Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 307–328

HALF-TWISTS AND EQUATIONS IN GENUS 2

Aline Aigon-Dupuy

École Polytechnique Fédérale de Lausanne, Institut de Mathématiques CH-1015 Lausanne, Switzerland; aline.aigon@epfl.ch

Abstract. The uniformization problem is to find equations for the algebraic curve associated to a given hyperbolic surface. If one can describe corresponding group actions both on the spaces of algebraic curves and hyperbolic surfaces, the whole orbits can be uniformized at the same time. We study here the action of a group generated by half-twists on the space of hyperbolic surfaces of genus 2 with a non-trivial involution and describe the corresponding action on the equations for the corresponding algebraic curves.

1. Introduction

The uniformization theorem of Poincaré and Koebe allows to assert that any compact connected Riemann surface of genus g > 1 is conformally equivalent to a quotient of the upper half-plane H by a Fuchsian group, i.e. a discrete subgroup of PSL_{2}(R). On the other hand, a Riemann surface is also an algebraic curve defined by an equation. The classical uniformization problem is to relate explicitly the two descriptions.

In this context, a new approach to tackle this question was initiated in [6], and developed in [2], [8] and [1].

It consists first in working inside families of surfaces with the idea that surfaces tiled with the same pattern by the same type of polygon must have equations of the same form. Then, in defining groups actions on those families. The two groups, one acting on the hyperbolic surfaces and the other on the algebraic curves, are not necessarily the same but there exists a correspondence between their actions. With this approach, whole orbits can be uniformized at the same time.

In this article, we generalize the D_{5} -action on the space of real genus 2 M - curves with a real involution ℳ_{R}^{(2,3,0)}(Z/2 × Z/2) described by Buser and Silhol in [6] to the complex family F_{2} of those Riemann surfaces having a non-trivial involution.

There are, nevertheless, two major differences in the way we tackle the problem in this paper. The first difference is in the method: we use a quotient, namely the Riemann sphere ramified over 5 points, while the authors considered coverings to define the D_{5} -action in [6]. Secondly, the complex situation is less rigid than the real one, in particular in [6] the hyperbolic description was guided by the natural choice of a pants decomposition and Fenchel–Nielsen coordinates given by the real structures. Here, we have to work with marked Riemann surfaces in the Teichmüller space for the hyperbolic description of our group action while the algebraic one is made with unmarked Riemann surfaces in the moduli space. In particular, the two groups are different. The first one, G_{Q} , is a group of transformations of a special type of hyperbolic quadrilateral and can be identified with the Teichmüller modular group of the sphere with 5 points removed. The second is the symmetric group S_{5} , giving the 5 points on the quotient a symmetric role, and naturally appears as a quotient of G_{Q} . These two actions correspond and we give here this correspondence in terms of equations and of generators for the Fuchsian groups (see Theorem 4.6 and Table 3).

2000 Mathematics Subject Classification: Primary 30F10; Secondary 14H15, 32G15.

Partially supported by the Swiss National Science Foundation, Contract 20-68181.02.

The fact that the two groups are different means that G_{Q} intersects the Teichmüller modular group of genus two surfaces and thus allows to interpret this difference in terms of Dehn twists.

But more interestingly and surprisingly, the action of the whole group G_{Q} can be interpreted in terms of half-twists (see Theorem 5.1). Thus, by merging Theorems 4.6 and 5.1, we obtain the main result of this paper which can be expressed as the following.

Theorem. Let S be a genus 2 Riemann surface having a non-trivial involution φ.

Then, on the one hand, the Fuchsian group has a set of generators of the form

(e_{1}e_{3})^{2}, e_{3}e_{2}e_{1}, e_{1}e_{3}e_{2}, e_{3}e_{4}e_{1}, e_{1}e_{3}e_{4},

where e_{i}, ∈ PSL_{2}(R), i = 1, …, 4, with tr(e_{i}) = 0, and tr(∏_{i=1}^{4} e_{i}) = 0, and on the other hand the underlying algebraic curve has a normalized equation of the form

y^{2} = (x^{2} − 1)(x^{2} − a)(x^{2} − b).

For such a surface S, the half-twists along certain geodesics lead to surfaces having a non-trivial involution. Sets of generators for the Fuchsian groups and normalized equations for these surfaces can be explicitly deduced from those of S.

We have, in particular, the correspondence between half-twists and changes of equation given in Table 1.

The action of G_{Q} = ⟨η_{1}, …, η_{4}⟩ for η_{i}, i = 1, …, 4 as in Table 1, induces an action of S_{5} on the underlying algebraic curves.

word representing

the (oriented) geodesic  ordered e_{i} ’s  parameters
under which the  for the Fuchsian  for equations
half-twist is made  group
none  (e_{1}, e_{2}, e_{3}, e_{4})  (a, b)
η_{1}  (e_{3}e_{2})^{2}  (e_{1}, e_{3}, e_{3}e_{2}e_{3}, e_{4})  (1 − a, b(1 − a)/(b − a))
η_{2}  (e_{4}e_{3})^{2}  (e_{1}, e_{2}, e_{4}, e_{4}e_{3}e_{4}) (a(1 − b)/(a − b), (1 − b))
η_{3}  (e_{1}e_{4})^{2}  (e_{1}e_{4}e_{1}, e_{2}, e_{3}, e_{1}) ((b − a)/(b − 1), b/(b − 1))
η_{4}  (e_{2}e_{3}e_{4})^{2}  (e_{2}e_{3}e_{4}e_{1}, e_{2}, e_{3}, e_{4})  (1 − a, 1 − b)

Table 1.

2. Notation and preliminaries

We recall briefly some classical definitions and notation.

Definition 2.1. Let S be a Riemann surface of genus 2, and τ be the hyperelliptic involution on S.

An automorphism φ ∈ Aut(S), with φ ≠ τ is said to be non-trivial.

For any hyperelliptic Riemann surface S, the hyperelliptic involution τ is in the center of Aut(S). The reduced automorphism group of S is then

Aut^{r}(S) = Aut(S)/τ.

The classification of Riemann surfaces of genus two in terms of their reduced automorphism group is due to Bolza ([4]). It is summarized Table 2 as well as the inclusions between families.

Except for F_{5} , every Riemann surface of genus two having a non-trivial automorphism has at least one non-trivial involution, and then belongs to F_{2} .

Let S be a Riemann surface, τ the hyperelliptic involution and φ a nontrivial involution on S. The involutions φ and φτ have two fixed points, say p_{1} and p_{2} for φ and q_{1} and q_{2} for φτ, that satisfy

φ(q_{1}) = q_{2}, τ(p_{1}) = p_{2}.

Lemma 2.2. Let S, τ, φ as before. The covering p_{φ}: S ⟶ S/⟨φ, τ⟩ ≃ P^{1}(C) is ramified over 5 points among which 3 are the images of the Weierstrass points and the last two are p = p_{φ}(p_{1}) = p_{φ}(p_{2}) and q = p_{φ}(q_{1}) = p_{φ}(q_{2}). Those five points and the triple of them that lift to the set of Weierstrass points determine S completely.

Proof. This follows directly from the fact that the surfaces of genus 2, are, as all the hyperelliptic algebraic curves, determined by their Weierstrass points.

Family Aut^{r} Classical form for the equation
F_{2}  Z/2Z y^{2} = (x^{2} − 1)(x^{2} − a)(x^{2} − b)
F_{4}  D_{2} y^{2} = (x^{2} − 1)(x^{2} − a)(x^{2} − 1/a)
F_{6}  D_{3}  y^{2} = x^{6} − 2 a x^{3} + 1
F_{12}  D_{6}  y^{2} = x^{6} + 1
F_{24}  S_{4}  y^{2} = x(x^{4} − 1)
F_{5}  Z/5Z  y^{2} = x^{5} − 1
\xymatrix{ & F_{4} \ar@{^{(}->}[dr] \\ F_{12} \ar@{^{(}->}[ur] \ar@{^{(}->}[dr] & F_{24} \ar@{^{(}->}[u] \ar@{^{(}->}[d] & F_{2} \\ F_{5} & F_{6} \ar@{^{(}->}[ur]}

Table 2.

