On the Space of Cyclic trigonal Riemann surfaces of genus 4

Milagros Izquierdo ^{∗}  Daniel Ying

May 25, 2005

Abstract. A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian groups, we describe the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

1 Introduction

A closed Riemann surface X which can be realized as a 3-sheeted covering of the Riemann sphere is said to be trigonal, and such a covering will be called a trigonal morphism. This is equivalent to the fact that X is represented by a curve given by a polynomial equation of the form:

y^{3} + yb(x) + c(x) = 0.

If b(x) ≡ 0 then the trigonal morphism is a cyclic regular covering and the Riemann surface is called cyclic trigonal. Trigonal Riemann surfaces have been recently studied (see [2] and [8]).

By Lemma 2.1 in [1], if the surface X has genus g ≥ 5, then the trigonal morphism is unique. The Severi-Castelnouvo inequality is used in order to prove such uniqueness, but this technique is not valid for small genera.

We develop some ad hoc methods to deal with small genera. Using the characterization of trigonality by means of Fuchsian groups ([3]), we obtain all possible cyclic trigonal Riemann surfaces of genus four. We also find that there is a uniparametric family of Riemann surfaces of genus four admitting several cyclic trigonal morphisms. The last result was shown with different calculations in [4], but we state it here for the sake of completeness.

∗Partially supported by the Swedish Research Council (VR)

2 Trigonal Riemann surfaces and Fuchsian groups

Let X_{g} be a compact Riemann surface of genus g ≥ 2. The surface X_{g} can be represented as a quotient X_{g} = D/Γ of the complex unit disc D under the action of a (cocompact) Fuchsian group Γ, that is, a discrete subgroup of the group G = Aut(D) of conformal automorphisms of D. The algebraic structure of a Fuchsian group and the geometric structure of its quotient orbifold are given by the signature of Γ:

s(Γ) = (g; m_{1}, ..., m_{r}).  (1)

The orbit space D/Γ is an orbifold with underlying surface of genus g, having r cone points. The integers m_{i} are called the periods of Γ and they are the orders of the cone points of D/Γ. The group Γ is called the fundamental group of the orbifold D/Γ.

A group Γ with signature (1) has a canonical presentation:

⟨x_{1}, ..., x_{r}, a_{1}, b_{1}, ..., a_{g}, b_{g}| x_{i}^{m_{i}}, i = 1, ..., r, x_{1}... x_{r}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1}... a_{g}b_{g}a_{g}^{−1}b_{g}^{−1}⟩

(2)

The last relation in the above presentation is called the long relation. The generators x_{1}, ..., x_{r}, are called the elliptic generators. Any elliptic element in Γ is conjugated to a power of some of the elliptic generators.

The hyperbolic area of the orbifold D/Γ coincides with the hyperbolic area of an arbitrary fundamental region of Γ and equals:

μ(Γ) = 2π(2g − 2 + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}})),  (3)

Given a subgroup Γ′ of index N in a Fuchsian group Γ, one can calculate the structure of Γ′ in terms of the structure of Γ and the action of Γ on the Γ′-cosets:

Theorem 1 ([12]) Let Γ be a Fuchsian group with signature (1) and canonical presentation (2). Then Γ contains a subgroup Γ′ of index N with signature

s(Γ′) = (h; m′_{11}, m′_{12}, ..., m′_{1s_{1}}, ..., m′_{r1}, ..., m′_{rs_{r}}).  (4)

if and only if there exists a transitive permutation representation θ : Γ → Σ_{N} satisfying the following conditions:

1. The permutation θ(x_{i}) has precisely s_{i} cycles of lengths less than m_{i}, the lengths of these cycles being m_{i}/m′_{i1}, ..., m_{i}/m′_{is_{i}}.

2. The Riemann-Hurwitz formula

μ(Γ′)/μ(Γ) = N.  (5)

The map θ : Γ → Σ_{N} is the monodromy of the covering f : D/Γ′ → D/Γ. Moreover Γ′ = θ^{−1}(Stb(1)), where 1, 2, …, N are the labels of the sheets of the covering f. See chapter 8 in [11] .

A Fuchsian group Γ without elliptic elements is called a surface group and it has signature (h; −). Given a Riemann surface represented as the orbit space X = D/Γ, with Γ a surface Fuchsian group, a finite group G is a group of automorphisms of X if and only if there exists a Fuchsian group Δ and an epimorphism θ : Δ → G with ker(θ) = Γ. The Fuchsian group Δ is the lifting of G to the universal covering π : D → D/Γ and is called the universal covering transformations group of (X, G).

Let Γ be a Fuchsian group with signature (1). Then the Teichmüller space T (Γ) of Γ is homeomorphic to a complex ball of dimension d(Γ) = 3g − 3 + r (see [10]). Let Γ′ ≤ Γ be Fuchsian groups, the inclusion mapping α : Γ → Γ′ induces an embedding T (α) : T (Γ) → T (Γ′) defined by [r] ↦ [rα]. See [7], [10] and [13].

The modular group Mod(Γ) of Γ is the quotient Mod(Γ) = Aut(Γ)/Inn(Γ), where Inn(Γ) is the normal subgroup of Aut(Γ) consisting of all inner automorphisms of Γ. The moduli space of Γ is the quotient M(Γ) = T (Γ)/Mod(Γ) endowed with the quotient topology.

A Fuchsian group Γ such that there does not exist any other Fuchsian group containing it with finite index is called a finite maximal Fuchsian group.

To decide whether a given finite group can be the full group of automorphism of some compact Riemann surface we will need all pairs of signatures s(Γ) and s(Γ′) for some Fuchsian groups Γ and Γ′ such that Γ′ ≤ Γ and d(Γ) = d(Γ′). The full list of such pairs of groups was obtained by Singerman in [13].

