Topological Classification of Conformal Actions on pq-Hyperelliptic Riemann Surfaces

A compact Riemann surface X of genus g>1 is said to be p-hyperelliptic if X admits a conformal involution ρ, for which X/ρ is an orbifold of genus p. If in addition X is q-hyperelliptic, then we say that X is pq-hyperelliptic. Here we study conformal actions on pq-hyperelliptic Riemann surfaces with central p- and q-hyperelliptic involutions.


Introduction
We say that the finite group G acts on a topological surface X if there exists a monomorphism ε : G → Hom + (X), where Hom + (X) is the group of orientation-preserving homeomorphisms of X. Two actions of finite groups G and G on X are topologically equivalent if the images of G and G are conjugate in Hom + (X). There are two reasons for the topological classification of finite actions rather than just the groups of homeomorphisms. Firstly, the equivalence classes of group actions are in 1-1 correspondence to conjugacy classes of finite subgroups of the mapping class group and so such a classification gives some information on the structure of this group. Secondly, the enumeration of finite group actions is a principal component of the analysis of singularities of the moduli space of conformal equivalence classes of Riemann surfaces of a given genus since this space is an orbit space of Teichmüller space by a natural action of the mapping class group, see [1].
The classification of conformal actions up to topological conjugacy is a classical problem which up to now was solved for surfaces of genera g = 2,3 in [2] (the paper omits one group for the genus 3), g = 4 in [3], for elliptic-hyperelliptic surfaces in [4] and for 2-hyperelliptic Riemann surfaces in [5]. Bujalance et al. [9] determined, for each g, the full automorphism groups of a hyperelleptic Riemann surface and Weaver classified their action in [7].

International Journal of Mathematics and Mathematical Sciences
A compact Riemann surface X of genus g ≥ 2 is said to be p-hyperelliptic if X admits a conformal involution ρ, called a p-hyperelliptic involution, such that X/ρ is an orbifold of genus p. This notion has been introduced by Farkas and Kra [6], where they also proved that for g > 4p + 1, the p-hyperelliptic involution is unique and central in the full automorphism group of X. A Riemann surface which is pand q-hyperelliptic simultaneously is called pq-hyperelliptic. In [8] it was shown that for 0 ≤ p ≤ q with pq = 0, such a surface of genus g ≥ 2 exists if and only if 2q − 1 ≤ g ≤ 2p + 2q + 1. Here we restrict our attention to conformal actions on pq-hyperelliptic Riemann surfaces whose pand q-hyperelliptic involutions are central in the full automorphism group. In particular, according to [8,Theorem 3.7], this class of surfaces contains all pq-hyperelliptic Riemann surfaces of genera g for 2 ≤ p < q < 2p and g > 3q + 1.
For commuting pand q-hyperelliptic involutions δ, ρ, let k be the genus of X/ ρ,δ . Then ρδ is a (g − p − q + 2k)-hyperelliptic involution and k is in the range 0 ≤ k ≤ (2p + 2q + 1 − g)/4. Let X p,q k denote a Riemann surface with central pand q-hyperelliptic involutions corresponding given k and let G be an automorphism group of X p,q k . The group G = G/ δ,ρ acts on the surface X p,q k / δ,ρ of genus k. Using the known classification of finite group actions on surfaces of low genera, we determine the presentation of G and next we lift G to the group acting on the surface X p,q k . The method is similar to that used in [4,5,9], however this time it involves many more calculations and the set of topological classes of actions is much bigger. For this reason, we restrict ourselves to k = 0,1,2 only. These are the only possible values of k corresponding g in range 2p + 2q − 10 ≤ g ≤ 2p + 2q + 1. We give the full topological classification of actions on surfaces X p,q k of such genera except X p,q 1 of genus 2p + 2q − 3 and X p,q 2 of genus 2p + 2q − 7 and decide which of them can be chosen to be full. For k = 0, we enlarge our assumption to g > 2q − 1. In the general case, the two exceptional surfaces need many more calculations than others and so we omit them. However, for the particular values of g, p, and q it is not difficult to complete the gap.