Corollary 2.3. Let r_{1}, …, r_{5} be five distinct points on P^{1}(C). There exist at most 10 different surfaces S_{j}, τ_{j}, φ_{j}, j = 1, …, 10, in the family F_{2} such that the coverings p_{φ_{j}} are ramified over the r_{i} ’s.

Proof. Each S_{j} correspond to the choice of a triple {r_{k}, r_{l}, r_{m}} of points among the r_{i} ’s that lift to the Weierstrass points of S_{j}.

3. Marked quadrilaterals

Let S, τ, φ as before. The hyperbolic structure on S induces, via p_{φ}, a structure of hyperbolic sphere with five cone points of angle π on the quotient S/⟨τ, φ⟩.

Such a surface can always be obtained by pasting the sides of a hyperbolic quadrilateral with interior angles adding up to π, as on Figure 1.

Figure 1.

This observation induces a particular presentation for the Fuchsian group of such a surface and motivates the following definitions.

Definition 3.1. An ordered system Q = (e_{1}, e_{2}, e_{3}, e_{4}), e_{i} ∈ PSL_{2}(R), is a marked quadrilateral if it satisfies the following conditions:

(i) tr(e_{i}) = 0, i = 1, …, 4. (ii) tr(∏_{i=1}^{4} e_{i}) = 0. (iii) The e_{i} ’s are positioned clockwise around the quadrilateral of vertices are the

fixed points of e_{1}e_{2}e_{3}e_{4} , e_{2}e_{3}e_{4}e_{1} , e_{3}e_{4}e_{1}e_{2} and e_{4}e_{1}e_{2}e_{3} (see Figure 2).

Remark 3.2. Using trace relations, one can easily show that the quadrilateral of (iii) above is a convex domain, delimited by the axes of the hyperbolic transformations e_{1}e_{2}e_{3} , e_{2}e_{3}e_{4} , e_{3}e_{4}e_{1} , e_{4}e_{1}e_{2} . As it is uniquely determined we will also denote it by Q.

Definition 3.3. We will denote by Q be the set of all marked quadrilaterals modulo the relation

(e_{1}, e_{2}, e_{3}, e_{4}) ∼ (e′_{1}, e′_{2}, e′_{3}, e′_{4}) ⟺ ∃ γ ∈ PSL_{2}(R), e′_{i} = γe_{i}γ^{−1}, i = 1, … 4.

Definition 3.4. Given a surface S_{0} of signature (0; 2, 2, 2, 2, 2), a quadrilateral fundamental domain for S_{0} is a marked quadrilateral Q = (e_{1}, e_{2}, e_{3}, e_{4}) such that Γ_{0}(Q) = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩ is a Fuchsian group for S_{0} .

We will denote by Q_{S_{0}} the set of all quadrilateral fundamental domains for S_{0} under the relation ∼.

Conversely, given Q ∈ Q, will denote by S_{0}(Q) the surface H/Γ_{0}(Q) = H/⟨e_{1}, e_{2}, e_{3}, e_{4}⟩.

Remarks 3.5. 1. As the e_{i} ’s are elliptic transformations of order 2, they completely determine their fixed points. So we may also denote by e_{i} the fixed point of e_{i} (notably on figures).

2. As a marked quadrilateral is defined up to direct isometry and separates into two triangles, it is also characterized by the following set of five lengths:

– the lengths l_{i} of the i-th sides given by:

cosh(\frac{1}{2}l_{i}) = \frac{1}{2}| tr(e_{i+1}e_{i+2}e_{i+3})|,

– the length l of the first diagonal given by

cosh(\frac{1}{2}l) = \frac{1}{2}| tr(e_{1}e_{2}e_{3}e_{4}e_{3}e_{4}e_{1}e_{2})| = \frac{1}{2}|(tr(e_{1}e_{2}))^{2} + (tr(e_{3}e_{4}))^{2} − 2|.

3.1. Transformations of a marked quadrilateral

Definition 3.6. (1) We define the following transformations on Q:

(i) the circular permutation:

σ_{0}: (e_{1}, e_{2}, e_{3}, e_{4}) ⟶ (e_{2}, e_{3}, e_{4}, e_{1}),

(ii) σ_{1} (see Figure 2):

σ_{1}(e_{1}, e_{2}, e_{3}, e_{4}) ⟶ (e_{3}, e_{2}, e_{3}e_{4}e_{1}e_{2}, e_{3}e_{4}e_{3}).

(2) We denote by G_{Q} , the group

G_{Q} = ⟨σ_{0}, σ_{1}⟩

Figure 2. The transformation σ_{1} .

Remarks 3.7. Let S_{0} be a hyperbolic surface of genus 0 with five cone points of angle π. Then σ_{0} and σ_{1} preserve Q_{S_{0}} .

The transformations σ_{0} and σ_{1} are of different nature. While σ_{0} only operates on the marking of Q but leaves the unmarked quadrilateral unchanged, σ_{1} is mainly devoted to changing the choice of the point among the five cone points on the sphere S_{0} which correspond to the vertices of Q.

For further use, we introduce the following transformations in G_{Q} : Note that σ_{2} and σ_{3} are of infinite order and that they do not have fixed

points in G_{Q} .

Proposition 3.8. Let S_{0} be a hyperbolic surface of signature (0; 2, 2, 2, 2, 2). G_{Q} acts transitively on Q_{S_{0}} .

Proof. Let Q and Q′ be two quadrilaterals of Q_{S_{0}} . We build a sequence of quadrilaterals of Q_{S_{0}} using elements of G_{Q} .

Let a_{1}, …, a_{4} be the four geodesic arcs in S_{0} corresponding to the sides of Q. The a_{i} ’s are oriented such that they all have the same source.

Start, if necessary, with a transformation of the form σ_{1}σ_{0}^{k} leading to a quadrilateral Q_{1} , such that the vertices of Q and Q_{1} correspond to the same point of S_{0} .

For each a_{i}, consider the number k_{i,Q_{1}} of connected components of a_{i} ∩ Q^{∘}_{1} . Let a_{1,1}, …, a_{1}, k_{1,Q_{1}} be the corresponding connected component.

We cut S_{0} along the sides of Q_{1} .

We treat the a_{i} ’s in the order given by the marking.

For a_{1} :

If k_{1,Q_{1}} = 0, then a_{1} corresponds to one of the sides of Q_{1} , and we go to a_{2} . If k_{1,Q_{1}} ≠ 0, we give the arcs a_{1,1}, …, a_{1,k_{1,Q_{1}}} the a_{1} ’s orientation. Using a transformation of the form σ_{0}^{k}, we get a quadrilateral Q_{2} such that

the source of a_{1,1} in Q_{2} is at the intersection of the first and the fourth sides. Its end point is then necessarily on the second or the third side of Q_{2} . Using σ_{2} in the first case and σ_{3} in the second one, we build a quadrilateral Q_{3} such

that the number k_{1,Q_{3}} of connected components of a_{1} ∩Q^{∘}_{3} is strictly smaller than k_{1,Q_{2}} = k_{1,Q_{1}} . Assume that one of the a_{1,k} ’s, say a_{1,k_{0}} penetrates into the triangle of sides a, b, c, where a is a part of a_{1,1} , b is a part of the side of Q_{3} which is not a side of Q_{3} , and c is a part of the side of Q_{2} which is not a side of Q_{3} . Then, the arc a_{1,k_{0}} must leave this triangle through b or c, the a_{1,k} ’s being disjoint. Thus, the arc is at the same time cut and pasted once. Therefore the number of connected components does not increase. As this situation is possible only in the case k_{0} > 1, we have k_{1,Q_{3}} < k_{1,Q_{2}} .

We treat similarly the path a_{2}, a_{3}, a_{4} , each construction respecting the preceding ones.

Lemma 3.9. Given a generic surface S_{0} of signature (0; 2, 2, 2, 2, 2), G_{Q} operates without fixed points on Q_{S_{0}} .

Proof. Let Q ∈ Q_{S_{0}} and σ ∈ G_{Q} such that σ · Q = Q. Then σ induces an isometry on S_{0} .

