) = 0.

For two rationals r_{1} = p_{1}/q_{1} and r_{2} = p_{2}/q_{2} in ℚ̂, where p_{j} and q_{j} are relatively prime integers, we assign the number D(r_{1}, r_{2}) = |p_{1}q_{2} − p_{2}q_{1}|. Note that this is independent of the particular choice of signs of p_{j} and q_{j}. The following statement will be useful in our argument below.

Lemma 5.3. Two rationals r_{1} and r_{2} are Farey neighbours if and only if D(r_{1}, r_{2}) = 1.

Proof. First assume that D(r_{1}, r_{2}) = 1. Then r_{1} and r_{2} can be written in the form r_{1} = p_{1}/q_{1} and r_{2} = p_{2}/q_{2} so that p_{1}q_{2} − p_{2}q_{1} = 1. The Möbius transformation g(z) = (p_{1}z + p_{2})/(q_{1}z + q_{2}) sends ∞ to r_{1} and 0 to r_{2}, which implies that the Farey triangle g(Δ) contains r_{1} and r_{2} as vertices. The converse can be seen similarly.

Let r_{1} and r_{2} be Farey neighbours with 0 ≤ r_{1} < r_{2} ≤ ∞. Then they are represented in the form r_{j} = p_{j}/q_{j}, where p_{j} and q_{j} are non-negative integers for j = 1, 2 and satisfy p_{2}q_{1} − p_{1}q_{2} = 1. We define r_{1} ⊕ r_{2} = r_{2} ⊕ r_{1} = (p_{1} + p_{2})/(q_{1} + q_{2}). Similarly, we define r_{1} ⊖ r_{2} = r_{2} ⊖ r_{1} = (p_{1} − p_{2})/(q_{1} − q_{2}). For these operations, one can show the following result.

Lemma 5.4. Let r_{1} and r_{2} be Farey neighbours with 0 ≤ r_{1} < r_{2} ≤ ∞. Then Δ(r_{1}, r_{2}, r_{1}⊕ r_{2}) and Δ(r_{1}, r_{2}, r_{1} ⊖ r_{2}) are Farey triangles sharing the side connecting r_{1} with r_{2} and the inequality r_{2} < r_{1} ⊕ r_{2} < r_{1} holds.

Conversely, any rational r with 0 < r < ∞ can uniquely be decomposed into the form r = r_{1} ⊕ r_{2} for rationals 0 ≤ r_{1} < r_{2} ≤ ∞ with r_{1} ∼ r_{2}. Furthermore if r ≠ 1 then level(r) = max{level(r_{1}), level(r_{2})} + 1 and level(r_{1}) ≠ level(r_{2}).

Proof. It is immediate to see that r_{1} ⊕ r_{2} ∼ r_{j} and r_{1} ⊖ r_{2} ∼ r_{j} for j = 1, 2 and that r_{1} < r_{1}⊕r_{2} < r_{2}. Hence, the first half of the assertion has been shown. To show the second half, assume that 0 < r < ∞ is a rational and let n = level(r). Then there is a sequence of Farey triangles Δ_{0} = Δ, Δ_{1}, …, Δ_{n} such that Δ_{k−1} and Δ_{k} have a common side for k = 1, …, n and that Δ_{n} has r as its vertex. Set Δ_{−1} = Δ′ for convenience. Note that Δ_{k} is of level k and is contained in the right half of the upper half plane ℍ for k = 0, 1, …, n.

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 17

Let r_{1} and r_{2} be the other two vertices of Δ_{n}. Since the side of Δ_{n} connecting r_{1} with r_{2} separates r from the left half of ℍ, r lies between r_{1} and r_{2}, and hence, the relation r = r_{1} ⊕ r_{2} follows. Assume now that r ≠ 1. Then n > 0 and Δ_{n−1} = Δ(r_{1}, r_{2}, r_{1} ⊖ r_{2}). In particular, max{level(r_{1}), level(r_{2})} ≤ n − 1. On the other hand, Δ_{n−1} shares a side joining r_{1} ⊖ r_{2} and, say, r_{2} with Δ_{n−2}. We see then that level(r_{1}) = n − 1 and that level(r_{2}) ≤ n − 2.

For each rational r with 0 < r < ∞, r ≠ 1, we define r^{∗} by r^{∗} = r_{1} ⊖ r_{2}, where r_{1} and r_{2} are as above. Then r^{∗} < min{r_{1}, r_{2}} or r^{∗} > max{r_{1}, r_{2}}. From the above proof, one can also see that max{level(r^{∗}), level(r_{2})} = n − 2 if level(r_{1}) = n − 1. Moreover, under the assumption that r_{1} < r_{2}, the inequality r_{2} < r^{∗} holds if and only if level(r_{1}) = n − 1.

Finally, we extend the above operations for all pairs (r_{1}, r_{2}) of Farey neighbours. When 0 ≤ −r_{1}, −r_{2} ≤ ∞, we set r_{1} ⊕ r_{2} = −(−r_{1}) ⊕ (−r_{2}) and r_{1} ⊖ r_{2} = −(−r_{1}) ⊖ (−r_{2}) unless {r_{1}, r_{2}} = {0, ∞}. Also let r^{∗} = −(−r)^{∗} for a rational r with 0 < −r < ∞ and r ≠ −1. For r = ±1, we set r^{∗} = ∓1.