Definition 2 A Riemann surface X is said to be trigonal if it admits a three sheeted covering f : X → ℂ̂ onto the Riemann sphere. If f is a cyclic regular covering then X is called cyclic trigonal. The covering f will be called the (cyclic) trigonal morphism.

The following result gives us a characterization of cyclic trigonal Riemann surfaces using Fuchsian groups:

Theorem 3 ([3]) Let X_{g} be a Riemann surface, X_{g} admits a cyclic trigonal morphism f if and only if there is a Fuchsian group Δ with signature

(0; \overbrace{3, ..., 3}^{g+2}) and an index three normal surface subgroup Γ of Δ, such that Γ uniformizes X_{g}.

By Lemma 2.1 in [1], if the surface X_{g} has genus g ≥ 5, then the trigonal morphism is unique. In this case, the cyclic trigonal morphism f is induced by a normal subgroup C_{3} in Aut(X_{g}). From now on we shall consider surfaces with genera smaller than 5.

3 Non-unique cyclic trigonal morphisms on Riemann surfaces

For genus four we use Theorems 1 and 3 to obtain the following method to find cyclic trigonal Riemann surfaces. Let G be the full automorphisms group of the surface X_{4}.

Let X_{4} = D/Γ be a Riemann surface of genus 4 uniformized by the surface Fuchsian group Γ, X_{4} admits a cyclic trigonal morphism f if and only if there is a maximal Fuchsian group Δ with signature (0; m_{1}, ..., m_{r}), a trigonal automorphism φ : X_{4} → X_{4}, such that ⟨φ⟩ ≤ G and an epimorphism θ : Δ → G with ker(θ) = Γ in such a way that θ^{−1}(⟨φ⟩) is a Fuchsian group with signature (0; 3, 3, 3, 3, 3, 3). Notice that the condition Γ to be a surface Fuchsian group imposes that the order of the image under θ of an elliptic generator x_{i} of Δ is the same as the order of x_{i}. Furthermore the trigonal morphism f is unique if and only if ⟨φ⟩ is normal in G.

Now, let θ : Δ → G be such an epimorphism and let |G| = N. Then s(Δ) = (0; m_{1(N)}, …, m_{r(N)}), where m_{i(N)} runs over the divisors of N. Applying the Riemann-Hurwitz formula we have that

2(g + N − 1) = ∑_{1(N)}^{r(N)} N \frac{(m_{i(N)} − 1)}{m_{i(N)}}.  (6)

Equation (6) and the list of maximal signatures ([13]) yield the following list of allowed signatures for genus 4.

Lemma 4 Let X_{4} = D/Γ be a Riemann surface of genus 4 uniformized by the surface group Γ, X_{4} admitting a cyclic trigonal morphism f. Then the maximal Fuchsian group Δ uniformizing the orbifold X/G must have one of the following signatures

|G| s(Δ)  |G| s(Δ)  |G| s(Δ)
3  (0; 3, 3, 3, 3, 3, 3), 6  (0; 2, 6, 6, 6),  6  (0; 2, 2, 3, 3, 3),
6  (0; 2, 2, 2, 3, 6),  6  (0; 2, 2, 2, 2, 2, 2), 12 (0; 2, 2, 2, 2, 2),
12 (0; 2, 2, 3, 6),  12 (0; 2, 3, 3, 3),  12 (0; 4, 6, 12),
18 (0; 2, 2, 2, 6),  24 (0; 3, 4, 6),  24 (0; 2, 2, 2, 4),
36 (0; 2, 4, 12),  36 (0; 2, 2, 2, 3),  42 (0; 2, 3, 42),
48 (0; 2, 3, 24),  54 (0; 2, 3, 18),  60 (0; 2, 3, 15),
72 (0; 2, 4, 6),  72 (0; 2, 3, 12),  90 (0; 2, 3, 10),
108 (0; 2, 3, 9),  120 (0; 2, 4, 5),  144 (0; 2, 3, 8),
252 (0; 2, 3, 7),

We study the existence of epimorphisms satisfying the diagram below

[figure: X_{4} ≅ ℍ/Γ; X_{4}/G ≅ ℍ/Δ; ℍ/Λ = X_{4}/⟨φ⟩; s(Λ) = (0; 3, 3, 3, 3, 3, 3)]

Existence of cyclic trigonal Riemann surfaces of genus 4 In the following we calculate all possible epimorphisms θ : Δ → G. We

separate the cases according to the order of the group G.

We use Theorem 1 to calculate θ^{−1}(⟨φ⟩). The Fuchsian groups Δ have signatures as in Lemma 4 and presentation (2).

1. |G| = 3. There are epimorphisms θ : Δ → C_{3} = ⟨a/a^{3} = 1⟩ where s(Δ) = (0; 3, 3, 3, 3, 3, 3). For instance, θ_{1}(x_{2i}) = a and θ_{1}(x_{2i−1}) = a^{−1}, 1 ≤ i ≤ 3, and θ_{2}(x_{i}) = a, 1 ≤ i ≤ 6.

Remark The cyclic trigonal Riemann surfaces of genus 4 form a space M_{4}^{3} of (complex) dimension d(Δ) = 0−3+6 = 3. By Theorem 2 in [6] this space consists of two disconnected components. One connected component is given by epimorphisms θ_{1}, that is, half the stabilizers of the cone points rotate in opposite directions. The other component is given by epimorphisms θ_{2}, that is, the stabilizers of all the cone points rotate in the same direction.

2. |G| = 6. First, consider the signature s(Δ_{1}) = (0; 2, 6, 6, 6). There are epimorphisms θ : Δ → C_{6} = ⟨a/a^{6} = 1⟩. The cone points in D/Ker(θ) are given by the action of θ(x_{2}), θ(x_{3}) and θ(x_{4}) on the ⟨a^{2}⟩-cosets. Each such element gives one cone point of order 3. Thus, the signature of θ^{−1}(⟨a^{2}⟩) = (1; 3, 3, 3) and the surfaces D/Ker(θ) are not trigonal.