The main results are presented in Theorems 3.4-3.7 and the supporting tables. In particular, Theorem 3.7 lists the actions on any pq-hyperelliptic Riemann surface of genus g in range 2p + 2q − 2 ≤ g ≤ 2p + 2q + 1 for 5 ≤ p < q < 2p − 3. As an example, we give the actions on 5-, 6-hyperelliptic Riemann surfaces of genus 20, 21, 22, and 23 (see Table 1.1). Every action is determined by the finite group of automorphisms G, the signature of a Fuchsian group Λ and a surface-kernel epimorphism θ : Λ → G defined by a so-called generating vector which is the sequence of the images of the canonical generators of Λ.
The group Λ has associated to it a fundamental region whose area μ(Λ), called the area of the group, is An abstract group Λ with the presentation (2.2) is isomorphic to a Fuchsian group with the signature (2.1) if and only if the right-hand side of (2.3) is greater than 0; in that case (2.1) is called a Fuchsian signature.
If Λ is a subgroup of finite index in a Fuchsian group Λ , then we have the Riemann-Hurwitz formula Let G be a finite group acting on a Riemann surface X of genus g > 1. If the canonical projection X → X/G is ramified at r points with multiplicities m 1 ,...,m r and γ is the genus of X/G, then the vector of numbers (γ : m 1 ,...,m r ) is called the branching data of G on X.
There is 1-1 correspondence between the set of generating vectors of G and the set of epimorphisms θ : Λ → G with torsion-free kernels. Two epimorphisms θ : Λ → G and θ : Λ → G define topologically equivalent actions if for some isomorphisms ϕ : G → G and ψ : Λ → Λ [2]. The relation of the equivalence of actions induces an equivalence relation on generating vectors in the sense that two such vectors are equivalent if they determine the epimorphisms which give rise to equivalent actions. We will need some pairs of isomorphisms of Fuchsian groups and abstract groups to demonstrate topological equivalence of actions, as in (2.7).
When G is generated by the element x and two central involutions ρ 1 , ρ 2 for which the assignment ω(x) = x m ρ k 1 ρ l 2 and ω(ρ j ) = ρ j defines an isomorphism ω : G → G , we will use the pair Ω k,l,m = (id Λ ,ω).
If G = Λ/Γ with σ(Λ) = (1;m 1 ,...,m r ) and for fixed integers α 1 ,α 2 ∈ {0, 1}, there exists s or w in range 1 ≤ s, w ≤ r such that θ(c) = ρ α1 1 ρ α2 2 or θ(d) = ρ α1 1 ρ α2 2 , where θ : Λ → G is a surface-kernel epimorphism, c = x 1 ··· x s and d = x w ··· x r , then by Υ α1,α2 and by Θ α1,α2 we will denote the pairs (υ,id G ) and (μ,id G ), respectively, where υ and μ are defined by Finally, in the case when m k = m l in the signature of Λ, we will use the pair Ψ k,l = (ψ k,l ,id G ), where ψ k,l is defined by the assignment ψ k,l (a j ) = a j , ψ k,l (b j ) = b j , and (2.8) Ewa Tyszkowska 7 The pairs Ψ k,l induce the equivalence of every two generating vectors admitting the same elements up to permutation of involutions ρ 1 , ρ 2 , ρ 3 , what allows us to write the generating vector in the form where u = u 1 + u 2 + u 3 and ρ ui i denotes the sequence ρ i , ui ...,ρ i . For better readability, we will separate the central involutions from the other elements of the generating vector by the vertical line. If in addition G is abelian, then any permutation of elements with the same orders of v provides an equivalent vector.

