Symplectic representations and Riemann surfaces with many automorphisms

Manfred Streit

Introduction. In this note I will describe a constructive method to calculate the representation of the automorphism group Aut(X) on the first homology group H_{1}(X) where X is a Riemann surface with many automorphisms. Therefore we describe a combinatorial method to calculate a basis of H_{1}(X), the matrices describing the action of Aut(X) on this basis and the intersection matrix S of this basis. Clearly this is equivalent to the calculation of a symplectic representation of Aut(X) of degree 2g where g ≥ 1 is the genus of X. There are various applications for which the explicit calculation of these matrices are of interest. One application is to get by a further splitting in two g–dimensional representation the action of Aut(X) on the abelian differetials of the first kind. In some cases there is also the possibility to calculate period matrices of the surface as fixed points of the action of the symplectic representation on the Siegel upper half plane, cf. [Bol,Schi]. We obtain the following generalization of a result in [Str]:

Theorem 1. The representation of Aut(X) for X with many automorphisms and genus g ≥ 1 on H_{1}(X) is a subrepresentation of the regular representation.

The proof of this result is fully constructive and the algorithm is implemented in GAP [Gap] and can be obtained from the author. Lemma 1 and Lemma 2 also appear in [Str] and are included for the convenience of the reader.

Part I

Let G be a finite group generated by two elements g_{0}, g_{1}, such that the inequality

q := \frac{1}{k_{0}} + \frac{1}{k_{1}} + \frac{1}{k_{∞}} ≤ 1  (1)

is satisfied, where k_{i} := ord(g_{i}), i = 0, 1, ∞ and g_{∞} := (g_{0}g_{1})^{−1}. This is the natural situation one can find for Riemann surfaces with many automorphisms, i.e. for any regular hypermap, see [JoSi, JSt, Wo]. We therefore want to describe the construction of a Riemann surface X admitting G as a (sub)group of conformal automorphisms. Let

Δ_{k_{0},k_{1},k_{∞}} := ⟨γ_{0}, γ_{1}, γ_{∞} | γ_{0}^{k_{0}} = γ_{1}^{k_{1}} = γ_{∞}^{k_{∞}} = γ_{0}γ_{1}γ_{∞} = id⟩

be a Fuchsian or plane triangle group. For the following we fix the geometric action of γ_{i}, i = 0, 1, ∞. We want them to be anticlockwise rotations with rotation angles 2π/k_{i} around the non accidental vertices of a canonical fundamental domain of the triangle group Δ_{k_{0},k_{1},k_{∞}}. We then define an homomorphism φ : Δ_{k_{0},k_{1},k_{∞}} ⟶ G, γ_{i} ⟼ g_{i}, i = 0, 1, ∞. The kernel of this epimorphism ker(φ) =: Γ defines a compact Riemann surface X taking X := Γ\ℌ for q < 1 or X := Γ\ℂ for q = 1. The genus g of X can be easily calculated by

2g − 2 := |G|(1 − q).  (2)

It is well known that Aut(X) ⊇ Δ_{k_{0},k_{1},k_{∞}}/Γ. In fact Aut(X) = Δ_{k_{0},k_{1},k_{∞}}/Γ is the generic situation as Aut(X) ⊃ Δ_{k_{0},k_{1},k_{∞}}/Γ holds only if g = 1 or ifΓ is normally contained in some triangle group which also contains Δ_{k_{0},k_{1},k_{∞}} and this is quite rare.

We now give a construction of the representation of Δ_{k_{0},k_{1},k_{∞}}/Γ on the first homology group H_{1}(X) as a subrepresentation of the regular representation. This is a generalization of the construction in [Str] because no further restrictions to the group G is necessary. Let V := ⟨e_{g} | g ϵ G⟩ be a |G|–dimensional ℂ–vector space. A basis {e_{g} | g ϵ G} is labeled by the group elements of G. We define two linear representations ρ, λ on V. ρ stands for multiplication from the right and for any group element h ϵ G we define ρ(h) on our basis {e_{g} | g ϵ G} by:

ρ(h) : e_{g} ⟼ e_{gh}, g ϵ G.

With λ we describe the analogous situation with multiplication from the left:

λ(h) : e_{g} ⟼ e_{hg}, g ϵ G.

Lemma 1 The eigenspaces V_{i} := Eig(ρ(g_{i}), 1) of ρ(g_{i}), i = 0, 1, ∞ to the eigenvalue 1 have dimension |G|/k_{i}. The space V_{i} is λ–invariant.

