Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 75–88

PARAMETRIZATION OF THE MODULI SPACE OF HYPERELLIPTIC AND SYMMETRIC RIEMANN SURFACES

A. F. Costa and E. Martínez

Universidad Nacional de Educación a Distancia, Departamento de Matemáticas Fundamentales Facultad de Ciencias, Senda del Rey, E-28040 Madrid, Spain

Abstract. Let S be a Riemann surface with genus g > 1, let φ be the hyperelliptic involution (i.e. φ is a conformal involution such that S/⟨φ⟩ has genus 0) and let σ be a symmetry of S (i.e. σ is an anticonformal involution of S). We obtain a finite set of real numbers which determines a canonical fundamental region of the NEC groups uniformizing S/⟨φ, σ⟩. The real numbers obtained in this way are used to parametrize the strata of equisymmetric hyperelliptic Riemann surfaces in the moduli space.

1. Introduction

A Riemann surface S is said to be hyperelliptic if and only if S admits an automorphism ϕ ∈ Aut^{+}(S) such that ϕ^{2} is the identity and the quotient S/⟨ϕ⟩ has genus 0. The automorphism ϕ is called the hyperelliptic involution. A Riemann surface S is said to be symmetric if and only if S admits an automorphism ψ ∈ Aut^{−}(S) with ψ^{2} the identity. In this case ψ is a symmetry of S and X = S/⟨ψ⟩ is a Klein surface. If the Riemann surface S is both hyperelliptic and symmetric then X is a hyperelliptic Klein surface.

Let p be the genus of S. If p ≥ 2, then S and X can be expressed as quotients of the hyperbolic plane D over surface groups F and Γ, respectively, F being a Fuchsian group of genus p and Γ a non-Euclidean crystallographic group (NEC group) of algebraic genus p.

Hyperelliptic Klein surfaces with boundary are characterized by means of NEC groups in [BEG] and non-orientable Klein surfaces without boundary in [BBGM].

The aim of this work is to parametrize the strata of the moduli space of hyperelliptic and symmetric Riemann surfaces S according to the different groups of automorphisms of the Klein surface X. It is done by means of the construction of a hyperbolic polygon R with right angles (unless one of them at most). R is a fundamental region for an NEC group Γ′ such that Γ′/Γ is isomorphic to the group of automorphisms of X.

1991 Mathematics Subject Classification: 30F10, 30F50.

Partially supported by CAICYT PB92-0716 and EC project CHRX-CT93-408.

2. Preliminaries on NEC groups

A non-Euclidean crystallographic group (NEC group in short) Γ is a discrete subgroup of isometries of the hyperbolic plane D with compact quotient D/Γ. Each NEC group Γ has a signature σ that has the following form [M]

(2.1)  σ : (g; ±; [m_{1}, …, m_{r}], {C_{1}, …, C_{k}}),

where C_{i} = (n_{i1}, …, n_{is_{i}}), C_{i} are called cycle-periods, n_{ij} link-periods and m_{i} proper periods.

The numbers in σ are non-negative integers; m_{i} and n_{ij} are greater than or equal to 2. The number g is the topological genus of the surface quotient D/Γ. This surface is orientable or not according as the sign in σ is ‘+’ or ‘−’ respectively. The algebraic genus of Γ is p = ηg + k − 1, where η = 2 in the orientable case or η = 1 otherwise.

If r = 0 or k = 0, we write in σ[−] or {−}, respectively. If the number s_{i} is zero for some i we denote C_{i} by (−).

We consider a Fuchsian group F as a particular case of NEC group. Since every element of F is an orientable isometry of the hyperbolic plane, F has the following (NEC) signature:

(g; +; [m_{1}, …, m_{r}], {−}).