Consider now the signature s(Δ_{2}) = (0; 2, 2, 3, 3, 3). There are possible epimorphisms from Δ_{2} onto both D_{3} and C_{6}. In each case, each of θ(x_{3}), θ(x_{4}) and θ(x_{5}) induces two cone points in D/Ker(θ). Thus s(θ^{−1}(C_{3})) = (1; 3, 3, 3), therefore these surfaces are trigonal with unique trigonal morphisms.

Consider the signature s(Δ_{3}) = (0; 2, 2, 2, 3, 6). There are epimorphisms θ : Δ_{3} → C_{6}. θ(x_{4}) induces two cone points, while θ(x_{5}) induces one conical point in D/Ker(θ). These surfaces are not trigonal since s(θ^{−1}(⟨a^{2}⟩)) = (1; 3, 3, 3).

The signature s(Δ_{4}) = (0; 2, 2, 2, 2, 2, 2) does not produce any trigonal surface since the orders of the elliptic elements of Δ_{4} are relative prime to 3.

Remark The cyclic trigonal surfaces X_{4} of genus 4 with automorphism group of order 6 are determined by the Fuchsian groups with s(Δ_{2}) = (0; 2, 2, 3, 3, 3), the universal covering transformation groups of (X_{4}, Aut(X_{4})). Therefore these surfaces form a disconnected subspace of M_{4}^{3} with (complex) dimension 0 − 3 + 5 = 2. Moreover, the surfaces with automorphism group Aut(X_{4}) = D_{3} lie in one of the connected components of M_{4}^{3} while the surfaces with Aut(X_{4}) = C_{6} lie in the other connected component.

3. |G| = 12. First, consider the signature s(Δ_{1}) = (0; 4, 6, 12). There are epimorphisms θ : Δ → C_{12} = ⟨a/a^{12} = 1⟩. The cone points in D/Ker(θ) are given by the action of θ(x_{2}) and θ(x_{3}) on the ⟨a^{4}⟩- cosets. θ(x_{2}) induces two cone points, while θ(x_{3}) induces one cone point in D/Ker(θ). Thus, the signature is s(θ^{−1}(⟨a^{4}⟩)) = (1; 3, 3, 3) and the surface D/Ker(θ) is not trigonal.

Consider the signature s(Δ_{2}) = (0; 2, 3, 3, 3). The only group of order 12 generated by three elements of order 3 is A_{4} = ⟨a, s/a^{3} = s^{2} = (as)^{3} = 1⟩ (see [5]). There are epimorphisms θ : Δ_{2} → A_{4}. For instance θ(x_{1}) = s, θ(x_{2}) = a, θ(x_{3}) = as and θ(x_{4}) = sas. Any element of order 3 in A_{4} leaves just one coset fixed when acting on the ⟨a⟩, ⟨sa⟩, ⟨as⟩ or the ⟨sas⟩-cosets, since all of them are conjugated. Then θ^{−1}(C_{3}) has signature (1; 3, 3, 3) and the corresponding surfaces are not trigonal.

Now, consider the signature s(Δ_{3}) = (0; 2, 2, 3, 6). There are possible epimorphisms from Δ_{3} onto both D_{6} and C_{6} × C_{2}. In each case, θ(x_{3}) induces four cone points and θ(x_{4}) induces two cone points in D/Ker(θ). Thus, these surfaces are trigonal with unique trigonal morphisms.

The signature s(Δ_{4}) = (0; 2, 2, 2, 2, 2) does not produce any trigonal surface since the orders of the elliptic elements of Δ_{4} are relative prime to 3.

Remark The cyclic trigonal surfaces X_{4} of genus 4 with automorphism group of order 12 are determined by the Fuchsian groups with s(Δ_{2}) = (0; 2, 2, 3, 6), the universal covering transformation groups of (X_{4}, Aut(X_{4})). Therefore these surfaces form a disconnected subspace of M_{4}^{3} with (complex) dimension 0−3+4 = 1. Moreover, the surfaces with automorphism group Aut(X_{4}) = D_{6} lie in one of the connected components of M_{4}^{3} while the surfaces with Aut(X_{4}) = C_{6} × C_{2} lie in the other connected component.

4. |G| = 18. Consider the signature s(Δ) = (0; 2, 2, 2, 6). There is no group of order 18 generated by three involutions and containing elements of order 6 ([5]). Hence there are no epimorphsims θ.

5. |G| = 24. First of all, the signature s(Δ_{1}) = (0; 2, 2, 2, 4) cannot give trigonal surfaces since the orders of the elliptic generators of Δ_{1} are relative prime to 3.

For the signature s(Δ_{2}) = (0; 3, 4, 6): The only groups of order 24 generated by one element of order 3 and one element of order 6 are A_{4} ×C_{2} and ⟨2, 3, 3⟩ = Q⋊C_{3} = ⟨a, s, t/a^{3} = t^{4} = s^{4} = (st)^{4} = 1, s^{2} = t^{2}, a^{2}sa = t, a^{2}ta = st⟩, the binary tetrahedral group, see [5]. There are no possible epimorphisms from Δ_{2} onto A_{4} × C_{2} since this group has no elements of order 4. There are epimorphisms θ : Δ_{2} → ⟨2, 3, 3⟩, for instance θ(x_{1}) = sta, θ(x_{2}) = s and θ(x_{3}) = s^{2}a^{2}. Now, the group ⟨2, 3, 3⟩ contains just one conjugacy class of subgroups of order 3, and one conjugacy class of elements of order 6, with representatives sta and s^{2}a^{2} respectively. Therefore it is sufficient to study the action of sta and s^{2}a^{2} on the (⟨a⟩)-cosets. The first action has orbits {[a], [s], [t^{3}]}, {[t], [s^{2}], [s^{3}]}, {[ts]} and {[st]}, the second action has orbits {[a], [s^{2}]} and {[s], [t^{3}], [st], [s^{3}], [t], [ts]}. Thus the signature of θ^{−1}(⟨a⟩) is (1; 3, 3, 3) and the corresponding surface is not trigonal.