Proof. Let \underline{x} = ∑ μ_{g}e_{g} an element of V_{i}. This can be expressed in the condition (A_{i})

(A_{i}) : ρ(g_{i})\underline{x} = \underline{x}

or equivalently

(A_{i}) : \underline{x} ϵ V_{i} ⟺ μ_{g} = μ_{gg_{i}} for all g ϵ G.

Then it is easy to see that V_{i} is generated by the orbit of e_{g_{i}} + … e_{g_{i}^{k_{i}}} under λ. As an induced representation V_{i} has the decomposition

⨁_{g̃ ϵ G/H_{i}}⟨e_{g̃g_{i}} + … + e_{g̃g_{i}^{k_{i}}} ⟩,

where H_{i} := ⟨g_{i}⟩, i = 0, 1, ∞ is the subgroup of generated by g_{i}. The character of this induced representation is denoted by χ_{i} := Ind(1, H_{i}). □

Lemma 2 The following statements are true for i ≠ j, i, j = 0, 1, ∞

i) V_{i} ∩ V_{j} = ⟨∑_{g ϵ G} e_{g}⟩ , ii) dim(V_{i} + V_{j}) = |G|/k_{i} + |G|/k_{j} − 1 , iii) The character of the restriction res(λ, V_{i} + V_{j}) of λ to V_{i} + V_{j} is χ_{i} + χ_{j} − 1.

Proof. It is enough to prove i) as ii) and iii) are simple consequences of i). Let \underline{x} = ∑ μ_{g}e_{g} ϵ V_{i}∩V_{j}. This implies that the conditions (A_{i}) and (A_{j}) hold simultaneously. We therefore find for \underline{x}:

μ_{g} = μ_{gg_{i}} for all g ϵ G
μ_{g} = μ_{gg_{j}} for all g ϵ G.

As G = ⟨g_{i}, g_{j}⟩ we find μ_{g} = μ_{id} for all g ϵ G. This proves the statements. □

Lemma 3: Let E be the |G| × |G| identity matrix. Then the following statements are true:

i) dim((E − ρ(g_{1}))V_{0}) = |G|/k_{0} − 1, ii) (E − ρ(g_{1}))V_{0} ⊥ V_{1} + V_{∞}, iii) dim((E − ρ(g_{1}))V_{0} + V_{1} + V_{∞}) = |G|(1/k_{0} + 1/k_{1} + 1/k_{∞}) − 2, iv)((E − ρ(g_{1}))V_{0} + V_{1} + V_{∞})^{⊥} has dimension 2g. λ restricted ((E − ρ(g_{1}))V_{0} + V_{1} + V_{∞})^{⊥} is isomorphic to the representation of Aut(X) on ℂ ⊗ H_{1}(X) and has the character 2 − χ_{0} − χ_{1} − χ_{∞}.

Proof. i) The kernel of (E−ρ(g_{1})) is V_{1}. From Lemma 2i) we know V_{0}∩V_{1} = ⟨∑ e_{g}⟩.

This shows the statement. ii) V_{0} has a basis

{∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}} | g̃ ϵ G/⟨g_{0}⟩} .

The image of V_{0} under (E − ρ(g_{1})) is generated by

{∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}} − ∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}g_{1}} | g̃ ϵ G/⟨g_{0}⟩} .

We prove orthogonality to V_{∞} first. Let ∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}} − ∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}g_{1}}, ∑_{r=1}^{k_{∞}} e_{gg_{∞}^{r}} be any two generating vectors of V_{0}, resp. V_{∞}.

⟨∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}} − ∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}g_{1}}, ∑_{r=1}^{k_{∞}} e_{gg_{∞}^{r}}⟩ = ∑_{s=1}^{k_{0}} ∑_{r=1}^{k_{∞}}⟨e_{g̃g_{0}^{s}}, e_{gg_{∞}^{r}}⟩ − ∑_{s=1}^{k_{0}} ∑_{r=1}^{k_{∞}}⟨e_{g̃g_{0}^{s}g_{1}}, e_{gg_{∞}^{r}}⟩

= ∑_{s=1}^{k_{0}} ∑_{r=1}^{k_{∞}}⟨e_{g̃g_{0}^{s}}, e_{gg_{∞}^{r}}⟩ − ∑_{s=1}^{k_{0}} ∑_{r=1}^{k_{∞}}⟨e_{g̃g_{0}^{s−1}g_{∞}^{−1}}, e_{gg_{∞}^{r}}⟩