The signature σ determines a canonical presentation of the group Γ [M], [W] which is given by the generators

x_{i}, e_{i}, c_{ij}, a_{i}, b_{i}, d_{i}, i = 1, …, r,

i = 1, …, k,
i = 1, …, k, j = 0, …, s_{i},
i = 1, …, g, if sign ‘+’,
i = 1, …, g, if sign ‘−’,

with the relations

x_{i}^{m_{i}} = 1,  i = 1, …, r,
c_{ij−1}^{2} = c_{ij}^{2} = (c_{ij−1}c_{ij})^{n_{ij}} = 1,  i = 1, …, k, j = 1, …, s_{i},
e_{i}^{−1}c_{i0}e_{i}c_{is_{i}} = 1,  i = 1, …, k,
∏_{i=1}^{r} x_{i} ∏_{i=1}^{k} e_{i} ∏_{i=1}^{g}[a_{i}, b_{i}] = 1  if sign +  or

∏_{i=1}^{r} x_{i} ∏_{i=1}^{k} e_{i} ∏_{i=1}^{g} d_{i}^{2} = 1  if sign − .

The area of Γ is the area of each fundamental region of Γ and it is calculated by

μ(Γ) = 2π(ηg + k − 2 + ∑_{i=1}^{r}(1 − \frac{1}{m_{i}} ) + \frac{1}{2} ∑_{i=1}^{k} ∑_{j=1}^{s_{i}}(1 − \frac{1}{η_{ij}} )).

A signature σ corresponds to an NEC group Γ if and only if μ(Γ) > 0.

If Γ is a subgroup of Γ′ with index N then the following relation between the areas holds

μ(Γ) = Nμ(Γ′).

An NEC group Γ is a surface group if it has the signature

(g; ±; [−], {(−) ⋯^{k} ⋯ (−)}),

and in this case we write the signature as

(2.2)  (g; ±; [−], {(−)^{k}}).

If k = 0 and the sign in σ is ‘+’ then Γ is a Fuchsian surface group.

Let Γ be an NEC group with signature σ. A canonical fundamental region for Γ found by Wilkie [W] is a hyperbolic polygon W with the sides labelled

\begin{matrix}ξ_{1}, ξ′_{1}, …, ξ_{r}, ξ′_{r}; & ε_{1}, γ_{10}, …, γ_{1s_{1}}, ε′_{i}, …, ε_{k}, γ_{k0}, …, γ_{ks_{k}}, ε′_{k}; \\ α_{1}, β′_{1}, α′_{1}, β_{1}, …, α_{g}, β′_{g}, α′_{g}, β_{g}, \end{matrix}

if sign ‘+’ and

ξ_{1}, ξ′_{1}, …, ξ_{r}, ξ′_{r}; ε_{1}, γ_{10}, …, γ_{1s_{1}}, ε′_{i}, …, ε_{k}, γ_{k0}, …, γ_{ks_{k}}, ε′_{k}; δ_{1}, δ_{1}^{∗}, …, δ_{g}, δ_{g}^{∗},

if sign ‘−’. The sides are identified in the following way

\begin{aligned}x_{i}(ξ′_{i}) & = ξ_{i}, \\ e_{i}(ε′_{i}) & = ε_{i}, \\ c_{ij}(γ_{ij}) & = γ_{ij}, \\ a_{i}(α′_{i}) & = α_{i}, \\ b_{i}(β′_{i}) & = β_{i}, \\ d_{i}(δ_{i}^{∗}) & = δ_{i}, \end{aligned}  \begin{aligned}i & = 1, …, r, \\ i & = 1, …, k, \\ i & = 1, …, k, j = 0, …, s_{i}, \\ i & = 1, …, g, \\ i & = 1, …, g, \\ i & = 1, …, g, \end{aligned}

where ‘∗’ denotes a non-orientable transformation.

The angles in the vertices of W are

\begin{aligned}⟨ξ_{i}, ξ′_{i}⟩ & = 2π/m_{i},  i = 1, …, r, \\ ⟨ε_{i}, γ_{i0}⟩ + ⟨γ_{is_{i}}, ε′_{i}⟩ & = π,  i = 1, …, k, \end{aligned}

and without a loss of generality we may assume that both angles are equal to \frac{1}{2}π,

⟨γ_{ij−1}, γ_{ij}⟩ = π/n_{ij}, i = 1, …, k, j = 1, …, s_{i},

and the sum of the remaining angles is 2π. We may assume W as a convex polygon, i.e. there is not an angle in W greater than π.