We now return to our case. Let X be a marked once-punctured torus with Teichmüller parameter τ. Let H : ℝ^{2} → ℝ^{2} = ℂ be the ℝ-linear map sending (1, 0) to 1 and (0, 1) to τ. Then H induces a homeomorphism h from the topological once-punctured torus Σ onto X = (ℂ \ L_{τ})/L_{τ}. The terminology above is translated, via h : Σ → X, to that for X with standard generator pair (θ_{0}, θ_{∞}) of the Fuchsian group Γ uniformizing X. For instance, a non-peripheral simple closed curve γ in X has slope r if and only if h_{∗}^{−1}(γ) has slope r. By Proposition 5.1, the conjugacy class of an element θ_{r} in Γ representing a non-peripheral simple closed curve of slope r is uniquely determined. In particular, the squared trace function Tr_{θ_{r}}^{2} : B_{2}(𝔻, Γ) → ℂ is determined by its slope. We now explain how to compute the trace functions Trθ̃_{r} for all r ∈ ℚ̂ and suitable representatives θ̃_{r} of θ_{r} in SU(1, 1) in a recursive way by using only the values for r = 0, 1 and ∞.

We begin by setting σ_{r} = Trθ̃_{r} for r = 0, 1, ∞, where θ̃_{0} and θ̃_{∞} are arbitrarily fixed representatives of θ_{0} and θ_{∞}, respectively, and θ̃_{1} = θ̃_{0}θ̃_{∞}. Define σ_{−1} by the formula σ_{−1} = σ_{0}σ_{∞} − σ_{1}. If we have defined the functions σ_{r} : B_{2}(𝔻, Γ) → ℂ for all r ∈ F̃(n), we then define σ_{r} for r ∈ F(n + 1) by the formula

σ_{r} = σ_{r_{1}}σ_{r_{2}} − σ_{r^{∗}}, (5.1)

where r_{1} and r_{2} are Farey neighbours satisfying r = r_{1} ⊕ r_{2}. Note here that r_{1}, r_{2} and r^{∗} are all in F̃(n). In this way, we can define the functions σ_{r} for all r ∈ ℚ̂.

Proposition 5.5. The mapping σ_{r} : B_{2}(𝔻, Γ) → ℂ gives the trace function for a nonperipheral simple closed curve of slope r in X for each r ∈ ℚ̂.

Proof. In order to prove the proposition, we construct elements θ̃_{r} in Γ̃ = P^{−1}(Γ) representing non-peripheral simple closed curves of slope r in a concrete way. Note that a similar construction can be found in [14]. We start with θ̃_{0}, θ̃_{∞} and θ̃_{1} = θ̃_{0}θ̃_{∞}, which represents a simple closed curve of slope 1. By Corollary 5.2, (θ_{0}, θ_{1}) and (θ_{1}, θ_{∞}) are standard generator pairs of Γ, where θ_{1} = P (θ̃_{1}). Suppose that we have defined θ̃_{r} ∈ Γ̃ for all r ∈ F̃_{+}(n) := F̃(n) ∩ [0, ∞] in such a way that the following properties are satisfied:

(1) θ_{r} represents a simple closed curve of slope r in X for r ∈ F̃_{+}(n), where θ_{r} = P (θ̃_{r}), and

(2) (θ_{r_{1}}, θ_{r_{2}}) is a standard generator pair for each pair of Farey neighbours r_{1}, r_{2} ∈ F̃_{+}(n) with 0 ≤ r_{1} < r_{2} ≤ ∞.

Then, for r ∈ F(n + 1) with 0 < r < ∞, we define θ̃_{r} by

θ̃_{r} = θ̃_{r_{1}} θ̃_{r_{2}},

where r_{1} and r_{2} are unique Farey neighbours with 0 ≤ r_{1} < r_{2} ≤ ∞ satisfying r = r_{1} ⊕ r_{2} (see Lemma 5.4). Note also, by Corollary 5.2, that (θ_{r_{1}}, θ_{r}) and (θ_{r}, θ_{r_{2}}) are standard generator pairs of Γ, where θ_{r} = P (θ̃_{r}). In particular, θ_{r} represents a simple closed curve with slope r. In this way, we extend θ̃_{r} for all r ∈ F̃_{+}(n + 1), while preserving the above properties (1) and (2). By induction, we extend θ̃_{r} for all r ∈ ℚ̂ ∩ [0, ∞].

For r ∈ ℚ̂ ∩ (−∞, 0), we can apply the same argument as above by replacing (θ_{0}, θ_{∞}) by (θ_{∞}, θ_{0}^{−1}). More precisely, we set θ̃′_{0} = θ̃_{∞}, θ̃′_{∞} = θ̃_{0}^{−1} and θ̃′_{1} = θ̃′_{0}θ̃′_{∞} = θ̃_{∞}θ̃_{0}^{−1}. Then θ̃′_{1} represents a simple closed curve of slope −1 in X. Also, in the same way as above, we let

θ̃′_{r} = θ̃′_{r_{1}} θ̃′_{r_{2}}

for r ∈ F(n + 1) with 0 < r < ∞ after defining θ̃′_{s} for all s ∈ F̃_{+}(n). Finally, we set θ̃_{−r} = θ̃′_{r} for r ∈ ℚ̂ ∩ (−∞, 0). Then we can easily see that θ_{−r} = P (θ̃_{−r}) represents a simple closed curve of slope −r in X.