3.2. Identification of G_{Q} with the modular group Γ_{0,5}. The group G_{Q} acts on sets of four generators of Fuchsian groups of a sphere with five punctures. It is then naturally linked with the modular group of a sphere with five points removed, Γ_{0,5} . More precisely, given a transformation σ ∈ G_{Q} , one can associate the isotopy class of h_{σ} to σ, where h_{σ} is a homeomorphism of the sphere preserving the set of five points and corresponding to the deformation from any quadrilateral Q to σ(Q) (the quadrilateral being simply connected) mapping the interior onto the interior and the i-th side onto the i-th side.

Define the following transformations of G_{Q} :

\begin{matrix}η_{1}: (e_{1}, e_{2}, e_{3}, e_{4}) & ⟶ (e_{1}, e_{3}, e_{3}e_{2}e_{3}, e_{4}), \\ η_{2}: (e_{1}, e_{2}, e_{3}, e_{4}) & ⟶ (e_{1}, e_{2}, e_{4}, e_{4}e_{3}e_{4}), \\ η_{3}: (e_{1}, e_{2}, e_{3}, e_{4}) & ⟶ (e_{1}e_{4}e_{1}, e_{2}, e_{3}, e_{1}), \\ η_{4}: (e_{1}, e_{2}, e_{3}, e_{4}) & ⟶ (e_{2}e_{3}e_{4}e_{1}, e_{2}, e_{3}, e_{4}). \end{matrix}

Then, with the topological interpretation below, if r_{1}, …, r_{4} are the points of S_{0}(Q) corresponding to the middle of the sides and r_{5} is the point corresponding to the vertices, each η_{i} corresponds to an homeomorphism φ_{i} where φ_{i} is the identity outside a disk D_{i} enclosing r_{i+1} and r_{i+2} (subscript modulo 5) and φ_{i} exchanges r_{i+1} and r_{i+2} . According to J. S. Birman (see [3, Theorem 4.5, p. 164 and Remark, p. 165]), this means that the set of geometric transformations {η_{1}, η_{2}, η_{3}, η_{4}} is a set of generators for Γ_{0,5} with the following full list of relations:

\begin{aligned}η_{i}η_{j} & = η_{j}η_{i}, |i − j| ⩾ 2, \\ η_{i}η_{i+1}η_{i} & = η_{i+1}η_{i}η_{i+1}, \\ η_{1}η_{2}η_{3}η_{4}^{2}η_{3}η_{2}η_{1} & = 1, \\ (η_{1}η_{2}η_{3}η_{4})^{5} & = 1. \end{aligned}

We remark without expanding the computations that the following correspondence between the σ_{i} ’s and the η_{i} ’s allows one to find a full list of relations for Γ_{0,5} with the minimal set of generators {σ_{0}, σ_{1}}:

\begin{aligned}η_{1} & = σ_{0}(σ_{1}^{2}σ_{0}^{3})^{3}σ_{0}^{3}, \\ η_{2} & = (σ_{1}^{2}σ_{0}^{3})^{3}, \\ η_{3} & = σ_{0}^{3}(σ_{1}^{2}σ_{0}^{3})^{3}σ_{0}, \\ η_{4} & = σ_{0}^{2}σ_{3}σ_{2}σ_{0}^{3}σ_{1}. \end{aligned}  \begin{aligned}σ_{0} & = η_{4}^{2}η_{3}η_{2}η_{1}, \\ σ_{1} & = η_{2}^{−1}η_{3}^{−1}η_{4}^{−2}η_{2}^{−1}η_{4}, \end{aligned}

4. The genus two coverings

4.1. The surface S_{Q}. Given Q ∈ Q, we construct a genus 2 cover S_{Q} of the sphere S_{0}(Q) as shown in Figure 3. In other words, we construct a set of generators for a Fuchsian group Γ_{Q} of signature (2, 0), namely Γ_{Q} = ⟨(e_{1}e_{3})^{2}, e_{3}e_{2}e_{1}, e_{1}e_{3}e_{2}, e_{3}e_{4}e_{1}, e_{1}e_{3}e_{4}⟩, from the generators (e_{1}, e_{2}, e_{3}, e_{4}) of the Fuchsian group Γ_{0}(Q) of S_{0}(Q). Note that the Weierstrass points of the surface S_{Q} correspond to the conjugacy classes in Γ_{Q} of the centers of the elliptic transformations

e_{2}, e_{1}e_{4}e_{1}, e_{4}, e_{1}e_{2}e_{1}, e_{1}e_{2}e_{3}e_{4}, e_{2}e_{3}e_{4}e_{1}

(and to the middles of the sides labelled 2, 3, 4, 5, 7, 8, 9, 10 and the vertices of the polygon on Figure 3).

Figure 3.

Note also that S_{Q} has two non-trivial involutions, the fixed points of the first being the conjugacy classes of e_{1} , and e_{3}e_{1}e_{3} , those of the second being e_{3} and e_{1}e_{3}e_{1} .

By analogy with Lemma 2.2, we have:

Proposition 4.1. Let S be a surface of F_{2} with a non-trivial involution φ and S_{0} = S/⟨φ, τ⟩. Choose Q ∈ Q_{S_{0}} such that the images p and q of the fixed points of φ and φτ via p_{φ} are on the first and the third side of Q.

Then S is isometric to S_{Q}.

Proposition 4.2. The map Q ∋ Q ⟼ S_{Q} is an injection of Q into the Teichmüller space of Riemann surfaces of genus 2, 𝒯_{2} .

Proof. We choose Q_{0} = (e_{1}^{0}, e_{2}^{0}, e_{3}^{0}, e_{4}^{0}) ∈ Q as a quadrilateral of reference. As a quadrilateral is simply connected, for any Q ∈ Q, there exists, up to isotopy, a unique homeomorphism η_{Q}: Q_{0} ⟶ Q such that η_{Q} ∘ e_{i}^{0} = e_{i}.

This condition ensures that η_{Q} is extendable to a homeomorphism η̂_{Q} such that

\xymatrix{Q_{0} \ar@{^{(}->}[d] \ar[r]^{η_{Q}} & Q_{0} \ar@{^{(}->}[d] \\ S_{Q_{0}} \ar[r]^{η̂_{Q}} & S_{Q}}

is commutative.

The couple (S_{Q}, η̂_{Q}) is then a marked Riemann surface. The injectivity follows from the construction of η̂_{Q}.

We will denote

F_{2}(Q) = {S_{Q}, Q ∈ Q} ⊂ 𝒯_{2}.

Corollary 4.3. The induced action of G_{Q} on F_{2}(Q) is generically fixed point free.

Remark 4.4. While F_{2} is a subspace of the space of isometry classes of Riemann surfaces of genus 2, ℳ_{2} , F_{2}(Q) is a subspace of the Teichmüller space of genus 2, 𝒯_{2} . It is well known that the moduli space ℳ_{2} is a quotient of 𝒯_{2} by the modular group, generated by Dehn twists. We will be concerned by this point of view in Section 5.

4.2. Equations for surfaces in F_{2}(Q)—induced action on F_{2} . The classical normalization of the equations for the surfaces of F_{2} under the form y^{2} = P (x) is the one given in Table 2. More precisely, given a surface S ∈ F_{2} , we choose the involutions φ and φτ so that they lift x ⟼ −x. We also impose that one of the Weierstrass points has coordinates (1, 0).

This choice is equivalent to the choice of a (global) coordinate x on the quotient S/⟨φ, τ⟩ such that

(1) the images of the fixed points of φ and φτ via p_{φ} are mapped onto 0 and ∞,

(2) the image of a pair of the Weierstrass points exchanged by φ and φτ is mapped via p_{φ} onto 1.

Given such an x, the remaining two points of S/⟨φ, τ⟩ are mapped upon a and b and

(4.4)  y^{2} = (x^{2} − a)(x^{2} − 1)(x^{2} − b)

is an equation of S.

Note that conditions (1) and (2) do not determine precisely the choice of the coordinate x, since x/a, x/b, 1/x, a/x, and b/x would also fufill them.