3. Surfaces

A Klein surface (a compact surface equipped with a dianalytic structure) of algebraic genus p ≥ 2 can be expressed as D/Γ where Γ is an NEC group with signature (2.2).

A group G of order N is a group of automorphisms of X if and only if there exists an NEC group Γ′ containing Γ as a subgroup of order N such that Γ′/Γ ≈ G. Let S = D/F a Riemann surface of genus p ≥ 2. S is hyperelliptic if and only if there exists a Fuchsian group F_{1} containing F as a subgroup of index 2 with signature

(0; +; [2^{2p+2}], {−}),

where 2^{2p+2} denotes 2p + 2 proper periods equal to 2 each one.

Let X = D/Γ be a Klein surface where Γ has signature (2.2). X is hyperelliptic if and only if there exists an NEC group Γ_{1} containing Γ as a subgroup of index 2, Γ_{1} having one of the following signatures [BEG], [BBGM]:

i) (0; +; [−], {(2^{2k})})  if g = 0, in this case k ≥ 3.
ii) (0; +; [2^{g+k}], {(−)}) if g > 0, k ≠ 0, sign ‘+’, then k = 1 or 2.
iii) (0; +; [2^{g}], {(2^{2k})}) if g > 0, k ≠ 0, sign ‘−’.
iv) (0; +; [2^{g}], {(−)})  if g ≥ 3, k = 0, sign ‘−’.
v) (1; −; [2^{g}], {−}  if g > 3, k = 0, sign ‘−’ in this case g must be even.

Table 1

The case v) corresponds to hyperelliptic surfaces admitting symmetries without fixed points, i.e. these surfaces correspond to purely imaginary curves. We are interested in the other cases because they provide real algebraic curves.

Let S = D/Γ a hyperelliptic and symmetric Riemann surface and let ψ be a symmetry. Then X = S/⟨ψ⟩ is a hyperelliptic Klein surface. Let G = Γ′/Γ = Aut(X) with order 2N. We have the following diagram that relates the different groups.

The signatures of Γ′ appears in the following table, where p = ηg + k − 1 [BBM], [BBGM]. The cases of the second column are as Table 1.

Hyperelliptic and symmetric Riemann surfaces 79

σ(Γ)  σ(Γ_{1})  σ(Γ′)  Aut(X) Case
g = 0,  i)  (0; +; [N], {(2^{2k/N})}),  Z_{N} × Z_{2}  1
k ≥ 3  N | k, N ≠ k
(0; +; [−], \{( \frac{1}{2} N, 2^{2k/N})\}), D_{N/2} × Z_{2} 2
N even, N | 2k
g ≠ 0, ii) (0; +; [2N, 2^{p/N}], {(−)}), Z_{2N} 3
k = 1, 2  N | p, N ≠ p
sign ‘+’ (0; +; [N, 2^{(p+1)/N}], {(−)}), Z_{N} × Z_{2} 4
N | p + 1, N ≠ p + 1
(0; +; [2^{r}], {(N, 2^{s})}), D_{N} 5
s = 2p/N + 2 − 2r, N even, N | 2p
(0; +; [2^{r}], {(N/2, 2^{s})}), D_{N/2} × Z_{2} 6

s = 2(p + 1)/N + 2 − 2r, N even, N | 2(p + 1)

g ≠ 0, iii) (0; +; [2N, 2^{r}], {(2^{s})}), Z_{2N} 7
k ≠ 0,  s = (2g + 2k − 2)/N − 2r, s ≠ 0, N | g − 1, N | k
sign‘−’ (0; +; [N, 2^{r}], {(2^{s})}, Z_{N} × Z_{2} 8
s = (2g + 2k)/N − 2r, s ≠ 0, N | g, N | k
(0; +; [2^{r}], {(N, 2^{s})}), D_{N} 9
s = (2g + 2k − 2)/N + 2 − 2r,
N even, N | 2k, N | 2(g − 1)
(0; +; [2^{r}], {(N/2, 2^{s})}),  D_{N/2} × Z_{2} 10