We are now at the final stage to show the assertion. The key is the general formula

tr AB + tr A^{−1}B = tr A tr B

for A, B ∈ SL(2, ℂ). Let σ̂_{r} = Trθ̃_{r} for r ∈ ℚ̂. Then σ̂_{r} = σ_{r} for all r ∈ F̃(0). We show the same statement for all r ∈ F̃(n) for each n by induction. Suppose that we have shown this up to n. Let r ∈ F(n + 1) and r_{1} and r_{2} be unique Farey neighbours with r_{1} < r_{2} satisfying r = r_{1} ⊕ r_{2}. Since r_{1}, r_{2}, r^{∗} = r_{1} ⊖ r_{2} ∈ F̃(n), we see that σ̂_{s} = σ_{s} for s = r_{1}, r_{2}, r^{∗}. As we observed above, r_{2} = r_{1} ⊕ r^{∗} if r_{2} < r^{∗}. In this case, θ̃_{r^{∗}} = θ̃_{r_{1}}^{−1}θ̃_{r_{2}}, and hence,

σ̂_{r}(φ) = tr χ̃_{φ}(θ̃_{r_{1}} θ̃_{r_{2}}) = tr χ̃_{φ}(θ̃_{r_{1}})tr χ̃_{φ}(θ̃_{r_{2}}) − tr χ̃_{φ}(θ̃_{r_{1}}^{−1}θ̃_{r_{2}})
= σ_{r_{1}}(φ)σ_{r_{2}}(φ) − σ_{r^{∗}}(φ) = σ_{r}(φ).

When r^{∗} < r_{1}, the relation r_{1} = r^{∗} ⊕ r_{2} holds, and hence, θ̃_{r^{∗}} = θ̃_{r_{1}} θ̃_{r_{2}}^{−1}. Since tr A^{−1}B = tr AB^{−1}, we can see the same relation σ̂_{r} = σ_{r} as above. We now conclude that Trθ̃_{r} = σ_{r} for all r ∈ ℚ̂ by induction.

6. McMullen’s bending coordinates of the Bers slice

Following McMullen [24], we describe the bending coordinates of the Teichmüller space T (Γ) of a once-punctured torus X = 𝔻/Γ. For precise definitions for the terminology below, see [24] or references cited there.

Considering the slope, we have identified the space of simple closed geodesics on X with ℚ̂ = ℚ∪{∞}. Extending this continuously, we obtain the homeomorphic identification of

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 19

the projectivized space ℙML(X) of measured laminations on X with the extended real line ℝ̂ ≅ S^{1}.

The region of discontinuity of the quasi-Fuchsian group χ_{φ}(Γ) for φ ∈ T (X) \ {0} consists of two quasidisks f_{φ}(𝔻) and ℂ̂\\overline{f_{φ}(𝔻)}, which are denoted by Ω_{φ}^{+} and Ω_{φ}^{−}, respectively. We remark that χ_{φ}(Γ) is never Fuchsian for φ ∈ T (Γ) \ {0}. Since f_{φ} is conformal on 𝔻, Ω_{φ}^{+}/χ_{φ}(Γ) is always conformally equivalent to X, whereas X_{φ}^{∗} := Ω_{φ}^{−}/χ_{φ}(Γ) varies. We denote by C_{φ} the hyperbolic convex hull of the limit set Λ_{φ} ⊂ ℂ̂ of χ_{φ}(Γ) in the hyperbolic three-space ℍ^{3}, where we identify ℂ̂ with the sphere at infinity of ℍ^{3}. Then the boundary ∂C_{φ} in ℍ^{3} consists of two connected components ∂C_{φ}^{±} facing to Ω_{φ}^{±}. We write M_{φ} for the hyperbolic three manifold ℍ^{3}/χ_{φ}(Γ). Then C_{φ}/χ_{φ}(Γ) is the convex core of M_{φ}, whose boundary consists of two connected components ∂C_{φ}^{±}/χ_{φ}(Γ). The nearest point retraction from Ω_{φ}^{±} to ∂C_{φ}^{±} induces the homotopy equivalence between Ω_{φ}^{±}/χ_{φ}(Γ) and ∂C_{φ}^{±}/χ_{φ}(Γ), and hence, ∂C_{φ}^{±}/χ_{φ}(Γ) are both topological punctured tori (see for example [5]). The surfaces ∂C_{φ}^{±}/χ_{φ}(Γ) are endowed with the hyperbolic metric by M_{φ} and both are pleated surfaces with the natural bending measures on their pleating loci. We denote by [pl^{±}(φ)] the projective class in ℙML(X) of the pleating locus of C_{φ}^{±}/χ_{φ}(Γ). Note that each connected component of ∂C_{φ}^{±} minus its pleating locus is contained in a hyperbolic plane in ℍ^{3}. We write ℓ_{pl^{−}(φ)}(M_{φ}) and ℓ_{pl^{−}(φ)}(X) for the hyperbolic length of the bending lamination pl^{−}(φ) on ∂C_{φ}^{−}/χ_{φ}(Γ) and on X, respectively. Now we can state a result of McMullen [24] on the bending coordinates of T (Γ).