As we want to describe the action of G_{Q} on F_{2}(Q) in terms of equations, we will, given a surface S_{Q}, make the choice of x precise by taking into account the geometry of Q as follows.

Let Q ∈ Q and let S_{0} = S_{0}(Q) the genus 0 surface obtained by gluing the sides of Q. We then choose a coordinate x_{Q} depending on the position of the cone points of S_{0} on Q. We denote by r_{1,Q}, …, r_{5,Q} these points in the order given by the marking of Q (in other words, if Q = (e_{1}, e_{2}, e_{3}, e_{4}), then r_{i,Q}, 1 ⩽ i ⩽ 4 is the conjugacy class of e_{i} in Γ_{0}(Q) = ⟨e_{1}, e_{2}, e_{3}, e_{4}⟩, and r_{5,Q} is the conjugacy class of e_{1}e_{2}e_{3}e_{4} ).

Figure 4.

More precisely, we choose x_{Q} such that:

(4.5)  x_{Q}(r_{1,Q}) = 0,  x_{Q}(r_{3,Q}) = ∞,  x_{Q}(r_{5,Q}) = 1.

We then call the couple

(a, b) = (x_{Q}(r_{2,Q}), (x_{Q}(r_{4,Q}))

normalized equation parameters for S_{Q}.

Let now Q ∈ Q, and σ ∈ G_{Q} . As S_{0}(Q) and S_{0}(σ(Q)) are isometric, we have

{r_{1,Q}, …, r_{5,Q}} = {r_{1,σ(Q)}, …, r_{5,σ(Q)}}.

This means that σ acts as a permutation on the set of the cone points of S_{0} . More precisely, we associate to σ the permutation σ̄ ∈ S_{5} defined for all Q ∈ Q and i ∈ {1, …, 5} by

r_{i,σ(Q)} = r_{σ̄^{−1}(i),Q}.

The map σ ⟼ σ̄ is a group homeomorphism as we have for all Q ∈ Q and i ∈ {1, …, 5}

r_{i,σ′σ(Q)} = r_{σ̄′^{−1}(i),σ(Q)} = r_{σ̄^{−1}σ̄′^{−1}(i),σ(Q)} = r_{\overline{(σ′σ)}^{−1}(i),σ(Q)}.

The image is the subgroup of S_{5} generated by the images of the generators σ_{0} and σ_{1} , for which we have

σ_{0}.(r_{1,Q}, r_{2,Q}, r_{3,Q}, r_{4,Q}, r_{5,Q}) = (r_{2,Q}, r_{3,Q}, r_{4,Q}, r_{1,Q}, r_{5,Q}); thus σ̄_{0} = (4, 3, 2, 1),

σ_{1}.(r_{1,Q}, r_{2,Q}, r_{3,Q}, r_{4,Q}, r_{5,Q}) = (r_{3,Q}, r_{2,Q}, r_{5,Q}, r_{4,Q}, r_{1,Q}); thus σ̄_{1} = (1, 5, 3). The map is surjective, since (4, 3, 2, 1) and (1, 5, 3) together generate S_{5} . Its kernel is the subgroup H_{Q} of those transformations such that for all Q ∈ Q and i ∈ {1, …, 5}

r_{i,Q} = r_{i,σ(Q)}.

The normalized equation parameters for S_{Q} depend only on the position of the cone points on Q. Then, for σ ∈ G_{Q} , it is σ̄ ∈ S_{5} rather than σ that acts on them. This action is as follows: the coordinates x_{Q} and x_{σ(Q)} on S_{0}(Q) = S_{0}(σ(Q)) are exchanged by the unique transformation A_{σ̄,Q} of P^{1} mapping x_{Q}(r_{1,σ(Q)}) onto 0, x_{Q}(r_{3,σ(Q)}) onto ∞ and x_{Q}(r_{1,σ(Q)}) onto 1, i.e., A_{σ̄,Q} is defined by:

\begin{aligned}A_{σ̄,Q}(z) & = (\frac{z − x_{Q}(r_{1,σ(Q)})}{z − x_{Q}(r_{3,σ(Q)})})(\frac{x_{Q}(r_{5,σ(Q)}) − x_{Q}(r_{3,σ(Q)})}{x_{Q}(r_{5,σ(Q)}) − x_{Q}(r_{1,σ(Q)})}) \\ & = (\frac{z − x_{Q}(r_{σ̄^{−1}(1),Q})}{z − x_{Q}(r_{σ̄^{−1}(3),Q})})(\frac{x_{Q}(r_{σ̄^{−1}(5),Q}) − x_{Q}(r_{σ̄^{−1}(3),Q})}{x_{Q}(r_{σ̄^{−1}(5),Q}) − x_{Q}(r_{σ̄^{−1}(1),Q})}) \end{aligned}

and we have

σ̄.(x_{Q}(r_{2}, Q), x_{Q}(r_{4}, Q)) = (x_{σ}(Q)(r_{2}, σ(Q)), x_{σ}(Q)(r_{4}, σ(Q)))
= (A_{σ̄,Q}(x_{Q}(r_{2,σ(Q)})), A_{σ̄,Q}(x_{Q}(r_{4,σ(Q)})))
= (A_{σ̄,Q}(x_{Q}(r_{σ̄^{−1}(2),Q})), A_{σ̄,Q}(x_{Q}(r_{σ̄^{−1}(4),Q}))).

For the generators σ_{0} and σ_{1} of G_{Q} , and a couple (a, b) of normalized equation parameters, we have

\begin{aligned}σ̄_{0} & = (4, 3, 2, 1), \\ Aσ̄_{0}(z) & = \frac{z − a}{z − b} \frac{1 − b}{1 − a}, \\ σ̄_{0}.(a, b) & = (\frac{1 − b}{1 − a}, \frac{a (1 − b)}{b (1 − a)}), \end{aligned}  \begin{aligned}σ̄_{1} & = (1, 5, 3), \\ Aσ̄_{1}(z) & = \frac{1}{1 − z} , \\ σ̄_{1}.(a, b) & = (\frac{1}{1 − a}, \frac{1}{1 − b}). \end{aligned}

We have thus proved:

Theorem 4.6. The action of G_{Q} on F_{2}(Q) (⊂ 𝒯_{2}) induces an action of the symmetric group S_{5} on F_{2} given in terms of parameters of equations by the generators

σ̄_{0}: (a, b) ⟼ (\frac{1 − b}{1 − a}, \frac{a (1 − b)}{b (1 − a)}),  σ̄_{1} : (a, b) ⟼ (\frac{1}{1 − a}, \frac{1}{1 − b}).

According to Corollary 2.3, given a generic surface of F_{2} , its quotient under its automorphism group has ten different genus two covers in F_{2} . A representative of each of these isomorphy classes is given in Table 3.

\begin{matrix}σ & σ̄ & σ̄.(a, b) \\ Id & Id & (a, b) \\ σ_{0} & (4, 3, 2, 1) & ((1 − b)/(1 − a), a(1 − b)/b(1 − a)) \\ σ_{1} & (1, 5, 3) & (1/(1 − a), 1/(1 − b)) \\ σ_{1}^{2} & (1, 3, 5) & ((−1 + a)/a, (−1 + b)/b)) \\ σ_{1}σ_{0} & (1, 4)(2, 5, 3) & ((−1 + a)/(−b + a), (b(−1 + a)/(−b + a)) \\ (σ_{1}σ_{0})^{2} & (2, 3, 5) & (1/(1 − a), −b/(−b + a)) \\ (σ_{1}σ_{0})^{3} & (1, 4) & ((b − a)/(b − 1), b/(b − 1)) \\ σ_{1}^{2}σ_{0} & (2, 3)(1, 4, 5) & ((b − a)/(b − 1), (a − b)/(a(b − 1)) \\ σ_{2} = σ_{0}^{2}(σ_{1}^{2}σ_{0}^{3})^{3}σ_{0}^{2} & (1, 2) & (a/(a − 1), (a − b)/(a − 1)) \\ σ_{3} = (σ_{0}σ_{1})^{3} & (3, 4) & (a(1 − b)/(a − b), 1 − b) \end{matrix}

Table 3.