6. |G| = 36. First, consider Fuchsian groups Δ_{1} with signature (0; 2, 4, 12). The only groups of order 36 containing elements of order 12 are C_{36}, C_{12} × C_{3}, (C_{3} × C_{3}) ⋊_{1} C_{4} = ⟨a, b, t/a^{3} = b^{3} = t^{4} = [a, b] = 1, t^{3}at = a^{−1}, t^{3}bt = b^{−1}⟩ and C_{3} ⋊C_{12} = ⟨a, b/a^{3} = b^{12} = b^{−1}aba = 1⟩. None of these groups are generated by elements of order 2 and 4. Hence there is no epimorphism from Δ_{1} onto a group of order 36 and there are no cyclic trigonal Riemann surfaces of genus 4 such that the quotient X_{4}/Aut(X_{4}) is uniformized by Δ_{1}.

Secondly, consider Fuchsian groups Δ_{2} with signature (0; 2, 2, 2, 3). The only group of order 36 generated by 3 involutions is D_{3} × D_{3} = ⟨a, b, s, t/a^{3} = b^{3} = s^{2} = t^{2} = [a, b] = [s, b] = [t, a] = (sa)^{2} = (tb)^{2} = 1⟩. The non-normal order three subgroups in D_{3} × D_{3} are ⟨ab⟩ and ⟨a^{2}b⟩ . Consider the epimorphism θ : Δ_{2} → D_{3} × D_{3} defined by θ(x_{1}) = s, θ(x_{2}) = tb, θ(x_{3}) = sta and θ(x_{4}) = a^{2}b. The action of θ(x_{4}) = a^{2}b on the (⟨ab⟩)-cosets has the following orbits: {[1], [a], [b]}, {[s]}, {[sb]}, {[sa]}, {[t]}, {[tb]}, {[ta]}, {[st], [stb], [sta]}. Then s(θ^{−1}(⟨ab⟩)) contains six periods equal to 3 and by the Riemann-Hurwitz formula s(θ^{−1}(⟨ab⟩)) = (0; 3, 3, 3, 3, 3, 3). In the same way the action of θ(x_{4}) = a^{2}b on the (⟨a^{2}b⟩)-cosets has the following orbits: {[1]}, {[b]}, {[ab]}, {[s], [sa^{2}], [sb]}, {[t], [tb], [ta^{2}]}, {[st]}, {[stb]}, {[sta^{2}]}. Again, s(θ^{−1}(⟨a^{2}b⟩)) contains six periods equal to 3 and then s(θ^{−1}(⟨a^{2}b⟩)) = (0; 3, 3, 3, 3, 3, 3). Thus the Riemann surfaces uniformized by Ker(θ) are cyclic trigonal Riemann surfaces that admit two different trigonal morphisms f_{1} : D/Ker(θ) → ℂ̂ and f_{2} : D/Ker(θ) → ℂ̂ induced by the subgroups ⟨ab⟩ and ⟨a^{2}b⟩ of D_{3} × D_{3}. The dimension of the family of surfaces D/Ker(θ) is given by the dimension of the space of groups Δ_{2} with s(Δ_{2}) = (0; 2, 2, 2, 3). This (complex)-dimension is 3(0) − 3 + 4 = 1. By theorem 1 in [6] this is the unique conjugacy class of order 3 subgroups of D_{3} × D_{3} inducing trigonal morphisms on the surfaces.

7. |G| = 42. There is no such epimorphism, since the only possible epimorphism from a Fuchsian group with signature (0; 2, 3, 42) should be onto C_{42}, which is not generated by elements of order 2 and 3.

8. |G| = 48. There is no epimorphism from a Fuchsian group Δ with signature (0; 2, 3, 24) onto a group of order 48. Otherwise there would be an epimorphism from a Fuchsian group with signature (0; 3, 3, 12) onto a group of order 24 generated by two elements of order 3. This is impossible because the only such group is the binary tetrahedral group, which does not have elements of order 12. See [5] and [13].

9. |G| = 54. There is no epimorphism from a Fuchsian group Δ with signature (0; 2, 3, 18) onto a group of order 54. Otherwise it would be an epimorphism from a Fuchsian group with signature (0; 3, 3, 9) onto C_{9} ⋊ C_{3}. This group is not generated by order three elements. See [5] and [13].

10. |G| = 60. Clearly there is no epimorphism from a Fuchsian group Δ with signature (0; 2, 3, 15) onto a group of order 60, since the only group of order 60 generated by elements of order 2 and 3 is A_{5}.

11. |G| = 72.