Theorem C ([24, Theorem 1.5]). The mapping

Φ : φ ↦ ([pl^{−}(φ)], \frac{ℓ_{pl^{−}(φ)}(M_{φ})}{ℓ_{pl^{−}(φ)}(X)})

gives a homeomorphism from T (Γ) \ {0} onto ℙML(X) × (0, 1) = ℝ̂ × (0, 1), where we identify ℙML(X) with ℝ̂ ≅ S^{1} as above.

For r ∈ ℝ̂, we set P_{r} = Φ^{−1}({r} × (0, 1)). These proper arcs in T (Γ) \ {0} will be called pleating rays. Especially for r ∈ ℚ̂, P_{r} is called a rational pleating ray of slope r. We remark that ℓ_{pl^{−}(φ)}(X) is constant on P_{r}. In particular, if r ∈ ℚ̂, we have ℓ_{pl^{−}(φ)}(X) = arccosh(|tr θ_{r}|/2) for φ ∈ P_{r}.

The next result guarantees computability of rational pleating rays up to the accessary parameter. For r ∈ ℚ̂, put

H_{r} := {φ ∈ B_{2}(𝔻, Γ) ; σ_{r}(φ) is real and σ_{r}(φ)^{2} > 4} = σ_{r}^{−1}((−∞, −2) ∪ (2, +∞))

and call it the hyperbolic locus of slope r. Note that 0 ∈ H_{r} for all r ∈ ℚ̂.

Theorem 6.1. The rational pleating ray P_{r}, r ∈ ℚ̂, is the unique connected component of H_{r} \ {0} with endpoint 0 on which σ_{r}(φ)^{2} is strictly smaller than σ_{r}(0)^{2}.

Besides Theorem C, we need the local pleating theorem due to Keen and Series [13, Theorem 8.1] for the proof of our theorem. We state it here in a specialized form for our aim.

Theorem D. Let φ_{0} ∈ P_{r} ∪ {0} for some r ∈ ℚ̂. Then there exists a neighbourhood U of φ_{0} in T (Γ) such that every element φ ∈ (H_{r} \ {0}) ∩ U satisfies either [pl^{−}(φ)] = r or [pl^{+}(φ)] = r.

Proof of Theorem 6.1. Let r ∈ ℚ̂. We may assume that σ_{r}(0) > 2. Take a point φ ∈ P_{r}, then the axis of χ_{φ}(θ_{r}) in ℍ^{3} lies on the boundary ∂C_{φ}^{−} which is invariant under χ_{φ}(θ_{r}). Therefore χ_{φ}(θ_{r}) does not rotate ℍ^{3} around the axis, namely, it is hyperbolic (see [13, Proposition 6.4] for details), which implies that P_{r} is contained in H_{r} \ {0}.

Recall the inequalities due to McMullen [24, Corollary 3.5]

(6.1)  ℓ_{pl^{−}(φ)}(X_{φ}^{∗}) < ℓ_{pl^{−}(φ)}(M_{φ}) < ℓ_{pl^{−}(φ)}(X) and
(6.2)  ℓ_{pl^{+}(φ)}(X) < ℓ_{pl^{+}(φ)}(M_{φ}) < ℓ_{pl^{+}(φ)}(X_{φ}^{∗})

for every φ ∈ T (Γ) \ {0}. Since ℓ_{pl^{−}(φ)}(M_{φ}) = arccosh(σ_{r}(φ)/2) for φ ∈ P_{r}, we see that σ_{r}(φ) → 2 as φ approaches the Bers boundary along P_{r} by Theorem C. This means that the endpoint of P_{r} corresponds to the cusp pinched along the simple closed geodesic of slope r. In particular, P_{r} is closed in H_{r} \ {0}.

In order to show that P_{r} is a connected component of H_{r} \ {0} ending at the basepoint 0 on which σ_{r} < σ_{r}(0), it suffices to see that P_{r} is open in H_{r} \{0}. Suppose that φ_{0} ∈ P_{r}. Note from (6.1) that 2 < σ_{r}(φ_{0}) < σ_{r}(0). Therefore, |σ_{r}| < σ_{r}(0) in a sufficiently small neighbourhood U of φ_{0} in T (Γ) \ {0}. By Theorem D, we can further choose U so that every φ ∈ H_{r} ∩ U satisfies [pl^{−}(φ)] = r or [pl^{+}(φ)] = r. If the latter case occurred, (6.2) would yield σ_{r}(0) < |σ_{r}(φ)|. However, this contradicts the choice of U. Therefore, only the first case occurs, namely, H_{r} ∩ U ⊂ P_{r}. Thus the openness follows. The uniqueness immediately follows from Theorems C and D.