Remarks 4.7. (1) While the action of G_{Q} on F_{2}(Q) is generically fixed point free, the induced action of S_{5} on F_{2} is such that there exists a subgroup of S_{5} , namely S_{{1,3}} × S_{{2,4,5}}, that preserves isometry classes.

(2) Call K_{Q} the group of transformations σ such that σ̄ ∈ S_{{1,3}} × S_{{2,4,5}}. It is neither normal in G_{Q} nor contains a normal subgroup of G_{Q} . Then S_{5} is the smallest group whose action on the set of genus two Riemann surfaces ramified over 5 points of P^{1} is transitive.

Table 4 gives a representative in K_{Q} for each element of S_{1,3} × S_{2,4,5} .

\begin{matrix}σ & σ̄ & σ̄.(a, b) & σ & σ̄ & σ̄.(a, b) \\ Id & Id & (a, b) & σ_{0}σ_{1}σ_{0} & (1, 3)(2, 5) & (a, a/b) \\ σ_{2}σ_{0}σ_{2}σ_{0}^{−1}σ_{2} & (2, 4) & (b, a) & σ_{0}σ_{1}^{2}σ_{0} & (1, 3)(4, 5) & (b/a, b) \\ σ_{0}^{2} & (1, 3)(2, 4) & (1/b, 1/a) & σ_{2}σ_{0}σ_{2}σ_{0}^{−1}σ_{2}σ_{0}^{−1}σ_{1}σ_{0} & (2, 5) & (a/b, 1/b) \\ σ_{0}^{2}σ_{2}σ_{0}σ_{2}σ_{0}^{−1}σ_{2} & (1, 3) & (1/a, 1/b) & σ_{2}σ_{0}σ_{2}σ_{0}^{−1}σ_{2}σ_{0}^{−1}σ_{1}^{2}σ_{0} & (4, 5) & (1/a, b/a) \\ σ_{0}^{−1}σ_{1}σ_{0} & (2, 5, 4) & (1/b, a/b) & σ_{2}σ_{0}σ_{2}σ_{0}^{−1}σ_{2}σ_{0}σ_{1}σ_{0} & (1, 3)(2, 4, 5) & (a/b, a) \\ σ_{0}^{−1}σ_{1}^{2}σ_{0} & (2, 4, 5) & (b/a, 1/a) & σ_{2}σ_{0}σ_{2}σ_{0}^{−1}σ_{2}σ_{0}σ_{1}^{2}σ_{0} & (1, 3)(2, 5, 4) & (b, b/a) \end{matrix}

Table 4.

Remark 4.8. In [6] the authors identify the space 𝒫_{2} of pairs of pants having the lengths of two boundary geodesics equal, with the space ℳ_{R}^{(2,3,0)}(Z/2 Z × Z/2 Z) of the real genus 2 curves with 3 real components and whose real automorphisms group contains Z/2 Z × Z/2 Z. More precisely, given a pair of pants P of 𝒫_{2} , they consider on the one hand the surface S_{P} obtained by gluing 2 copies of P with twist 0 on each component, and on the other hand the real algebraic curve whose real components correspond to the boundary components of P. The surface S_{P} clearly belong to F_{2} as the isometry exchanging the two boundary components of same lengths on P can be extended to an isometry φ of S_{P} . The involution φ can be normalized in (x, y) ⟼ (−x, y) and the algebraic curve has equation y^{2} = (x^{2} − a)(x^{2} − 1)(x^{2} − b), 0 < a < 1 < b < 1.

The authors then define (see [6, 5.17]) a D_{5} -action on 𝒫_{2} both in terms of the lengths (l_{1}, l_{2}) (where a pair of pants has one boundary component of length 2l_{1} , and two of lengths 2l_{2} ) and of the parameters (a, b) of real equation above. We will show that our Theorem 4.6 is in fact a generalization of this result.

Let P be the pair of pants given by length (l_{1}, l_{2}). We denote, as in [6], by l̂_{1} the length of the common perpendicular to the two boundary components of length 2 l_{2} , and by l̂_{2} that of the common perpendicular arcs to the boundary of length 2 l_{1} and each of those of length 2 l_{2} . Those arcs cut P into two copies of a right-angled hyperbolic hexagon given by the lengths (l_{1}, l̂_{2}, l_{2}, l̂_{1}, l_{2}, l̂_{2}) in cyclic order. Each copy of this hexagon can be cut into two isometric mirror pentagons along the common perpendicular to the sides of length l_{1} and l̂_{1} (see Figure 5).

The hyperbolic surface S_{P} is isometric to the surface S_{Q_{P}} where Q_{P} is the quadrilateral obtained from P as on Figure 5.

Figure 5.

It is easily shown, using, for example, trigonometric formulas in triangles and trace relations (given in [5]), that Q_{P} is given by the following trace relations:

(4.8)

2 | tr(e_{2}e_{3})| = | tr(e_{1}e_{3})| | tr(e_{1}e_{2})| = 4 cosh(h_{1}) cosh(\frac{1}{2} l̂_{1}),
2 | tr(e_{2}e_{3})| = | tr(e_{1}e_{3})| | tr(e_{1}e_{2})| = 4 cosh(h_{1}) cosh(\frac{1}{2} l̂_{1}),
2 | tr(e_{2}e_{3}e_{4})| = | tr(e_{1}e_{2})| | tr(e_{3}e_{4}e_{1})| = 4 cosh(\frac{1}{2} l̂_{1}) cosh(l_{2}),
2 | tr(e_{4}e_{1}e_{2}e_{1})| = | tr(e_{3}e_{4}e_{1})| | tr(e_{3}e_{1}e_{2})| = 4 cosh(l_{2}) cosh(l̂_{2}),
2 | tr(e_{4}e_{1}e_{2}) | = | tr(e_{3}e_{4})| | tr(e_{3}e_{1}e_{2})| = 4 cosh(l̂_{2}) cosh(\frac{1}{2}l_{1}),
2 | tr(e_{1}e_{4})| = | tr(e_{3}e_{4})| | tr(e_{1}e_{3})| = 4 cosh(\frac{1}{2}l_{1}) cosh(h_{1}).

Conversely if a marked quadrilateral Q = (e_{1}, e_{2}, e_{2}, e_{4}) verifies relations (4.8), one can build from Q the right-angled hexagon as in Figure 5 and thus the pair of pants P. This pair of pants P is then given by the lengths

(l_{1}, l_{2}) = (arccosh(\frac{1}{2}| tr(e_{3}e_{4})|^{2} − 1), arccosh(| tr(e_{3}e_{4}e_{1})|)).

From the algebraic point of view, the fact that the boundary components of P correspond to the real components of the algebraic curve

y^{2} = (x^{2} − 1)(x^{2} − a)(x^{2} − b), 0 < a < 1 < b

ensures that the coordinate induced on the quotient S_{P} /⟨φ, τ⟩ via (x, y) ⟼ x^{2} is in fact the coordinate x_{Q_{P}} defined by (4.5). In particular (a, b) are the normalized equation parameters for S_{Q_{P}} .

Let us now consider the transformations of G_{Q} :

\begin{matrix}ϱ_{1} : (e_{1}, e_{2}, e_{3}, e_{4}) & ⟼ (e_{2}e_{3}e_{4}e_{1}, e_{2}, e_{1}, e_{1}e_{3}e_{1}), \\ ϱ_{2} = σ_{0}^{2} : (e_{1}, e_{2}, e_{3}, e_{4}) & ⟼ (e_{3}, e_{4}, e_{1}, e_{2}). \end{matrix}

Straightforward computations show that ϱ_{1} and ϱ_{2} preserve relations (4.8), that ϱ_{1} is of order 5, and that the subgroup ⟨ϱ_{1}, ϱ_{2}⟩ of G_{Q} is isomorphic to the dihedral group D_{5} .