s = (2g + 2k)/N + 2 − 2r, N even, N | 2k, N | 2g

g > 3, iv) (0; +; [N, 2^{g/N]}, {(−)}), Z_{2N} 11
k = 0,  N | g − 1, N ≠ g − 1, or N | g, N ≠ g, N even
g odd, (0; +; [2N, 2^{(g−1)/N}], {(−)}), Z_{N} × Z_{2} 12
sign ‘−’  N | g, N ≠ g, N even, N ≠ 2
(0; +; [2^{r}], {(N/2, 2^{s})}), D_{N} 13
s = 2g/N − 2r + 2,
N | 2(g − 1), N even or N | 2g, N ≠ 4t
(0; +; [2^{r}], {(N, 2^{s})}),  D_{N/2} × Z_{2} 14
s = 2(g − 1)/N − 2r + 2, N | 2g, 4 | N.

Table 2

These fourteen signatures can be put in five different classes: A : (0; +; [N], {(2^{2k/N})}), B : (0; +; [−], {(N/2, 2^{2k/N+2})}), C : (0; +; [M, 2^{α}], {(−)}), D : (0; +; [2^{α}], {(M, 2^{β})}), E : (0; +; [M, 2^{α}], {(2^{β})})

which correspond to the cases from Table 2: 1, 3, 4, 11 and 12, 5, 6, 9, 10, 13 and 14, 7 and 8,

respectively, and M, α, and β take different values according to the different cases.

For each one of these signatures we will construct a hyperbolic polygon R that will be a fundamental region for the group Γ′ and the lengths of some sides of R will parametrize the strata of the moduli space of the hyperelliptic and symmetric Riemann surfaces S.

About the general construction of a right-angled hyperbolic polygon, with n sides, see [CM].

4. Surfaces and polygons

Class A. For this class Γ′ has the following signature:

(4.1)  (0; +; [N], {(2^{2k/N})})

where N | k and N ≠ k.

We take as parameters the lengths of the sides γ_{1}, …, γ_{s−1} , where s = 2k/N, and we construct a right-angled polygon P_{A}

The length of the side λ is determined by γ_{1}, …, γ_{s−1} . Now we construct a hyperbolic pentagon with an angle α = 2π/N and four right angles, such that the length of the opposite side to the angle α be λ and the two sides of the angle α be equal. In this condition the pentagon is unique. We join this pentagon with the polygon P_{A} by pasting the sides ‘λ’ and so we obtain a polygon R_{A} that is a fundamental region for the group Γ′ with the signature (4.1). See next figure.

Class B. The signature of Γ′ is in this case

(4.2)  (0; +; [−], {(N/2, 2^{2k/N+2})}),

where N is even and N | 2k.

We take as parameters the lengths of the sides γ_{2}, …, γ_{s−1} , where s = 2k/N+ 2, and we construct the right-angled polygon P_{B}. See next figure.

Now we construct a hyperbolic quadrilateral with an angle α = 2π/N and three right angles, such that the lengths of the opposite side of α be λ. This quadrilateral is unique. We join the quadrilateral with P_{B} pasting the sides labelled ‘λ’ and we obtain the following fundamental region R_{B} for a group Γ′ with signature (4.2).

Class C. Let Γ′ be an NEC group with signature

(4.3)  (0; +; [M, 2^{r}], {(−)}),

where M and r take different values according to the different cases from Table 2. Let W a fundamental region of Wilkie for Γ′ where ∑_{i=0}^{r+1} θ_{i} = 2π.