Figure 1. Bers embedding of the once-punctured square torus with pleating rays of level ≤ 4 Figure 2. Bers embedding of a once-punctured “long” rectangular torus (λ = 0.001)

We emphasize that the set described in the above theorem can be computed, at least in a numerical way, by tracing the real locus of the function σ_{r} in the decreasing direction from the origin. Figure 1 and 2 were created in this way. The above theorem shows also that the entire function σ_{r} has no branch point on the pleating ray P_{r} and at the origin. Furthermore, H. Miyachi [26] recently proved that σ_{r} has no branch point at the cusp corresponding to θ_{r}. Summarizing the above, we obtain the next result.

Corollary 6.2. Each rational pleating ray P_{r} in T (Γ) with its endpoints is a regular analytic simple closed arc and σ_{r}(φ)^{2} is positive real and decreases to 4 when φ moves along the ray toward the cusp on the Bers boundary of T (Γ) corresponding to slope r.

We take a closer look at the analytic structure of pleating rays for the later use.

Theorem 6.3. Any two of rational pleating rays intersect transversally at the basepoint unless the tangent vectors have opposite directions there.

The above assertion seems true even when we allow all pleating rays. We, however, content ourselves here with the above form since we will require only the case of rational pleating rays in this paper. See the final part of this section explaining how to numerically compute the value of accessary parameter and a standard pair of generators of a Fuchsian group.

We should comment on the similar result for the Maskit embedding of the Teichmüller space of once-punctured tori (see [14], [25] and [38]). The Maskit embedding is a deformation space of terminal regular b-groups of type (1, 1) and it is realized as an unbounded domain in ℍ, where ∞ plays the same role as the base point in the Bers embedding even though no groups correspond to ∞. Rational pleating rays can also be considered for the Maskit embedding; among other things, Keen and Series [14, Proposition 3.2] showed that for a point μ = s + it of the rational pleating ray of slope p/q, the real part s tends to 2p/q as t → ∞. This result implies that the asymptotic behaviour of rational pleating rays at the base point (∞ for the Maskit embedding) characterizes them.

To prove Theorem 6.3, we need a preliminary result connecting the real analytic structure of the Teichmüller space with the complex analytic one, which might be of independent interest.

Let γ be a hyperbolic element in an arbitrarily given Fuchsian group Γ. We denote by ℓ_{γ} : T (Γ) → ℝ_{+} the length function on T (Γ) corresponding to γ, namely, ℓ_{γ}(φ) is hyperbolic length of the closed geodesic corresponding to γ in the Riemann surfaces Ω_{φ}^{−}/χ_{φ}(Γ), where Ω_{φ}^{−} = ℂ̂ \ \overline{f_{φ}(𝔻)}. Let ε_{γ}(φ) be the logarithm of multiplier of χ_{φ}(γ) with positive real part for φ ∈ T (Γ). Therefore, σ_{γ}^{2}(φ) = tr^{2}χ_{φ}(γ) = 2 cosh(ε_{γ}(φ)) + 2 and Re ε_{γ}(φ) > 0. If we knew the conformal mapping g_{φ} from 𝔻 onto Ω_{φ}^{−}, we could relate ℓ_{γ} to ε_{γ} by the formula

tr^{2}[g_{φ}^{−1} ∘ χ_{φ}(γ) ∘ g_{φ}] = 2 cosh ℓ_{γ}(φ) + 2.

It is, however, difficult to manage g_{φ}, in general. At least, we can say something at the origin.

Lemma 6.4. For a fixed φ ∈ B_{2}(𝔻, Γ), the formula

\frac{d}{dt}ℓ_{γ}(tφ)|_{t=0} = Re \frac{d}{dt}ε_{γ}(tφ)|_{t=0}

holds, where the differentiation is taken with respect to the real parameter t.

Proof. We refer the reader to [10] and [27] as standard textbooks for the basic facts used herein. In this proof, we take advantage of the lower half-plane ℍ^{∗} instead of the unit disk 𝔻, so that we regard Γ as a Fuchsian group acting on ℍ. This procedure is harmlessly done, as usual, through the Cayley transform z ↦ −i(1 + z)/(1 − z).

Let μ be a Beltrami differential for Γ in ℍ with ∥μ∥_{∞} < 1. We extend μ to be 0 in ℍ^{∗}. On the other hand, let μ^{∗} denote the Beltrami coefficient for Γ defined by

μ^{∗}(z) = \begin{cases}μ(z), & z ∈ ℍ, \\ \overline{μ(z̄)}, & z ∈ ℍ^{∗}. \end{cases}

We denote by F^{μ} and F_{μ} the quasiconformal self-homeomorphism of ℂ with Beltrami coefficients μ and μ^{∗}, respectively, normalized so as to fix 0 and 1. Note that F_{μ} maps ℍ onto itself while F^{μ} maps ℍ^{∗} conformally onto the quasidisk F^{μ}(ℍ^{∗}). (Note also that F^{μ} ∘ F_{μ}^{−1} is a conformal mapping from ℍ onto F^{μ}(ℍ).) Therefore, F_{μ}ΓF_{μ}^{−1} is a Fuchsian group while F^{μ}Γ(F^{μ})^{−1} is a Kleinian group acting on F^{μ}(ℍ^{∗}). It is known that the mapping Φ : μ ↦ S_{F^{μ}|_{ℍ^{∗}}} is a holomorphic submersion from the open unit ball of the Banach space of Beltrami differentials for Γ onto T (Γ).