= ∑_{s=1}^{k_{0}} ∑_{r=1}^{k_{∞}}⟨e_{g̃g_{0}^{s−1}}, e_{gg_{∞}^{r}}⟩ − ∑_{s=1}^{k_{0}} ∑_{r=1}^{k_{∞}}⟨e_{g̃g_{0}^{s−1}g_{∞}^{−1}}, e_{gg_{∞}^{(r−1)}}⟩

The first equation is a consequence of the group relation g_{0}g_{1}g_{∞} = id. In the last equation we have changed the summation index in the second sum from r to r − 1. The entries in the occuring sums are always equal to 0 or 1. We find

⟨e_{g̃g_{0}^{s−1}}, e_{gg_{∞}^{r}}⟩ = 1 ⟺ g̃g_{0}^{s} = gg_{∞}^{r} ⟺ g̃g_{0}^{s−1}g_{∞}^{−1} = gg_{∞}^{r−1}
⟺ −⟨e_{g̃g_{0}^{s−1}g_{∞}^{−1}}, e_{gg_{∞}^{r−1}}⟩ = −1.

Therefore we have

⟨∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}} − ∑_{s=1}^{k_{0}} e_{g̃g_{0}^{s}g_{1}}, ∑_{r=1}^{k_{∞}} e_{gg_{∞}^{r}}⟩ = 0.

The orthogonality to V_{1} follows with the same type of argument. This shows ii). iii) This is a simple consequence of ii) together with equality (2). iv) There are two ways of proving the statements of iv). The first method uses a theorem of Chevalley and Weil [ChW] which enables us to calculate how often a given irreducible character χ occurs in Aut(X) on ℂ⊗H_{1}(X). It turns out to be the same as the scalar product(χ, 2 − (χ_{0} + χ_{1} + χ_{∞})). A detailed proof of this can be found in [Str]. The second way is to construct a basis of ((E −ρ(g_{1}))V_{0} +V_{1} +V_{∞})^{⊥} and give it a direct geometric interpretation as a homology basis of the ℤ module H_{1}(X) ie. as closed curves on X. We do this in part II and give also a method to construct the intersection matrix for this basis. iv) follows from this construction. □

Part II

To explain how to calculate a basis of the space ((E − ρ(g_{1}))V_{0} + V_{1} + V_{∞})^{⊥} it is more convenient to work with permutations. Let (g_{1}, …, g_{|G|}) be a fixed list of group elements. Associated with this list we define a basis e_{i} := e_{g_{i}}, i = 1 … |G|. The regular representations λ, ρ can then be expressed by permutations:

λ(g) ⟷ α_{g} with gg_{i} = g_{α_{g}(i)} and
ρ(g) ⟷ π_{g} with g_{i}g = g_{π_{g}(i)}.

for all g, g_{i} ϵ G. For the following we only need the group generators g_{0}, g_{1}, g_{∞} which we have introduced in part I. Note that the indices 0, 1, ∞ are introduced to refer to the so called Belyi functions and have nothing to do with the ordering of the above list where the same group elements appear again with a different label. The triple (π_{0}, π_{1}, π_{∞}) is called a regular hypermap. The triple (α_{0}, α_{1}, α_{∞}) is the automorphism group of the regular hypermap. It is known that the regular hypermap carries all the information one needs to construct a connected fundamental domain of the group Γ from a fundamental domain of the triangle group Δ_{k_{0},k_{1},k_{∞}} [JoSi]. This is equivalent to construct a decomposition of the surface X = Γ\ℌ into triangles. One can proceed as follows:

1. Take 2|G| triangles △_{1}, …, △_{2|G|} and label the vertices of each triangle △_{i} with 0, 1, ∞ in positive sense whenever i is odd and in negative sense when i is even. In order to construct a fundamental domain we have to identify the edges of these triangles in saying for instance the triangle i and the triangle j have a common edge of type (0, 1). Another way to give the same information is to start a walk around the vertex 0 of the triangle i in say positive sense and list all the triangles we are passing by in order of appearance i, i_{1}, i_{2}, …, i_{k}. Remark that an even triangle is always followed by an odd triangle and an odd triangle by an even.

2. The situation we find for regular hypermaps allows us to start to construct regular 2k_{0}–gons in putting all the triangles together with a common vertex 0. The order of appearance is given by the permutation π_{0}. In positive sense we meet the triangles

[2i − 1, 2i, 2π_{0}(i) − 1, 2π_{0}, 2π_{0}^{2}(i) − 1, 2π_{0}^{2}(i), …, 2π_{0}^{k_{0}−1}(i) − 1, 2π_{0}^{k_{0}−1}(i)].