We draw the orthogonal lines to γ from the points X_{i}, i = 0, 1, …, r. The side γ becomes divided in different parts labelled γ_{0}, γ_{1}, …, γ_{r+1} , according to the ‘opposite’ angle θ_{i}. Let λ_{i} be the length from X_{i} to γ as it is shown in the next figure

We cut by λ_{r} and we transform the region W in the new region

(W − (λ_{r}, ξ_{r}, ξ′_{r}, γ_{r+1}) ∪ x_{r+1}(λ_{r}, ξ_{r}, ξ′_{r}, γ_{r+1})).

In the next step we cut the new region by λ_{r−1} and apply x_{r−1} in a similar way and so on. In the last step we obtain the polygon P_{C} that is a fundamental region for the group Γ′

This region is a hyperbolic polygon with 2r + 3 sides, an angle α = 2π/M and the remaining angles are equal to \frac{1}{2}π. The lengths of the sides are λ_{0} (two sides), 2λ_{i} (i = 1, …, r), γ_{i} (i = 1, …, r) and γ_{0} + γ_{r+1} .

Like we saw for polygons of Class A, we may construct a hyperbolic rightangled polygon taking as parameters 2λ_{1}, γ_{2}, 2λ_{2}, γ_{3}, …, 2λ_{r}, and a hyperbolic pentagon with four right angles and the fifth angle equal to α.

Class D. Let Γ′ be an NEC group with signature

(4.4)  (0; +; [2^{r}], {(M, 2^{s})}),

where M, r > 0 and s > 0 take different values according to the respective cases from Table 2.

A Wilkie fundamental region in this case is W where ∑_{i=1}^{r+1} θ_{i} = 2π.

We draw the orthogonal lines to γ_{0} from X_{i}, i = 1, …, r. Let us denote the distance from X_{i} to γ_{0} by λ_{i}. The side γ_{0} becomes divided in several parts labelled γ_{1}^{∗}, γ_{2}^{∗}, …, γ_{r+1}^{∗}, according to the ‘opposite’ angle θ_{i}.

In a similar way to Class C we cut by λ_{r} and we transform the region W in the new region

(W − (ξ′_{r}, ε, γ_{1}^{∗}, λ_{r})) ∪ x_{r}(ξ′_{r}, ε, γ_{1}^{∗}, λ_{r}).

Repeating this process for x_{r−1}, x_{r−2}, …, x_{1} we obtain the polygon P_{D}

This polygon has an angle α = π/M and the remaining angles are equal to \frac{1}{2}π. The lengths of the sides are 2λ_{i}, i = 1, …, r, γ_{i}^{∗}, i = 1, …, r − 1, γ_{1}, γ_{2}, …, γ_{s+1}, γ_{r}^{∗}.

We may construct a right-angled polygon with sides

2λ_{1}, γ_{2}^{∗}, …, 2λ_{r}, γ_{s+1} + γ_{r}^{∗}, γ_{s}, …, γ_{3}, a, b, c,

where the lengths a, b, c are determined by the lengths of the remaining angles. Now we construct a hyperbolic quadrilateral with three right angles, the fourth angle equal to α and the opposite side to α with length b. Joining this quadrilateral to the right-angled polygon by pasting the sides ‘b’ we obtain the region P_{D} for the group Γ′ with signature (4.4)

Class E. The group Γ′ has the following signature

(4.5)  (0; +; [M, 2^{r}], {(2^{s})}),

where M, r > 0 and s > 0 take different values according to the respective cases from Table 2.

A Wilkie fundamental region for Γ′ is in this case with ∑_{i=0}^{r+1} θ_{i} = 2π. See the above figure.

We draw the orthogonal lines from X_{0}, X_{1}, …, X_{r} to γ_{0} . This last side becomes divided in parts labelled γ_{0}^{∗}, γ_{1}^{∗}, …, γ_{r+1}^{∗}, according to the ‘opposite’ angle θ_{i}. Let λ_{i} be the segment from X_{i} to γ_{0} . We cut the region W by λ_{r} and we transform W in a new region

(W − (λ_{r}, γ_{r}^{∗}, ε, ξ′_{r})) ∪ x_{r}(λ_{r}, γ_{r}^{∗}, ε, ξ′_{r}).