i) Consider Fuchsian groups Δ_{1} with signature (0; 2, 3, 12). The group Δ_{1} contains the group Λ_{1} with signature s(Λ_{1}) = (0; 3, 3, 6) as a subgroup of index 2 (see [13]). Any epimorphism θ_{1} : Δ_{1} → G_{72} is an extension of an epimorphism ϕ_{1} : Λ_{1} → G_{36}. The existence of epimorphisms ϕ_{1} obliges the group G_{36} to be generated by two elements of order 3. The only group of order 36 generated by elements of order 3 is A_{4} × C_{3} = ⟨a, b, s/a^{3} = b^{3} = s^{2} = [a, b] = [s, b] = (as)^{3} = 1⟩. It contains four conjugacy classes of subgroups of order 3 with representatives ⟨a⟩, ⟨ab⟩, ⟨a^{2}b⟩ and ⟨b⟩, where ⟨b⟩ is central. Consider the epimorphism ϕ_{1} : Λ_{1} → A_{4} × C_{3} defined as ϕ_{1}(z_{1}) = ba, ϕ_{1}(z_{2}) = bsa, ϕ_{1}(z_{3}) = bsas. As ⟨b⟩ is central, the action of any element of order 3 on the (⟨b⟩)-cosets gives four orbits, each with three cosets. The action of any element of order 6 has six orbits, with 2 cosets each. Then ϕ_{1}(z_{3}) = bsas induces six periods of order 3 in ϕ_{1}^{−1}(⟨b⟩). By the Riemann-Hurwitz formula s(ϕ_{1}^{−1}(⟨b⟩)) = (0; 3, 3, 3, 3, 3, 3). Therefore, a cyclic trigonal Riemann surface X_{4} will be uniformized by Kerθ_{1}, where θ_{1} : Δ_{1} → G_{72} as above and G_{72} an extension of degree 2 of A_{4} × C_{3} containing elements of order 12 . This extension is Σ_{4} × C_{3} = ⟨a, b, \overline{s}/a^{3} = b^{3} = (\overline{s})^{2} = [a, b] = (a\overline{s})^{4} = [\overline{s}, b] = 1⟩, with ⟨b⟩ central in Σ_{4} × C_{3}. The epimorphism θ_{1} : Δ_{1} → Σ_{4} × C_{3} given by θ_{1}(x_{1}) = \overline{s}, θ_{1}(x_{2}) = ab, θ_{1}(x_{3}) = a^{2}\overline{s}b yields a trigonal Riemann surface X_{4} = D/Ker(θ_{1}) with trigonal morphism induced by ⟨b⟩. But by Theorem 1 in [6] this trigonal morphism is unique. ii). Consider Fuchsian groups Δ_{2} with signature (0; 2, 4, 6). The group Δ_{2} contains the group Λ_{2} with signature s(Λ_{2}) = (0; 3, 4, 4) as a subgroup of index 2 (see [13]). Any epimorphism θ_{2} : Δ_{2} → G_{72} is an extension of an epimorphism ϕ_{2} : Λ_{2} → G_{36}. The existence of epimorphisms ϕ_{2} obliges the group G_{36} to be generated by two elements of order 4. The only group of order 36 generated by elements of order 4 is (C_{3} × C_{3}) ⋊_{2} C_{4} = ⟨a, b, t/a^{3} = b^{3} = t^{4} = [a, b] = 1, t^{3}at = b, t^{3}bt = a^{−1}⟩. It contains two conjugacy classes of subgroups of order 3 with representatives ⟨a⟩ and ⟨ab⟩. Consider the epimorphism ϕ_{2} : Λ_{2} → (C_{3} × C_{3}) ⋊ C_{4} defined as ϕ_{2}(y_{1}) = a, ϕ_{2}(y_{2}) = at, ϕ_{2}(y_{3}) = bt^{3}. The action of ϕ_{2}(y_{1}) = a on the (⟨a⟩)-cosets has the following orbits: {[1]}, {[b]}, {[t^{2}]}, {[t], [at], [a^{2}t]}, {[t^{3}], [at^{3}], [a^{2}t^{3}]}, {[bt^{2}]}, {[b^{2}]}, {[b^{2}t^{2}]}. The action of a on the (⟨b⟩)-cosets has the same cycle-structure. Then ϕ_{2}^{−1}(⟨a⟩) and ϕ_{2}^{−1}(⟨b⟩) have six periods of order 3, induced by ϕ_{2}(y_{1}) = a. By the Riemann-Hurwitz formula s(ϕ_{2}^{−1}(⟨a⟩)) = s(ϕ_{2}^{−1}(⟨b⟩)) = (0; 3, 3, 3, 3, 3, 3). Therefore a cyclic trigonal Riemann surface Y_{4} with non-unique trigonal morphism will be uniformized by Ker(θ_{2}), where θ_{2} : Δ_{2} → G_{72} as above and G_{72} is an extension of degree 2 of (C_{3} × C_{3}) ⋊ C_{4} containing just two conjugacy classes of subgroups of order 3. This extension is (C_{3} × C_{3}) ⋊ D_{4} = ⟨a, b, t, s/a^{3} = b^{3} = t^{4} = s^{2} = (st)^{2} = [a, b] = (sa)^{2} = (sb)^{2} = 1, t^{3}at = b, t^{3}bt = a^{2}⟩. The epimorphism θ_{2} : Δ_{2} → (C_{3} × C_{3}) ⋊ D_{4} given by θ_{2}(x_{1}) = s, θ_{2}(x_{2}) = ta, θ_{2}(x_{3}) = stb yields the required trigonal Riemann surface Y_{4} = D/Ker(θ_{2}) with non-unique trigonal morphisms. The trigonal morphisms f_{1} : D/Ker(θ_{2}) → ℂ̂ and f_{2} : D/Ker(θ_{2}) → ℂ̂ are induced by the conjugated subgroups ⟨a⟩ and ⟨b⟩ of (C_{3}×C_{3})⋊D_{4}.

12. |G| = 90. There is no epimorphism from a Fuchsian group Δ with signature (0; 2, 3, 10) onto a group of order 90. Otherwise it would be an epimorphism from a Fuchsian group Δ̂ with signature (0; 5, 5, 5) onto C_{15}, which is impossible. Notice that groups with signature (0; 5, 5, 5) are normal subgroups of index 6 in groups with signature (0; 2, 3, 10) (see [13]).

13. |G| = 108. As in Case 9, there is no epimorphism θ from a Fuchsian group Δ with signature (0; 2, 3, 9) onto a group of order 108. Notice that groups with signature (0; 3, 3, 9) are non-normal subgroups of index 4 in groups with signature (0; 2, 3, 9) (see [13]).