Using the above correspondence between this description and that given in [6], and the expression of ϱ̄_{1} and ϱ̄_{2} in terms of normalized equation parameters, we get

\begin{aligned}ϱ_{1}.(l_{1}, l_{2}) & = (2 h_{1}, l̂_{2}), \\ ϱ_{2}.(l_{1}, l_{2}) & = (l̂_{1}, l̂_{2}), \end{aligned}  \begin{aligned}ϱ̄_{1}.(a, b) & = (\frac{b (1 − a)}{b − a}, \frac{b}{b − a}), \\ ϱ̄_{1}.(a, b) & = (\frac{1}{b}, \frac{1}{a}). \end{aligned}

These are exactly the generators for D_{5} as given in [6].

Finally, we note that in [6], each orbit under D_{5} consisted of five a priori different isometry classes (i.e., complex isomorphy classes) while the actions of G_{Q} and S_{5} involve 10 different surfaces. The remaining five surfaces were in fact obtained by transporting the D_{5} -action from 𝒫_{2} onto a space of Riemann surfaces with a half-twist. The next section is devoted to showing that the action of G_{Q} can in fact be completely interpreted in term of half-twists.

Examples 4.9. We give here some exact examples. The first two surfaces are isometric, and thus complex isomorphic as algebraic curves, to surfaces that can be found in [6]. However, for both examples the reduced automorphism groups contain at least two different involutions. For each of them, the involutions considered here and in [6] are such that the genus 0 quotient are not isometric. This in particular implies that the genus two curves described here are not real isomorphic to those in [6] and that the transformed surfaces are not isometric to those under the D_{5} -action in [6].

I have never found Example 3 in the literature.

1. Let Q_{0} be the totally regular quadrilateral, i.e., such that σ_{0}(Q_{0}) = Q_{0} , defined in terms of length of its sides and first diagonal by:

cosh(l_{i}) = 3 + 2 \sqrt{2} , cosh(l) = 4 \sqrt{2} + 5.

Then the pair (a, b) of normalized equation parameters for S_{Q_{0}} must satisfy

\overline{σ_{0}}.(a, b) = (a, b),  i.e.,  a = −i and b = i or a = i and b = −i,

and thus y^{2} = (x^{2} − 1)(x^{4} + 1) is an equation for S_{Q_{0}} . As an algebraic curve, S_{Q_{0}} is complex (but not real) isomorphic to that of equation

y^{2} = (x^{2} − 1)(x^{2} − (3 + 2\sqrt{2} ))(x^{2} − (3 − 2\sqrt{2} )).

2. Let Q_{1} be the quadrilateral such that Q_{1} = σ_{1}(Q_{1}), then the surface S_{Q_{1}} must satisfy

\overline{σ_{1}}.(a, b) = (a, b) i.e., \{a, b\} = \{\frac{1}{2}(1 + i \sqrt{3} ), \frac{1}{2}(1 − i \sqrt{3} )\}.

3. Let Q_{2} be the quadrilateral given by (in terms of hyperbolic cosine of length of the sides and of the first diagonal)

(3 + 2\sqrt{3} , 3 + 2\sqrt{3} , 3 + 2\sqrt{3} , 3 + 2\sqrt{3} , 3 + 2\sqrt{3} ).

Note that Q_{2} is obtained by gluing two copies of a triangle with interior angles (\frac{1}{6}π, \frac{1}{6}π, \frac{1}{6}π) we obtain Q_{2} . This implies that D_{4} ⊂ Aut(S_{Q_{2}}) (and thus belongs to the family F_{4} of Table 2) and that the surface S_{Q_{2}}/D_{4} has an automorphism of order 3. We then have

b = \frac{1}{a}, \frac{a + \frac{1}{a}}{2} = ±i\sqrt{3} .

From easy but technical considerations (see [1] for more details) on real structures and on the position of the unit circle on the quotient, one can also deduce that in fact \frac{1}{2}(a + 1/a) = i\sqrt{3} , and that the normalized equation parameters are:

(a, b) = (a, \frac{1}{a}) = (i(\sqrt{3} − 2), i(\sqrt{3} + 2)).

As the surface belongs to F_{4} , one can find only six a priori non-isometric transformed surfaces under the actions of G_{Q} and S_{5} . We give them in terms of length of the quadrilaterals and in normalized equation parameters in Table 5.

Note that the two last surfaces have the real structure induced by x ⟼ 1/x̄. Note also the two first ones have the real structures induced by x ⟶ i (x̄−i)/(x̄+i) (I wish to thank R. Silhol for this last remark). They are thus isomorphic as they are conjugated.

One can easily verify that the third and the fourth have no real structures. Exact examples without real structure are very rare in the literature.

σ  cosh(l_{1})  cosh(l_{2})  cosh(l_{3}) cosh(l_{4})  cosh(l)
Id  3 + 2\sqrt{3}  3 + 2\sqrt{3}  3 + 2\sqrt{3} 3 + 2\sqrt{3} 3 + 2\sqrt{3}
σ_{0}  3 + 2\sqrt{3}  3 + 2\sqrt{3}  3 + 2\sqrt{3} 3 + 2\sqrt{3} 11 + 6\sqrt{3}
σ_{1}  7 + 4\sqrt{3}  1 + \sqrt{3}  3 + 2\sqrt{3} 5 + 3\sqrt{3} 9 + 5\sqrt{3}
σ_{1}^{2}σ_{0}  3 + 2\sqrt{3}  1 + \sqrt{3}  7 + 4\sqrt{3} 5 + 3\sqrt{3} 5 + 3\sqrt{3}
σ_{2}  11 + 6\sqrt{3}  3 + 2\sqrt{3}  3 + 2\sqrt{3} 3 + 2\sqrt{3} 3 + 2\sqrt{3}
(σ_{1}σ_{0})^{3}  51 + 30\sqrt{3}  3 + 2\sqrt{3}  3 + 2\sqrt{3} 3 + 2\sqrt{3} 21 + 12\sqrt{3}
σ  a  b
Id  i(\sqrt{3} − 2)  i(\sqrt{3} + 2)
σ_{0}  −i(\sqrt{3} + 2)  −i(\sqrt{3} − 2)
σ_{1}  1 − i(\sqrt{3} + 2)  1 − i(\sqrt{3} − 2)
σ_{1}^{2}σ_{0} 1 + i(\sqrt{3} − 2)  1 + i(\sqrt{3} + 2)
σ_{2}  1 − i(\sqrt{3} + 2) (1 + i(\sqrt{3} + 2))^{−1}
(σ_{1}σ_{0})^{3} 1 + i(\sqrt{3} − 2) (1 − i(\sqrt{3} − 2))^{−1}

Table 5.

5. Twist and half-twists

Recall that K_{Q} is the subgroup of transformations σ in G_{Q} such that σ̄ as in Section 4.2 belongs to S_{1,3} × S_{2,4,5} . As for any quadrilateral Q the surfaces S_{Q} and S_{σ(Q)} are isometric but correspond to a priori different points in Teichmüller space 𝒯_{2} , they are linked by an element of the modular group.

On the other hand, K_{Q} ⊊ G_{Q} , and the main result of this section is that G_{Q} is in fact generated by half-twists in the following sense . Let Q = (e_{1}, e_{2}, e_{3}, e_{4}) ∈ Q. An oriented geodesic of the surface S_{Q} is represented by a word m(e_{1}, e_{2}, e_{3}, e_{4}) in the letters e_{1}, e_{2}, e_{3} , and e_{4} up to conjugacy in Γ_{Q}. The word m then represents a free homotopy class for surfaces in {S_{Q}, Q ∈ Q}.

Let σ ∈ G_{Q} . We will say that σ corresponds to a half-twist (or a Dehn twist) along a geodesic m if given any Q = (e_{1}, e_{2}, e_{3}, e_{4}), one can go from S_{Q} to S_{σ(Q)} by a half-twist (or a Dehn twist) along the geodesic m(e_{1}, e_{2}, e_{3}, e_{4}) of S_{Q}. Of course, as on the one hand the elements of G_{Q} preserve F_{2}(Q) and on the other hand only the twists along the geodesics stable under aut(S) preserve F_{2}(Q), m(e_{i}) has to be conjugated in the Fuchsian group Γ_{Q} to e_{1}m(e_{i})e_{1} or e_{1}m(e_{i})^{−1}e_{1} . We will also consider simultaneous Dehn twists along m(e_{i}) and e_{1}m(e_{i})e_{1} , in the case where the two corresponding geodesics are disjoint.