For a given φ, we take a Beltrami differential μ in such a way that the Fréchet derivative of Φ at the origin sends μ to φ.

Let γ ∈ Γ be a hyperbolic element. By suitable Möbius conjugate, we may assume that γ is expressed in the form γ(z) = e^{ε_{0}}z, where ε_{0} > 0 is a constant. Set γ^{t} = F^{tμ}∘γ∘(F^{tμ})^{−1} and γ_{t} = F_{tμ} ∘ γ ∘ F_{tμ}^{−1} for t ∈ ℝ small enough. Then, by definition, these Möbius

TEICHMÜLLER SPACE OF A ONCE-PUNCTURED TORUS 23

transformations can be written as

γ^{t}(z) = e^{ε(t)}z and γ_{t}(z) = e^{ε^{∗}(t)}z.

We now recall Gardiner’s variation formula for the multiplier of a hyperbolic Möbius transformation under the quasiconformal mappings, [6]. As Miyachi remarked in [26], the formula is still valid not only for Fuchsian groups but also for Kleinian groups. Gardiner’s formula reads as

\frac{d}{dt}ε(t)|_{t=0} = ∫∫ _{A_{+}} \frac{μ(z)}{z^{2}}dxdy, and

\frac{d}{dt}ε^{∗}(t)|_{t=0} = ∫∫ _{A} \frac{μ^{∗}(z)}{z^{2}}dxdy = ∫∫ _{A_{+}} (\frac{μ(z)}{z^{2}} + \frac{\overline{μ(z)}}{z̄^{2}}) dxdy,

where A = {1 < |z| < e^{ε}} and A_{+} = A ∩ ℍ. Hence, we obtain (ε^{∗})′(0) = 2Re ε′(0).

From the definitions, the basic relations 2ℓ_{γ}(Φ(tμ)) = ε^{∗}(t) and ε_{γ}(Φ(tμ)) = ε(t) follow. Noting that Φ(tμ) = tφ + O(|t|^{2}) as t → 0, we get the desired relation between ℓ_{γ} and ε_{γ}.

Proof of Theorem 6.3. The argument and notation herein will be based on a paper of McMullen, [24]. Let r ∈ ℚ̂. Since P_{r} is a real locus of an analytic function defined in ℂ, the unit direction vector of P_{r} at the origin,

Θ_{r} = lim_{φ→0 in P_{r}} ∥φ∥_{𝔻}^{−1}φ,

is well defined. We show that the mapping r ↦ Θ_{r} is injective on ℚ̂.

First we observe that Θ_{r} is parallel to the tangent vector \frac{d}{dt}gr_{tγ}(X^{∗}) at t = 0, where gr_{tγ}(X^{∗}) is the point in T (Γ) corresponding to the time t grafting of the mirror image X^{∗} = X_{0}^{∗} of X along the simple closed geodesic representing a hyperbolic element γ ∈ Γ of slope r. Let u be the real part of the holomorphic function ε_{γ} : T (Γ) → ℂ given above. By the relation σ_{γ}^{2}(φ) = 2 cosh(ε_{γ}(φ)) + 2, the vector Θ_{r} has the opposite direction to the gradient ∇u at the origin because u is harmonic. On the other hand, McMullen’s theorem [24, Theorem 3.8] states the formula

\frac{d}{dt}gr_{tγ}(X^{∗}) = −∇ℓ_{γ}(X^{∗}),

where the gradient is taken with respect to the Weil-Petersson metric. In our case, the Teichmüller space T (Γ) is one-dimensional, and therefore, the Weil-Petersson (Kähler) metric is conformally equivalent to the Euclidean metric. Now, the formula ∇ℓ_{γ} = ∇u deduced from Lemma 6.4 connects these facts to conclude the above claim.

To show the injectivity, we next use the fact that grafting and twisting along γ are related by the relation \frac{d}{dt}gr_{tγ}(X^{∗}) = i· \frac{d}{dt}tw_{tγ}(X^{∗}) at the origin with a suitable orientation for γ (see the proof of Theorem 3.8 in [24]). Therefore, the (positive) earthquake path and the pleating locus (with respect to γ) form the right angle at the basepoint. Hence, it suffices to show that different earthquake paths starting from the basepoint have different directions. This assertion has been proved by Kerckhoff [15, Theorem 3.5] in a more general setting.

We conclude this section with a comment on the numerical computation of the accessory parameter and the Fuchsian group uniformizing a given once-punctured torus.