14. |G| = 120. The signature s(Δ) = (0; 2, 4, 5) cannot give trigonal surfaces since the orders of the elliptic generators of Δ are relative prime

to 3.

15. |G| = 144. There is no epimorphism Δ with signature (0; 2, 3, 8) onto a group of order 144 since there is no epimorphism θ : Δ̂ → C_{24} where s(Δ̂) = (0; 2, 8, 8). C_{24} is the only group of order 24 containing elements of order 8 (see [5]). Notice that the groups with signature s(Δ̂) = (0; 2, 8, 8) are non-normal subgroups of index 6 in groups Δ with s(Δ) = (0; 2, 3, 8) (see [13]).

16. |G| = 252. The are no Riemann surfaces of genus 4 with exactly 252 automorphisms, otherwise there would be an epimorphism θ : Δ → G_{252}, with s(Δ) = (0; 2, 3, 7). A possible epimorphism θ : Δ → G_{252}, is an extension of a epimorphism ϕ : Δ̂ → C_{7} ⋊ C_{4} where s(Δ̂) = (0; 2, 7, 7) (see [13]). Such an epimorphism ϕ does not exist since no group of order 28 is generated by two elements of order 7 (see [5]).

We summarize the above results in the following theorems:

Theorem 5 ([4]) There is a uniparametric family of Riemann surfaces X_{4}(λ) of genus 4 admitting several cyclic trigonal morphisms. The surfaces X_{4}(λ) have G = Aut(X_{4}(λ)) = D_{3} × D_{3} and the quotient Riemann surfaces X_{4}(λ)/G are uniformized by the Fuchsian groups Δ with signature s(Δ) = (0; 2, 2, 2, 3). There is one Riemann surface Y_{4} in the family with automorphism group Aut(Y_{4}) = (C_{3} × C_{3}) ⋊ D_{4} and the quotient Riemann surface Y_{4}/Aut(Y_{4}) is uniformized by the Fuchsian group \overline{Δ} with signature s(\overline{Δ}) = (0; 2, 4, 6).

Theorem 6 The space M_{4}^{3} of cyclic trigonal Riemann surfaces of genus 4 form a disconnected subspace of dimension 3 of the moduli space M_{4}.

1. The subspace ^{6}M_{4}^{3} of M_{4}^{3} formed by Riemann surfaces of genus 4 with automorphism group of order 6 is a disconnected space of dimension 2 determined by the Fuchsian groups Δ′′ with s(Δ′′) = (0; 2, 2, 3, 3, 3). The automorphism group of the Riemann surfaces is either C_{6} or D_{3}.

2. The subspace ^{12}M_{4}^{3} of M_{4}^{3} formed by Riemann surfaces of genus 4 with automorphism group of order 12 is a disconnected space of dimension 1 determined by the Fuchsian groups Δ′ with s(Δ′) = (0; 2, 2, 3, 6). The automorphism group of the Riemann surfaces is either C_{2} × C_{6} or D_{6}.

3. The subspace ^{36}M_{4}^{3} of M_{4}^{3} formed by Riemann surfaces X_{4}(Δ) of genus 4 with automorphism group of order 36 is a space of dimension 1 detemined by the Fuchsian groups Δ with s(Δ) = (0; 2, 2, 2, 3). The automorphism group of the Riemann surfaces is D_{3} × D_{3} and the surfaces admit non-normal trigonal morphisms.

4. There are exactly 2 cyclic trigonal Riemann surfaces X_{4} and Y_{4} of genus 4 with automorphism groups of order 72.

(i) X_{4} has a normal trigonal morphism, Aut(X_{4}) = S_{4} × C_{3} and X_{4}/S_{4} × C_{3} uniformized by the Fuchsian group Δ_{1} with s(Δ_{1}) = (0; 2, 3, 12).

(ii) Y_{4} has non-normal trigonal morphisms, Aut(Y_{4}) = (C_{3}×C_{3})⋊D_{4} and Y_{4}/(C_{3}×C_{3})⋊D_{4} uniformized by the Fuchsian group Δ_{2} with s(Δ_{2}) = (0; 2, 4, 6).

Theorem 7 The different subspaces in Theorem 6 satisfy the following inclusion relations:

1. The space ^{12}M_{4}^{3} is a subspace of ^{6}M_{4}^{3}.

2. The space ^{36}M_{4}^{3} is a subspace of ^{6}M_{4}^{3}.

3. The surface X_{4} given in Theorem 6.4 belongs to the space ^{12}M_{4}^{3}.

4. The surface Y_{4} given in Theorem 6.4 belongs to ^{36}M_{4}^{3} ⋂ ^{12}M_{4}^{3}.

Proof.

1. In fact the Riemann surfaces D/Δ′′ with s(Δ′′) = (0; 2, 2, 3, 3, 3) are double coverings of the the Riemann surfaces D/Δ′ with s(Δ′) = (0; 2, 2, 3, 6). Consider the map ϕ′′ : Δ′ → Σ_{2} defined by ϕ′′(x′_{1}) = (1, 2), ϕ′′(x′_{2}) = ϕ′′(x′_{3}) = 1_{d}, ϕ′′(x′_{4}) = (1, 2). By Theorem 1, ϕ′′(x′_{1}) induces no cone points, ϕ′′(x′_{2}) induces two cone points of order 2, ϕ′′(x′_{3}) two cone points of order 3 and ϕ′′(x′_{4}) one of order 3. Thus ϕ′′ is the required monodromy of the covering D/Δ′′ → D/Δ′, with Δ′′ = ϕ^{′′−1}(Stb(1)).