Now we cut by λ_{r−1} and so on. Repeating the process we finally obtain a polygon with an angle α = 2π/M, the remaining angles equal to \frac{1}{2}π and sides labelled in counterclockwise

γ_{0}^{∗}, γ_{1}, …, γ_{s}, x_{0}x_{1} ⋯ x_{r}(λ_{r}), x_{0}x_{1} ⋯ x_{r−1}(λ_{r}),
x_{0}x_{1} ⋯ x_{r−1}(γ_{r−1}^{∗}), x_{0}x_{1} ⋯ x_{r−1}(λ_{r−1}),
x_{0}x_{1} ⋯ x_{r−2}(λ_{r−2}), x_{0}x_{1} ⋯ x_{r−2}(γ_{r−2}^{∗}), …, x_{0}x_{1}(λ_{1}),
x_{0}(λ_{1}), x_{0}(γ_{1}^{∗}), x_{0}(λ_{0}), λ_{0}.

We may construct a hyperbolic right-angled polygon with parameters

γ_{1}, γ_{2}, …, γ_{s}, 2λ_{r}, γ_{r−1}^{∗}, 2λ_{r−1}, γ_{r−2}^{∗}, …, 2λ_{1},

and the three remaining sides a, b, c, are determined by these parameters.

The length b and the angle α determine a unique pentagon with the condition that the two sides of σ be equal. We join this pentagon to the right-angled polygon by pasting the sides ‘b’ and we finally obtain the region P_{E} that is a fundamental region for the group Γ′ with signature (4.5).

5. Parametrization of the moduli space

Let Γ be an abstract surface Fuchsian group of genus p. We consider Ω = Aut(D) as a topological group. The Weil space of Γ with respect to Ω is the set

R(Γ) = {group of monomorphisms r: Γ → Ω

such that r(Γ)is a surface Fuchsian group}.

The Teichmüller space of Γ is the orbit space T (Γ) = R(Γ)/ Aut(Ω). The modular space of Γ is Mod(Γ) = Aut(Γ)/ Inn(Γ) and the moduli space M(p) is T (Γ)/ Mod(Γ). We may also define M(p) as the set of complex structures on surfaces of genus p modulo orientation preserving diffeomorphisms [SS].

In this section we want to obtain a parametrization of the equisymmetric strata of the subspace of M(p) given by hyperelliptic and symmetric Riemann surfaces of genus p. Now we need to know in our first definition of moduli space which is the subspace to be parametrized.

We are interested in the parametrization of the Fuchsian groups F such that D/F is a symmetric and hyperelliptic Riemann surface modulo conjugation by elements of Ω and elements of Mod(F). If F uniformizes a hyperelliptic and symmetric Riemann surface then F < Γ′ and Γ′ has one of the signatures from Table 2.

Given a group Γ′ in some cases it is possible to obtain several groups F, with F < Γ′ such that D/F is a hyperelliptic and symmetric Riemann surface. In fact if Γ′/F ≃ G, each epimorphism θ: Γ′ → G such that θ has a kernel which is a surface Fuchsian group of genus p and θ induces a diagram as one of Section 3 and then D/ ker θ is a symmetric and hyperelliptic Riemann surface.

Let E(F, Γ′) be the set of epimorphisms

θ: Γ′ → G ≃ Γ′/F,

such that ker θ is a hyperelliptic and symmetric Riemann surface modulo the following equivalence relation:

θ ≈ θ′  ⟺  ker θ′ = ker θ.

Each element θ of E(F, Γ′) gives us a way to construct a fundamental region R_{F} of F as union of n (= index of F in Γ′) copies of a fundamental region R′ of Γ′ and the sides of R_{F} identified according to some rules given by θ. Let us note that E(F, Γ′) is a finite set.