3. We now have to deal with |G|/k_{0} many 2k_{0}–gons to construct our fundamental domain. The only remaining edges are of type (1, ∞) as all vertices of type 0 are mid points of the 2k_{0}–gons. The permutation π_{1} encodes all the information we need to identify those edges which are to be identified to obtain either a connected fundamental domain or even the surface itself. To construct the fundamental domain we would start with any 2k_{0}–gon. An anticlockwise walk around a vertex of type 1 of say an even triangle 2i in this 2k_{0}–gon brings us to the triangle 2π_{1}(i) − 1. So we get an identification with another 2k_{0}–gon and we glue them along this (1, ∞)–edge together which leaves us with one remaining 2k_{0}–gon less. It is quite clear that we can repeat this procedure as long as we have remaining 2k_{0}–gons and that we finish with a connected fundamental domain of our group Γ. But we have not used all the information we can get from π_{1} as it also reveals all the side identifications of the so constructed fundamental domain. In the following picture we show all the adjacent triangles of the triangle pair △_{2i−1}, △_{2i} and it is obvious that this is enough information to construct the surface X.

2i − 1  2i 2π_{0}(i) − 1 2π_{∞}^{−1}(i) − 1 2π_{0}^{−1}(i) 2π_{∞}(i) 0 1  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Algorithmically the gluing together process can be described by inserting lists like

[2π_{∞}^{−1}(i) − 1, 2π_{∞}^{−1}(i), 2π_{0}(π_{∞}^{−1}(i)) − 1, …]

which represents a 2k_{0}–gon into a starting list [2i − 1, 2i, …]. The result is

[2i − 1, 2i, [2π_{∞}^{−1}(i) − 1, 2π_{∞}^{−1}(i), 2π_{0}(π_{∞}^{−1}(i)) − 1 …] …].  (3)

Repeating this process until we have placed all the 2k_{0}–lists we will end up with a list L with entries either natural numbers or lists of natural numbers and this characterization holds at each level. This simply means that every entry of each list which is a list contains natural numbers and lists of natural numbers. This list L clearly is a precise description of the fundamental domain 𝔉_{Γ}.

4. Now we associate to each triangle △_{i} a directed edge c_{i} joining 1 and ∞.

0  0 1  1 ∞ ∞ 2i − 1  2i >  <_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

With the help of this directed edges we get a new interpretation of the list L. Reading the indices appearing in L from the left to the right and ignoring the different levels of the list we obtain a walk around the fundamental domain 𝔉_{Γ}. Therefore we need to go −c_{i} when i appears and when i is even and c_{i} if i appears and i is odd. If we realize this walk on the surface X we have a 0–homotopic closed curve which can be thought to be the border of the fundamental domain 𝔉_{Γ} cancelling subsequently edges if they differ only by their orientation. According to the list (3) a part of the border of the fundamental domain 𝔉_{Γ} would look as follows.

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5. In order to describe a basis of H_{1}(X) we consider all the side identification which we obtain through π_{∞}. We know that the (1, ∞)–edge of the triangle △_{2j−1} has to be identified with the (1, ∞)–edge of the triangle △_{2π_{∞}(j)}. Therefore we find a closed curve w on X which is represented by a curve w_{2j−1} cutting through 𝔉_{Γ} and connecting the midpoints of the (1, ∞)–edges of both of the triangles. The convention to start at a mid point of a (1, ∞)–side is not necessary. One also could start at the ∞–point of the triangle △_{2j−1} as shown by the picture below as both curves are freely homotopic. But then it is obvious that this curve w_{2j−1} is homotopically equivalent to a curve ∑ ±c_{i} along the border of 𝔉_{Γ}. Which way to go can be easily read off by the list L. We just have to go all the c_{i} with i standing in L in between 2j − 1 and 2π_{∞}(j). It is also obvious that any closed curve on X can be deformed such that it meets the border of 𝔉_{Γ} only transversally. Therefore {w_{2j−1}|1 = 1, …, |G|} contains a basis of H_{1}(X).