In the above, we have assumed that we know the value of the accessory parameter. However, in practical calculations, we need to know it in advance. We now indicate how to do that by using our methods developed above. Let λ ∈ ℂ \ {0, 1} be given. Instead of (4.5), we consider the differential equation

2y′′ + \{\frac{1}{2z^{2}(z − 1)^{2}} + \frac{1}{2(z − λ)^{2}} + \frac{t}{z(z − 1)(z − λ)}\} y = 0. (6.3)

Let α_{0} and α_{∞} be the same as in Section 4 and let L̂_{t}[α_{r}] be the transition matrix of (6.3) along the path α_{r} for r = 0, ∞. We set A(t) = L̂_{t}[α_{0}]M_{z_{0}}(A_{0}) and B(t) = L̂_{t}[α_{∞}]M_{z_{0}}(A_{∞}), where M_{z_{0}}(A_{r}) is given in Section 4. We consider the three entire functions F_{0}(t) = tr A(t), F_{∞}(t) = tr B(t) and F_{1}(t) = tr A(t)B(t). Note that the Möbius group Γ(t) generated by A(t) and B(t) is conjugate to a subgroup of PSL(2, ℝ) if and only if the values F_{0}(t), F_{∞}(t) and F_{1}(t) are all real. In particular, provided that the monodromy homomorphism Γ → Γ(t) corresponds to a point in T (Γ), the quasi-Fuchsian group Γ(t) is Fuchsian if and only if F_{0}(t), F_{∞}(t) and F_{1}(t) are all contained in (−∞, −2) ∪ (2, +∞). Since Γ(c(λ)) is Möbius conjugate with the original Fuchsian group Γ uniformizing the once-punctured torus X with the Teichmüller parameter τ satisfying λ(τ) = λ by Theorem 4.3, such a point t is a candidate for the value of the accessory parameter c(λ). Note also that Γ(t) is never Fuchsian for t with 0 < |t − c(λ)| < 2/∥ψ_{0}∥_{Y} by the Ahlfors-Weill theorem (see Section 2).

Hence, if we are given an initial point t_{0} which is sufficiently close to c(λ), we can construct a sequence t_{n} tending to the value c(λ) as follows. By Theorem 6.3, we can choose two of F_{0}, F_{∞}, F_{1}, say F_{0} and F_{1}, so that the corresponding pleating rays P_{0} and P_{1} are transversal at the basepoint. Then the point t = c(λ) will be determined (at least locally) as the intersection of the real loci of functions F_{0} and F_{1}. Set

t_{2j+1} = t_{2j} − i\frac{Im F_{0}(t_{2j})}{F′_{0}(t_{2j})} and
t_{2j+2} = t_{2j+1} − i\frac{Im F_{1}(t_{2j+1})}{F′_{1}(t_{2j+1})}

for j = 0, 1, …. Then, in a similar way to Newton’s method, the sequence t_{n} converges to c(λ) if t_{0} is sufficiently close to c(λ) (see, for example, [29, § 3.5.2]). Note here that the transversality of the pleating rays guarantees the convergence of t_{n}.

We make a few technical remarks. Since it is difficult to calculate the derivative of F_{r}, practically we replace it by a suitable difference quotient in the above formulae like the Secant method. It is typical to use (F_{r}(t_{n}) − F_{r}(t_{n−2}))/(t_{n} − t_{n−2}) as the difference quotient provided that t_{n} − t_{n−2} is sufficiently small.

It is difficult to give the initial point t_{0} a priori for a given λ. However, if we know the value of c(λ_{0}) for some λ_{0} (e.g., λ_{0} = 1/2), we may choose a finite sequence λ_{1}, λ_{2}, …, λ_{n} = λ so that λ_{j} and λ_{j−1} are close enough for j = 1, …, n. Then, we could compute c(λ_{j}) by using the value of c(λ_{j−1}) as the initial point t_{0} for λ_{j}. In this way, we could reach λ after n-times of this procedure.

Using the above method, we have drawn the graph of the function c(λ) in 0 < λ < 1/2 in Figure 3. Note that Hempel [8] has obtained the asymptotic formula

c(λ) = \frac{1}{2} − \frac{π^{2}}{2(− log |λ|)^{2}} (1 + o(1))

as λ → 0 in ℂ.

Figure 3. Graph of the accessory parameter c(λ) in 0 < λ < 1/2

7. Relationship with Heun’s differential equation

In this section, we explain that our differential equation (4.5) can be translated to Heun’s differential equation in the standard way. It may be more advantageous to use the latter form because that is more widely studied and the behaviour of coefficients is tamer than the former near the singularities (see [31] and its extensive references).

Heun’s equation has the form

u′′ + (\frac{c_{0}}{z} + \frac{c_{1}}{z − 1} + \frac{c_{2}}{z − λ}) u′ + \frac{abz − q}{z(z − 1)(z − λ)}u = 0,

where c_{0}, c_{1}, c_{2}, a, b and q are complex parameters satisfying the relation

c_{0} + c_{1} + c_{2} = a + b + 1.

More generally, a linear differential equation of the form u′′ + P u′ + Qu = 0 can be transformed locally to the form y′′ + Ry = 0 by putting u(z) = exp(− ∫^{z} P (ζ)dζ/2)y(z). In the case when c_{0} = c_{1} = c_{2} = a = b = 1 and q = (1 − t − c(λ))/2, Heun’s equation takes the form

u′′ + (\frac{1}{z} + \frac{1}{z − 1} + \frac{1}{z − λ}) u′ + \frac{2z + t + c(λ) − 1}{2z(z − 1)(z − λ)}u = 0 (7.1)

and the above-mentioned transformation

u = \frac{y}{\sqrt{z(z − 1)(z − λ)}}

reduces Heun’s equation to our equation (4.5). It is an interesting fact that the factor ω(z) := 1/\sqrt{z(z − 1)(z − λ)} forms a basis of the one-dimensional vector space of Abelian differentials on the torus defined as the algebraic curve w^{2} = z(z −1)(z −λ). In particular, (ω ∘ A) · A′ = ±ω holds for an automorphism A of Y = ℂ \ {0, 1, λ}.