5.1. Half-twists. We show that the generators for G_{Q} given in Section 3.2 act on F (Q) as half-twists.

Theorem 5.1. The transformation η_{1}: (e_{1}, e_{2}, e_{3}, e_{4})⟶(e_{1}, e_{3}, e_{3}e_{2}e_{3}, e_{4}) is a half-twist along the geodesic represented by the word (e_{3}e_{2})^{2} = w^{1}(e_{1}, e_{2}, e_{3}, e_{4}).

The transformation η_{2}: (e_{1}, e_{2}, e_{3}, e_{4}) ⟶ (e_{1}, e_{2}, e_{4}, e_{4}e_{3}e_{4}) is a half-twist along the geodesic represented by the word (e_{4}e_{3})^{2} = w^{2}(e_{1}, e_{2}, e_{3}, e_{4}).

The transformation η_{3}: (e_{1}, e_{2}, e_{3}, e_{4}) ⟶ (e_{1}e_{4}e_{1}, e_{2}, e_{3}, e_{1}) is a half-twist along the geodesic represented by the word (e_{1}e_{4})^{2} = w^{3}(e_{1}, e_{2}, e_{3}, e_{4}).

The transformation η_{4}: (e_{1}, e_{2}, e_{3}, e_{4}) ⟶ (e_{2}e_{3}e_{4}e_{1}, e_{2}, e_{3}, e_{4}) is a half-twist along the geodesic represented by the word (e_{2}e_{3}e_{4})^{2} = w^{4}(e_{1}, e_{2}, e_{3}, e_{4}).

The group G_{Q} = ⟨η_{1}, η_{2}, η_{3}, η_{4}⟩ is thus generated by half-twists.

Proof. We treat the four cases with the same argument. Namely, for any Q, we exhibit pants decompositions 𝒟_{i} and 𝒟′_{i} on S_{Q} and S_{η_{i}(Q)} , respectively, such that:

(1) The four pairs of pants obtained by cutting S_{Q} along 𝒟_{i} and S_{η_{i}(Q)} along 𝒟′_{i} are isometric.

(2) The three geodesics of 𝒟_{i} on S_{Q} and that of 𝒟′_{i} on S_{η_{i}(Q)} correspond to the same decomposition according to the marking: if Q = (e_{1}, e_{2}, e_{3}, e_{4}) and η_{i}(Q) = (e_{1}^{i}, e_{2}^{i}, e_{3}^{i}, e_{4}^{i}), there exist three words in four letters w^{i} = w_{1}^{i}, w_{2}^{i} and w_{3}^{i} such that

𝒟_{i} = (w_{1}^{i}(e_{j}), w_{2}^{i}(e_{j}), w_{3}^{i}(e_{j})), 𝒟′_{i} = (w_{1}^{i}(e_{j}^{i}), w_{2}^{i}(e_{j}^{i}), w_{3}^{i}(e_{j}^{i})).

Conditions (1) imply in particular that the length parts of the Fenchel–Nielsen coordinates of S_{Q} and S_{η_{i}(Q)} associated to the pants decompositions 𝒟_{i} and 𝒟′_{i}, respectively, are the same. With condition (2), this means that the transformation η_{i} corresponds to a change of these Fenchel–Nielsen parameters that affects only their twist part. In other words, η_{i} corresponds to a product of twist deformations (not necessarily Dehn twists) along the geodesics of the decomposition 𝒟_{i} of the marked Riemann surface S_{Q}.

It remains to show that the values of the twist parameters are 0 for w_{2}^{i} and w_{3}^{i} and \frac{1}{2} for w_{1}^{i}.

The first part is achieved by considering geodesics crossing w_{2}^{i} and w_{3}^{i} and observing that their word expression is unchanged by η_{i}.

To show that the twist is \frac{1}{2} on w_{1}^{i}, we use the underlying algebraic curves observing that η_{i}^{2} belongs to K_{Q} while η_{i} does not.

We take:

for η_{1} : w_{1}^{1}(e_{i}) = (e_{3}e_{2})^{2} , w_{2}^{1}(e_{i}) = e_{1}e_{2}e_{3} , w_{3}^{1}(e_{i}) = e_{2}e_{3}e_{1} , for η_{2} : w_{1}^{2}(e_{i}) = (e_{4}e_{3})^{2} , w_{2}^{2}(e_{i}) = e_{3}e_{4}e_{1} , w_{3}^{2}(e_{i}) = e_{1}e_{3}e_{4} , for η_{3} : w_{1}^{3}(e_{i}) = (e_{1}e_{4})^{2} , w_{2}^{2}(e_{i}) = e_{3}e_{1}e_{4} , w_{3}^{2}(e_{i}) = e_{1}e_{4}e_{3} , for η_{4} : w_{1}^{4}(e_{i}) = (e_{2}e_{3}e_{4})^{2} , w_{2}^{4}(e_{i}) = e_{2}e_{3}e_{4}e_{3} , w_{3}^{4}(e_{i}) = e_{3}e_{2}e_{3}e_{4} .

Now consider the geodesic represented by the word (e_{2}e_{3}e_{4})^{2} . It crosses the geodesics represented by w_{2}^{1}(e_{i}) and w_{3}^{1}(e_{i}) (or w_{2}^{2}(e_{i}) and w_{3}^{2}(e_{i})) but not the one represented by w_{1}^{1}(e_{i}) (or w_{1}^{2}(e_{i})) and its word expression is invariant by occurrences of η_{1} (or η_{2} ). In particular, its length is unchanged. According to A. Douady in [7, Exposé 7], this means that the twist parameter along the geodesic represented by w_{2}^{1}(e_{i}) and w_{3}^{1}(e_{i}) when applying η_{1} (or w_{2}^{2}(e_{i}) and w_{3}^{2}(e_{i}) when applying η_{2} ) is zero.

The same arguments for the geodesics represented by (e_{2}e_{1}e_{4})^{2} for η_{3} and (e_{2}e_{2})^{2} for η_{4} shows the nullity of the twist parameter on w_{2}^{3}(e_{i}) and w_{3}^{3}(e_{i}) when applying η_{3} and on w_{2}^{4}(e_{i}) and w_{3}^{4}(e_{i}) when applying η_{4} .

Remark 5.2. Theorems 4.6 and 5.1 together allow to see some half-twists and Dehn twists on the equation of the associated algebraic curve.

Remark 5.3. As working on groups with representation by generators and relations is not an easy thing, we do not have a precise idea of what is possible in terms of twists with elements of G_{Q} .

Note for example that simultaneous half-twists along disjoint geodesics exchanged by the non-trivial involutions of surfaces S_{Q} do not correspond to elements of G_{Q} in general. We give an example that illustrates this fact. Let P ∈ 𝒫_{2} , and S_{P} the genus two surface constructed from P as in (4.8). According to [6], for a generic P, the transforms of S_{P} under G_{Q} either have a real structure with 2 components or a real structure with 3 components. Now consider the surface S̃_{P} obtained from S_{P} by making simultaneous half-twists along the two real components exchanged by the involution of S_{P} . Then, one can show that for a generic P, S_{P} has no real structure with more than one component, and thus does not belong to the transforms of S_{P} . See [1] for more details.

5.2. Dehn twists along the sides of the quadrilateral. There are several motivations to not consider only Dehn twists as double half-twists in our situation. First, as mentionned in Remark 5.3, simultaneous half-twists are not allowed in general while we will see below an example of a correspondence between an element of G_{Q} and simultaneous Dehn twists.

Another motivation to deal with Dehn twists separately is their difference in nature with half-twists: the Dehn twists are defined on the topological surface T_{2} of genus two, while half-twists only make sense on the (marked) Riemann surfaces. In particular, the composition of different transformations corresponding to halftwists can only be considered as successive operations on successively different Riemann surfaces. On the other hand, it makes sense to compose Dehn twists even along geodesics which are not disjoint on the topological surface T_{2} .