2. In fact the Riemann surfaces D/Δ′′ with s(Δ′′) = (0; 2, 2, 3, 3, 3) are 6-sheeted coverings of the the Riemann surfaces D/Δ with s(Δ) = (0; 2, 2, 2, 3). Consider the maps ϕ_{1} : Δ → Σ_{2} defined by ϕ_{1}(x_{1}) = ϕ_{1}(x_{2}) = (1, 2), ϕ_{1}(x_{3}) = ϕ_{1}(x_{4}) = 1_{d} and ϕ_{2} : Λ → Σ_{3} defined by ϕ_{2}(y_{1}) = (1, 2), ϕ_{2}(y_{2}) = (1, 3), ϕ_{2}(y_{3}) = (1, 2, 3), ϕ_{2}(y_{4}) = 1_{d}, where Λ = ϕ_{1}^{−1}(Stb(1)). By Theorem 1, s(Λ) = (0; 2, 2, 3, 3). Now ϕ_{2}(y_{1}) and ϕ_{2}(y_{2}) induce one cone point of order 2 each, and ϕ_{2}(y_{4}) induces three cone points of order 3. Therefore the composition map ϕ_{2} · ϕ_{1} is the required monodromy of the covering D/Δ′′ → D/Δ. Again Δ′′ = ϕ_{2}^{−1}(Stb(1)). The monodromy ϕ_{2} · ϕ_{1} yields the action of D_{3} × D_{3} on the ⟨ab, st⟩-cosets, where ab, st are defined in Case 6.2

The space ^{36}M_{4}^{3} belongs to the connected component of the space M_{4}^{3} consisting of the Riemann surfaces with half the stabilizers of the cone points rotating in opposite directions since ⟨ab, st⟩ = D_{3}. We conjecture here that ^{36}M_{4}^{3} is a connected subspace.

3. The Riemann surface D/Δ_{1} with s(Δ_{1}) = (0; 2, 3, 12) is a 6-sheeted covering of the Riemann surfaces D/Δ′ with s(Δ′) = (0; 2, 2, 3, 6). Consider the representation ϕ : Δ_{1} → Σ_{6} defined by ϕ(\overline{x}_{1}) = (2, 4)(3, 5), ϕ(\overline{x}_{2}) = (1, 2, 3)(4, 5, 6), ϕ(\overline{x}_{3}) = (1, 5, 6, 2)(3, 4). By Theorem 1, ϕ(\overline{x}_{1}) induces two cone points of order 2, ϕ(\overline{x}_{2}) induces no cone points and ϕ(\overline{x}_{3}) induces one cone point of order 3 and one of order 6, then s(Δ′) = s(ϕ^{−1}(Stb(1))) = (0; 2, 2, 3, 6). Thus, the map ϕ is the required monodromy of the covering D/Δ′ → D/Δ_{1}. The monodromy ϕ yields the action of Σ_{4} × C_{3} on the C_{6} × C_{2}-cosets, where C_{6} × C_{2} = ⟨b, \overline{s}, (a\overline{s})^{2}⟩ as in Case 11.i)

As C_{6} × C_{2} is a subgroup of Σ_{4} × C_{3} this surface belongs to the connected component of the space M_{4}^{3} consisting of the Riemann surfaces with the stabilizers of all the cone points rotating in the same direction.

4. First of all the Riemann surface Y_{4} = D/Δ_{2} with s(Δ_{2}) = (0; 2, 4, 6) is a 6-sheeted covering of the Riemann surfaces D/Δ′ with s(Δ′) = (0; 2, 2, 3, 6). Consider the representation ϕ : Δ_{2} → Σ_{6} defined by ϕ(\overline{y}_{1}) = (1, 4)(2.3)(5, 6), ϕ(\overline{y}_{2}) = (2, 4, 5, 6)(1, 3), ϕ(\overline{y}_{3}) = (1, 2, 5)(3, 4). By Theorem 1, ϕ(\overline{y}_{1}) induces no cone points, ϕ(\overline{y}_{2}) induces one cone point of order 2 and ϕ(\overline{y}_{3}) induces one cone point of order 3, one cone point of order 6 and one of order 2, then s(Δ′) = s(ϕ^{−1}(Stb(1))) = (0; 2, 2, 3, 6). Thus, the map ϕ is the required monodromy of the covering D/Δ′ → D/Δ_{2}. The monodromy ϕ yields the action of (C_{3} × C_{3}) ⋊ D_{4} on the D_{6}-cosets, where D_{6} = ⟨a, s, t^{2}⟩ as in Case 11.ii).

Secondly, the Riemann surface Y_{4} = D/Δ_{2} with s(Δ_{2}) = (0; 2, 4, 6) is a double covering of the the Riemann surfaces D/Δ with s(Δ) = (0; 2, 2, 2, 3). By Theorem 1, the map τ : Δ → Σ_{2} defined by τ(\overline{x}_{1}) = 1_{d}, τ(\overline{x}_{2}) = τ(\overline{x}_{3}) = (1, 2) is the required monodromy of the covering D/Δ → D/Δ_{2} with Δ = τ^{−1}(Stb(1)). Notice that τ(\overline{x}_{1}) induces two cone points of order 2, τ(\overline{x}_{2}) the third cone point of order 2 and τ(\overline{x}_{3}) one cone point of order 3. The monodromy τ yields the action of (C_{3} × C_{3}) ⋊ D_{4} on the D_{3} × D_{3}-cosets, where D_{3} × D_{3} = ⟨a, b, s, t^{2}⟩ as in Case 11.ii).

Remark The Riemann surfaces D/Δ′ with s(Δ′) = (0; 2, 2, 3, 3, 3) cannot be coverings of the the Riemann surfaces D/Δ with s(Δ) = (0; 2, 2, 2, 3). Hence the space ^{36}M_{4}^{3} is not a subspace of ^{12}M_{4}^{3}.

Some results in this paper form part of Ying’s Licentiate Thesis [14]. We thank A. F. Costa for useful comments.