Therefore, for a solution u of (7.1) and A ∈ Aut(Y), we can see that A_{1/2}^{∗}u := (u ∘ A)(A′)^{1/2} is a solution of (7.1) again.

We now set

F̃_{z_{1}} = (\begin{matrix}u′_{1} & u_{1} \\ u′_{2} & u_{2} \end{matrix}) ,

where u_{1} and u_{2} are the fundamental solutions of (7.1) at z_{1}, namely, the solutions satisfying the initial conditions u_{1}(z_{1}) = 0, u′_{1}(z_{1}) = 1 and u_{2}(z_{1}) = 1, u′_{2}(z_{1}) = 0, respectively, and for [α] ∈ Y [z_{1}, z_{2}] we denote by H̃_{[α]} the germ at z_{2} obtained by analytic continuation of the fundamental matrix F̃_{z_{1}} along the path α in Y. In the same way as in the case of (4.5), we also define the transition matrix L̃[α] = L̃_{t}[α] by the relation H̃_{[α]} = L̃[α]F̃_{z_{2}} for [α] ∈ Y [z_{1}, z_{2}]. Then, the similar relations to (i), (ii) and (iii) in Section 3 hold for these quantities. Noting the elementary identity

ω (\begin{matrix}y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix}ω(z_{1}) & 0 \\ ω′(z_{1}) & ω(z_{1}) \end{matrix}) (\begin{matrix}u_{1} \\ u_{2} \end{matrix}) = K(z_{1}) (\begin{matrix}u_{1} \\ u_{2} \end{matrix})

for the fundamental solutions of (4.5) and (7.1) at z_{1}, where

K(z) = ω(z) (\begin{matrix}1 & 0 \\ μ(z) & 1 \end{matrix}) , μ(z) = −\frac{1}{2} (\frac{1}{z} + \frac{1}{z − 1} + \frac{1}{z − λ}) ,

we have the fundamental relation F_{z_{1}}K = K(z_{1})F̃_{z_{1}} near z_{1} ∈ Y. By analytic continuation, we obtain H_{[α]}K(z_{2}) = K(z_{1})H̃_{[α]} for [α] ∈ Y [z_{1}, z_{2}], in particular,

L[α] = K(z_{1})L̃[α]K(z_{2})^{−1}. (7.2)

Let z_{1} be a point in Y = ℂ\{0, 1, λ}. We now take curves ε_{1}, ε_{2}, ε_{3} and ε_{4} in Y starting from z_{1}, rounding 0, 1, λ and ∞ once anticlockwise and ending at z_{1}, respectively, so that π_{1}(Y, z_{1}) = ⟨[ε_{1}], [ε_{2}], [ε_{3}], [ε_{4}]⟩ and [ε_{1}][ε_{2}][ε_{3}][ε_{4}] = 1. Then, the next result immediately follows.

Theorem 7.1. Let l_{t} and l̃_{t} be monodromy homomorphisms of differential equations (4.5) and (7.1) from π_{1}(Y, z_{1}) into SL(2, ℂ), respectively. We then have l_{t}[ε_{j}] = −l̃_{t}[ε_{j}] for j = 0, 1, 2, 3. In particular, l_{t} = l̃_{t} on the canonical image of the Fuchsian group Γ_{0} = Γ∩Γ′.

Corollary 7.2. The monodromy group is discrete for (4.5) if and only if so is that for (7.1).

Since L[α_{r}]M_{z_{0}}(A_{r}) is conjugate with L̃[α_{r}]M̃_{z_{0}}(A_{r}) in GL(2, ℂ), where M̃_{z_{0}}(A_{r}) = K(A_{r}(z_{0}))^{−1}M_{z_{0}}(A_{r})K(z_{0}) for r = 0, ∞, we obtain the following statement from Theorem 4.3.

Theorem 7.3. For suitable choices of representatives θ̃_{0} of θ_{0} and θ̃_{∞} of θ_{∞} in SU(1, 1),

χ̃_{tφ_{0}}(θ̃_{0}) = L̃_{t}[α_{0}]M̃_{z_{0}}(A_{0}), and χ̃_{tφ_{0}}(θ̃_{∞}) = L̃_{t}[α_{∞}]M̃_{z_{0}}(A_{∞})

for any t ∈ ℂ up to SL(2, ℂ)-conjugacy, where φ_{0} = p_{2}^{∗}(ψ_{0}).

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Department of Mathematics, Osaka City University, Osaka, 558-8585 Japan

  E-mail address: komori@sci.osaka-cu.ac.jp

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

  E-mail address: sugawa@math.sci.hiroshima-u.ac.jp