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6. If we wish to choose a basis of H_{1}(X) in {w_{2j−1}|1 = 1, …, |G|} we best consider the intersection matrix S̃ = (w_{2i−1}·w_{2j−1})_{i,j=1,... ,|G|}. Like every intersection matrix S̃ is skew symmetric. From our construction it is clear that w_{2i−1} and w_{2j−1} meet at most once. Therefore the entries of S are either 0 or ±1. (This answers a question of Rodriguez and Riera [RiRo]. They observed that in all known examples of explicitly constructed intersection matrices the entries only consist of 0, ±1. But this automatically happens if you choose to construct a homology basis in the described manner, i.e. as paths along the border of a fundamental domain.) Again the list L gives all the information one needs to calculate the intersection number as the list gives an ordering of the border of 𝔉_{Γ}. The whole task of choosing a basis therefore is to choose a submatrix S of S̃ with maximal rank which clearly is 2g where g is the genus of X. Therefore we can define a basis v_{k} := w_{2j_{k}−1}, k = 1, …, 2g where S = (w_{2j_{k}−1} · w_{2j_{l}−1})_{k,l=1,... ,2g} = (v_{k} · v_{l})_{k,l=1,... ,2g}.

7. We now return to the notations of part I to give the connection of the construction above to the space ((E−ρ(g_{1}))V_{0}+V_{1}+V_{∞})^{⊥}. We map the curves c_{i}, i = 1, …, 2|G| into the space ⟨e_{g}|g ϵ G⟩. This will be done by the definition c_{2i−1} ↔ e_{g_{i}}. Note that c_{2i−1} = c_{2π_{∞}(i)}. By this definition we can map H_{1}(X) into the space ⟨e_{g}|g ϵ G⟩. An interesting space is V_{1}. A generating element of V_{1} is of type say e_{g_{i}} + e_{g_{i}g_{1}} + ⋯ + e_{g_{i}g_{1}^{k_{1}−1}}. This element is the image of a directed star c_{2i−1} + c_{2π_{1}(i)−1} + ⋯ + c_{2π_{1}^{k_{1}−1}(i)−1}.

[figure: 1; 2i − 1; 2π_{1}(i) − 1; 2π_{1}^{2}(i) − 1; 2π_{1}^{3}(i) − 1; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

These are all directed edges which start at the point of type 1 of the triangle △_{2i−1}. Let w be the image of w ϵ H_{1}(X) under the above embedding. The scalar product ⟨w, e_{g_{i}}+e_{g_{i}g_{1}}+⋯+e_{g_{i}g_{1}^{k_{1}−1}}⟩ simply counts how often any of the edges of the directed star c_{2i−1} +c_{2π_{1}(i)−1} +⋯+c_{2π_{1}^{k_{1}−1}(i)−1} are used counting a going in negatively and a going out with a positive sign. Therefore ⟨w, e_{g_{i}} + e_{g_{i}g_{1}} + ⋯ + e_{g_{i}g_{1}^{k_{1}−1}}⟩ = 0 as w is closed. We see that w is an element of V_{1}^{⊥}. The same argument shows that w ϵ V_{∞}^{⊥}. Therefore the images of the above constructed basis v_{1}, …, v_{2g} ϵ H_{1}(X) automatically fall in (V_{1} + V_{∞})^{⊥}. The generating elements of (E − ρ(g_{1}))V_{0} are of type e_{g_{i}} − e_{g_{i}g_{1}} + e_{g_{i}g_{0}} − e_{g_{i}g_{0}g_{1}} + … + e_{g_{i}g_{0}^{k_{0}−1}} − e_{g_{i}g_{0}^{k_{0}−1}g_{1}}. This is the directed border of a 2k_{0}–gon from which we have bild up the fundamental domain 𝔉_{Γ}. Therefore the associated closed curve in H_{1}(X) is 0. The basis v_{1}, …, v_{2g} project therefore uniquely and well defined to the vector space elements v_{1}, …, v_{2g} in ((E −ρ(g_{1}))V_{0} +V_{1} +V_{∞})^{⊥} with a well defined intersection matrix S. This finally shows iv) of Lemma 3 and Theorem 1.

References

[Bol] O. Bolza, On Binary Sextics with Linear Transformation onto themselves. Amer. J. Math. 10 (1888).

[ChW] C. Chevalley, A. Weil, Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers. Abh. math. Sem. Univ. Hamburg 10 (1934), 358-361. [GAP] Groups, Algorithm and Programming, Computer algebra system, RWTH Aaachen.

[JSt] G.A. Jones, M. Streit, Galois Groups, Monodromy Groups and Cartographic Groups in Leila Schneps: Geometric Galois actions: 2. Dessins d’Enfants, Mapping Class Groups and Moduli. Lecture Notes of the Lond. Math. Soc. 243 Cambridge University Press (1997).

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