4 Appendix: Groups of order 36 and 72

Groups of order 36 1. C_{36} = ⟨u|u^{36} = 1⟩

2. C_{18} × C_{2} = ⟨s, u|s^{2} = u^{18} = [s, u] = 1⟩

3. C_{12} × C_{3} = ⟨s, u|s^{3} = u^{12} = [s, u] = 1⟩

4. C_{6} × C_{6} = ⟨s, u|s^{6} = u^{6} = [s, u] = 1⟩

5. (C_{2} × C_{2}) ⋊ C_{9} = ⟨a, s, t|a^{9} = s^{2} = t^{2} = [s, t] = 1, a^{8}sa = t, a^{8}ta = st⟩ 6. A_{4} × C_{3} = ⟨a, b, s/a^{3} = b^{3} = s^{2} = [a, b] = [s, b] = (as)^{3} = 1⟩

7. C_{9} ⋊ C_{4} = ⟨a, t|a^{9} = t^{4} = 1, t^{3}at = a^{8}⟩

8. D_{18} = ⟨a, s|a^{18} = s^{2} = (sa)^{2}⟩

9. (C_{3} ×C_{3})⋊_{1} C_{4} = ⟨a, b, t|a^{3} = b^{3} = t^{4} = [a, b] = 1, t^{3}at = a^{2}, t^{3}bt = b^{2}⟩ 10. T × C_{3} = C_{12} ⋊ C_{3} = ⟨a, b, t|a^{3} = b^{4} = t^{3} = [a, t] = [b, t] = 1, b^{3}ab =

a^{2}⟩ = ⟨a, b|a^{3} = b^{12} = b^{−1}aba = 1⟩

11. (C_{3} × C_{3}) ⋊_{2} C_{4} = ⟨a, b, t|a^{3} = b^{3} = t^{4} = [a, b] = 1, t^{3}at = b, t^{3}bt = a^{2}⟩ 12. ⟨3, 3, 3, 2⟩ × C_{2} = ⟨a, b, s, t|a^{3} = b^{3} = s^{2} = t^{2} = [a, b] = [a, t] = [b, t] =

[s, t] = 1, sas = a^{2}, sbs = b^{2}⟩

13. D_{3} × C_{6} = ⟨a, s, t|a^{6} = s^{2} = t^{3} = (sa)^{2} = [a, t] = [s, t] = 1⟩

14. D_{3} ×D_{3} = ⟨a, b, s, t|a^{3} = b^{3} = s^{2} = t^{2} = (st)^{2} = [a, b] = [a, t] = [b, s] =

(sa)^{2} = (tb)^{2}⟩ Groups of order 72 There are 50 groups of order 72. However, we only list the groups that

are interesting for this work:

1. Σ_{4} × C_{3} = ⟨t, a, b|t^{2} = a^{3} = b^{3} = (ta)^{4} = [t, b] = [a, b] = 1⟩

2. (C_{3} × C_{3}) ⋊_{8} C_{8} = ⟨a, b, t|a^{3} = b^{3} = [a, b] = t^{8} = 1, t^{7}at = b, t^{7}bt = ab⟩ 3. ((C_{3} × C_{3}) ⋊_{1} C_{4}) × C_{2} = ⟨a, b, t, s|a^{3} = b^{3} = [a, b] = t^{4} = s^{2} = [a, s] =

[b, s] = [t, s] = 1, t^{3}at = a^{2}, t^{3}bt = b^{2}⟩

4. ((C_{3} × C_{3}) ⋊_{2} C_{4}) × C_{2} = ⟨a, b, t, s|a^{3} = b^{3} = [a, b] = t^{4} = s^{2} = [a, s] = [b, s] = [t, s] = 1, t^{3}at = b, t^{3}bt = a^{2}⟩

5. (C_{3} ×C_{3})⋊_{1} D_{4} = ⟨a, b, s, t|a^{3} = b^{3} = [a, b] = t^{4} = s^{2} = (st)^{2} = [s, a] = [s, b] = 1, t^{3}at = b, t^{3}bt = a⟩ = T ⋊ C_{6} 6. (C_{3} × C_{3}) ⋊ D_{4} = ⟨a, b, s, t|a^{3} = b^{3} = [a, b] = t^{4} = s^{2} = (st)^{2} = (sa)^{2} =

[s, b] = 1, t^{3}at = b, t^{3}bt = a^{2}⟩

7. (C_{3} × C_{3}) ⋊_{1} Q = ⟨a, b, s, t|a^{3} = b^{3} = s^{4} = t^{4} = [a, b] = [b, t] = [a, s] = (ts)^{2} = 1, t^{2} = s^{2}, t^{−1}at = a^{−1}, s^{−1}bs = b^{−1}⟩

8. (C_{3} × C_{3}) ⋊_{2} Q = ⟨a, b, s, t|a^{3} = b^{3} = s^{4} = t^{4} = [a, b] = [a, s] = [b, t] = 1, t^{2} = s^{2}, s^{−1}ts = t^{−1}, s^{−1}bs = b^{−1}, t^{−1}at = a^{−1}⟩

9. (C_{3} × C_{3}) ⋊_{3} Q = ⟨a, b, s, t|a^{3} = b^{3} = s^{4} = t^{4} = [a, b] = [a, b] = [b, t] = [a, s] = 1, t^{2} = s^{2}, s^{−1}ts = t^{−1}, s^{−1}bs = b^{−1}⟩

10. (C_{3} × C_{3}) ⋊_{4} Q = ⟨a, b, s, t|a^{3} = b^{3} = s^{4} = t^{4} = [a, b] = [a, t] = [b, t] = 1, t^{2} = s^{2}, s^{−1}bs = b^{−1}, s^{−1}as = a^{−1}⟩

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M. Izquierdo, Matematiska institutionen, Linköpings universitet, 581 83 Linköping, Sweden e-mail: miizq@mai.liu.se

D. Ying, Matematiska institutionen, Linköpings universitet, 581 83 Linköping, Sweden e-mail: dayin@mai.liu.se