Let σ be a signature from Table 2 and column σ(Γ′). Let P_{σ} be the set of classes of congruence of polygons that are fundamental regions of a group with signature σ and with the geometrical conditions described in Section 4. Each element of P_{σ} provides a group Γ′ with signature σ and a set of epimorphisms E(F, Γ′). Let P_{σ} × E_{σ} be the set of pairs given by the class of congruence of polygons and the set of classes of epimorphisms corresponding to the group Γ′ having as fundamental region a polygon of the given class. Each pair of P_{σ} × E_{σ} provides in a natural way a group F such that D/F is a hyperelliptic and symmetric Riemann surface and then an element of M_{h,s}(p) (subspace of hyperelliptic and symmetric Riemann surfaces of genus p). We have in this way a parametrization:

π: P_{s} × E_{s} → M_{h,s}(p).

Theorem. If the signature σ has at most one proper period then π is a finite-to-one map.

Proof. Let S be a surface representing a point of M_{h,s}(p) in the image of P_{s}× E_{s} by π. Let G be the subgroup of Aut(S) such that S/G ≃ D/Γ′ where Γ′ has signature σ and there is θ ∈ E_{σ} such that θ: Γ′ → G and ker θ uniformizes S. If Γ′ has no proper periods then Γ′ only admits one element of P_{σ} as a fundamental region. Since Aut(S) is finite there is only a finite number of NEC groups with signature as Γ′ containing the Fuchsian group which uniformizes S. In this case π is a finite-to-one map.

If there is a proper period in the signature σ then the orbifold D/Γ′ has a unique conic point and the polygons of P_{σ} are obtained from different ways by cutting on the orthogonal line from the conic point to the boundary of D/Γ′. Since the number of such ‘cuts’ is finite then the number of elements in P_{σ} for a given group Γ′ is finite. As E_{σ} is finite for a given group Γ′ we obtain again that π is a finite-to-one map.

If there are more than one proper period in σ then π^{−1} can be an infinite set as we may see in the following example.

Let σ be the signature (0; +; [2, 2]{(2, 2, 2)}). Then there are infinite elements of P_{σ} corresponding to the same Γ′ and so they give by π the same element in M_{h,s}(p). To see it let us consider the element P ∈ P_{σ} represented by the following polygon:

We cut by the orthogonal line to ξ′ and γ_{3,3} ; by the orthogonal line from X_{1} to γ_{3,3} and by the orthogonal line from X_{2} to ξ_{1} as it is showed in the next figure. Identifying the sides properly we obtain a new element P_{1} ∈ P_{σ} that is, in general, different from P.

88 A. F. Costa and E. Martínez

We can repeat this process as many times as we want and we obtain different elements of P_{σ} providing by π the same element of M_{h,s}(p).

References

[BBGM] Bujalance, E., J.A. Bujalance, G. Gromadzki and E. Martínez: The groups of automorphisms of non-orientable hyperelliptic Klein surfaces without boundary. - Lecture Notes in Math. 1398, Springer-Verlag, 1989, 43–51.

[BBM] Bujalance, E., J.A. Bujalance, and E. Martínez: On the automorphism group of hyperelliptic Klein surfaces. - Michigan Math. J. 35, 1988, 361–368.

[BEG] Bujalance, E., J.J. Etayo, and J.M. Gamboa: Hyperelliptic Klein surfaces. - Quart. J. Math. Oxford (2) 36, 1985, 141–157.

[CM] Costa, A.F., and E. Martínez: On hyperbollic right-angled polygons. - Geom. Dedicata 58, 1995, 313–326.

[M] Macbeath, A.M.: The classification of non-Euclidean crystallographic groups. - Canad. J. Math. 19, 1967, 14–32.

[SS] Seppälä, M., and T. Sorvali: Geometry of Riemann Surfaces and Teichmller Spaces. - North-Holland Math. Stud., 1992.

[W] Wilkie, H.C.: On non-Euclidean crystallographic groups. - Math. Z. 91, 1966, 87–102.

Received 11 August 1995