We will show that in a way the group law in the mapping class group and in the subgroup K_{Q} and G_{Q} are reversed.

Let Q = (e_{1}, e_{2}, e_{3}, e_{4}) ∈ Q, we choose S_{Q} as a model for T_{2} . As S_{Q} is a hyperbolic surface, each homotopy class of a closed path c on T_{2} is represented by a unique geodesic of S_{Q}, i.e., the conjugacy class of a word w in the letters e_{1}, e_{2}, e_{3} and e_{4} . We will denote by τ_{w(e_{i}}) the change of the marking corresponding to the twist along c on T_{2} .

We have then

Proposition 5.4. Let τ = τ_{w_{1}(e_{i})}∘⋯∘τ_{w_{k_{1}} (e_{i})} and τ′ = τ_{m′_{1}(e_{i})}∘⋯∘τ_{m′_{k_{2}} (e_{i})} be two products of Dehn twists along geodesics of S_{Q}, for any Q = (e_{1}, e_{2}, e_{3}, e_{4}) ∈ Q.

Assume that there exist transformations σ and σ′ in K_{Q} such that for any Q = (e_{1}, e_{2}, e_{3}, e_{4}) ∈ Q,

S_{σ(Q)} = τ(S_{Q}) and S_{σ′(Q)} = τ′(S_{Q}).

Then

(i) τ ∘ τ′(S_{Q}) = S_{σ′∘σ(Q)} . (ii) For σ̃ ∈ K_{Q} , and τ σ̃ = τ_{w_{1}(σ̃(e_{i}))} ∘ ⋯ ∘ τ_{w_{k_{1}} (σ̃(e_{i}))} , we have

τσ̃(S_{Q}) = S_{σ̃^{−1}σσ̃(Q)}.

Proof. It is well known that if α is an homeomorphism of a surface T_{2} , and c is a homotopy class of a simple closed path on T_{2} , then τ_{α(c)} = ατ_{c}α^{−1} (see for example [3]).

Point (i) is a direct consequence of this fact. We have

τ ∘ τ′(S_{Q}) = τ ∘ (τ_{w_{1}(e_{i})} ∘ ⋯ ∘ τ_{w_{k′} (e_{i})})(S_{Q})
= τ ∘ ((τ^{−1} τ_{τ(m′_{1}(e_{i}))} τ) ∘ ⋯ ∘ (τ^{−1} τ_{τ(m′_{k′} (e_{i}))} τ)) (S_{Q})
= (τ_{τ(m′_{1}(e_{i}))} ∘ ⋯ ∘ τ_{τ(m′_{k′} (e_{i}))}) ∘ τ (S_{Q})
= (τ_{τ(m′_{1}(e_{i}))} ∘ ⋯ ∘ τ_{τ(m′_{k′} (e_{i}))}) (S_{σ(Q)})

= (τ_{m′_{1}(σ(e_{i}))} ∘ ⋯ ∘ τ_{m′_{k′} (σ(e_{i}))}) (S_{σ(Q)}) = τ′_{σ}(S_{σ(Q)}) = S_{σ′∘σ(Q)}.

(ii) As σ̃ belongs to K_{Q} , there exists τ̃ ∈ Γ_{2} such that τ̃(S_{Q}) = S_{σ̃(Q)} and τ̃(w_{k}(e_{i})) = τ̃(w_{k}(σ̃(e_{i}))), and the result follows from (i).

Figure 6.

We end by showing in an example how, given a Dehn twist or a product of Dehn twists along geodesics exchanged by the involutions of S_{Q}, Q = (e_{1}, e_{2}, e_{3}, e_{4}), one can recover “by hand” the corresponding transformation of G_{Q} .

Consider the topological model on the right-hand side of Figure 6 for S_{Q}. Let τ_{2} = τ_{(e_{3}e_{4}e_{1})} ∘ τ_{(e_{1}e_{3}e_{4})} = τ_{(e_{1}e_{3}e_{4})} ∘ τ_{(e_{3}e_{4}e_{1})} . We obtain the correspon-

dence between τ_{2} and the transformation σ = σ_{0}σ_{2}σ_{3}σ_{0}^{3}σ_{1}σ_{0} using the techniques developed in Section 3 as follows. We first determine the homotopy classes of the images under τ_{2} of the geodesics of S_{Q} corresponding to the sides of the different copies of Q. The corresponding geodesics are mapped, in the quotient S_{0}(Q) = S_{0}(σ(Q)), onto the sides of the quadrilateral fundamental domain σ(Q).

Then, if one cuts the quotient along the sides of Q, the sides of σ(Q) appear as geodesic arcs on Q (see Figure 7).

Figure 7.

328 Aline Aigon-Dupuy

The transformation σ is then built using the technique indicated in the proof of Proposition 3.8.

Using Theorem 5.1, the twist τ_{1} along (e_{2}e_{3}e_{4})^{2} corresponds to η_{4}^{2} and Lemma 5.4(ii) we get the following correspondences for the Dehn twists along geodesics corresponding to the sides of the different copies of Q in S_{Q}:

τ_{1} = τ_{(e_{2}e_{3}e_{4})^{2}} corresponds to η_{4}^{2}: (e_{1}, e_{2}, e_{3}, e_{4}) ⟼ ((e_{2}e_{3}e_{4})^{2}, e_{2}, e_{3}, e_{4}),

τ_{2} = τ_{(e_{3}e_{4}e_{1})} ∘ τ_{(e_{1}e_{3}e_{4})} = τ_{(e_{1}e_{3}e_{4})} ∘ τ_{(e_{3}e_{4}e_{1})} corresponds to σ: (e_{1}, e_{2}, e_{3}, e_{4}) ↦ (e_{1}, e_{3}e_{4}e_{1}e_{2}, e_{3}, e_{4}),

τ_{3} = τ_{(e_{4}e_{1}e_{2})^{2}}τ_{1}^{σ_{0}^{2}(e_{i})} corresponds to σ_{0}^{2}η_{4}^{2}σ_{0}^{2}: (e_{1}, e_{2}, e_{3}, e_{4}) ⟼ (e_{1}, e_{2}, (e_{4}e_{1}e_{2})^{2}e_{3}, e_{4}),

τ_{4} = τ_{(e_{1}e_{2}e_{3})} ∘ τ_{(e_{3}e_{1}e_{2})} = τ_{(e_{3}e_{1}e_{2})} ∘ τ_{(e_{1}e_{2}e_{3})} = τ_{2}^{σ_{0}^{2}(e_{i})} corresponds to σ_{0}^{2}σσ_{0}^{2}: (e_{1}, e_{2}, e_{3}, e_{4}) ⟼ (e_{1}, e_{2}, e_{3}, e_{1}e_{2}e_{3}e_{4}).

Acknowledgements. I would like to express my gratitude to Peter Buser and Robert Silhol for many helpful suggestions.

References

[1] Aigon, A.: Transformations hyperboliques et courbes algébriques en genres 2 et 3. - Ph.D. thesis, Université Montpellier II, 2001.

[2] Aigon, A., and R. Silhol: Hyperbolic hexagons and algebraic curves in genus 3. - J. London Math. Soc. (2) 66, 2002, 671–690.

[3] Birman, J. S.: Braids, Links, and Mapping Class Groups. - Ann. Math. Stud. 82, Princeton Univ. Press, Princeton, N.J., 1974.

[4] Bolza, O.: On binary sextics. - Amer. J. Math. 10, 1888, 47–70.

[5] Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. - Progr. Math. 106, Birkhäuser Boston Inc., Boston, MA, 1992.

[6] Buser, P. and R. Silhol: Geodesics, periods, and equations of real hyperelliptic curves. - Duke Math. J. 108, 2001, 211–250.

[7] Fathi, A. et al.: Travaux de Thurston sur les surfaces. - Astérisque, 66, Séminaire Orsay; with an English summary, Société Mathématique de France, Paris, 1979.

[8] Silhol, R.: Hyperbolic lego and algebraic curves in genus 2 and 3. - In: Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), Contemp. Math. 311, Amer. Math. Soc., Providence, RI, 2002, 313–334.

Received August 28, 2003