The actions of finite groups on pq-hyperelliptic Riemann surfaces
A conformal involution ρ of a Riemann surface X of genus g > 1 is called a p-hyperelliptic involution if X/ρ has genus p. For simplicity, we will say that ρ is a p-involution. Here we study conformal actions on Riemann surfaces admitting central pand q-involutions simultaneously for some integers q ≥ p. In particular, this class of surfaces contains all pqhyperelliptic Riemann surfaces of genus g for any integers p, q, g in range 2 ≤ p < q < 2p and g > 3q + 1. The product of commuting pand q-involutions is a t-involution, where the possible values of t are given in the following lemma which is a consequence of [8,Theorem 3.4].

Then there exists a Riemann surface of genus g admitting commuting pand q-involutions whose product is a t-involution if and only if
k denote a Riemann surface of genus g with central pand q-involutions whose product is a (g − p − q + 2k)-involution for some fixed k in the range 0 ≤ k ≤ (2p + 2q + 1 − g)/4. The group Z 2 ⊕ Z 2 generated by the pand q-involutions of X p,q k can be represented as Δ/Γ for some Fuchsian group Δ, which by Proposition 2.1 has the signature (k;2, g+3−4k ... ,2). Thus the dimension d of the corresponding locus in the moduli space is 6(k − 1) + 2(g + 3 − 4k) = 2g − 2k. So by the inequality g ≥ 2q − 1 and the restrictions on k, we obtain the following lemma.
Lemma 3.2. For q ≥ p, the pq-hyperelliptic locus in the moduli space corresponding to classes of surfaces admitting central pand q-involutions is a finite union of manifolds of dimensions ranging between 3(p − 1) and 2g.
Given an integer g ≥ 2, a group G is said to be a g − (p, q,k)-hyperelliptic subgroup if there exists a surface X p,q k of genus g such that G ⊆ Aut(X p,q k ) and the pand q-involutions belong to G. In such a case X p,q k = Ᏼ/Γ for some Fuchsian surface group of the orbit genus g and G = Λ/Γ for some Fuchsian group Λ, say with the signature τ = (γ;m 1 ,...,m r ), containing Δ as a normal subgroup. Assume that G = Λ/Δ has order N. Let θ and π be the canonical epimorphisms from Λ onto G and G onto G and θ = πθ. Let p i be the order of θ(x i ). Then by Proposition 2.1, m i / p i = 2 or m i / p i = 1 and (2.6) is satisfied.
of the elliptic generators of Λ corresponding to p i = 1 onto ρ j for some assignment μ : We describe g − (p, q,k)-subgroups corresponding to g in range 2p + 2q − 10 ≤ g ≤ 2p + 2q + 1, where according to Lemma 3.1, k takes one of the values 0, 1, 2. In order to find the complete list of such groups acting on a surface of given genus, first of all we determine all the possible signatures of Λ, up to permutation of periods, however any such permutation gives rise to the same sets of topological types of actions. For, we find Ewa Tyszkowska 9 1} such that r i s i = 0 and θ maps u j of elliptic generators with p i = 1 onto ρ j , where u j are given by (3.2). So we examine all the possible values of r i , s i and choose those for which the numbers u j are integers for some 0 ≤ p ≤ q with pq = 0. In this way, we obtain the connection between the number of periods m i = 2 corresponding to p i = 1 and the values of p and q. To simplify the calculations we list in Table 3.1 the values of F p , F q , and F t corresponding the considered surfaces, where F μ( j) = 2g + 2 − 4μ( j) denotes the number of fixed points of μ( j)-involution for the assignment μ : Having all candidates for the signature of Λ, we determine the presentations of the corresponding groups G = G/ ρ 1 ,ρ 2 by inspecting groups of automorphisms of a genus k surface and choosing those of them which admit generators of orders p i not leading to a contrary with the values of r i and s i (some p i may be equal to 1). The generating vector of G given in Tables 3.2, 3.5, and 3.9 according as k = 2,1 or 0 determines, up to topological equivalence, how an epimorphism θ : Λ → G maps the canonical generators of Λ, except the elliptic elements corresponding to p i = 1 which are mapped onto 1. A generating vector of G can be written as 1} and ρ ui i denotes ρ i , ui ...,ρ i . In order to determine the set of relations in the presentation of G, we assume that any relation R θ a 1 ,..., θ b γ , θ x 1 ,..., θ x r = 1 (3.4)  Following the program outlined above we classify the finite group actions on surfaces X p,q k of genera g in range 2p + 2q − 10 ≤ g ≤ 2p + 2q + 1 except X 1 of genus g 5 and X 2 of genus g 9 since the two cases require lifting all groups of automorphisms of a genus 1 or a genus 2 surface, respectively, and the corresponding sets of the topological classes of Ewa Tyszkowska 11 actions are very big. Additionally, for k = 0, we extend the assumption to g ≥ 2q − 1. The results can be applied for any pq-hyperelliptic Riemann surface of genus g > 3q + 1 with p < q.

Classification of conformal actions on a surface X p,q 2
Theorem 3.4. The topological type of the action on a surface X p,q 2 of genus g 10 , g 11 , or g 12 is determined by the group of automorphisms G, the signature of Λ, and the generating vector listed in Tables 3.3 and 3.4, where ν denotes a permutation of the set {p, q}. Proof. Let G be an automorphism group of a surface X p,q 2 of genus g 10 , g 11 , or g 12 and let ρ 1 , ρ 2 , and ρ 3 = ρ 1 ρ 2 denote the central involutions of G. According to the classification of finite group actions on a genus 2 surface given up to topological equivalence by Broughton [2] and the solutions of (3.1) for k = 2, the corresponding group G is isomorphic to one of the groups listed in Table 3 Table 3.1, we conclude that u j can be integers only for a surface of genus g 12 if there is exactly one t-involution among the elements θ(x i ) pi with p i = 1 corresponding to p i = 4. However, this condition cannot be satisfied because p i = 4 for i = 2,3, and θ(x 2 ) p2 = θ(x 3 ) p3 . Indeed, since G ∼ = x, y : x 4 , y 3 , x y x −1 y and the epimorphism θ is defined by θ(x 1 ) = y, θ(x 2 ) = ( x y) −1 , θ(x 3 ) = x, and θ(x j ) = 1 for j = 4,...,r, it follows that any generating vector of G has a form v = (yρ k1 1 ρ l1 2 ,(xy) −1 ρ k2 1 ρ l2 2 ,xρ k3 1 ρ l3 2 | ρ u1 1 ,ρ u2 2 ,ρ u3 3 ) for x and y belonging to π −1 (x) and π −1 ( y), respectively, and k i ,l i ∈ {0, 1}, which implies that θ( Let us consider the remaining cases from Table 3.2. First suppose that G is isomorphic to Z n = x . Then G is generated by element x ∈ π −1 ( x) and two central involutions ρ 1 , ρ 2 . Since x n = ρ δ1 1 ρ δ2 2 for some δ 1 ,δ 2 ∈ {0, 1}, it follows that G is isomorphic to Z n ⊕ Z 2 ⊕ Z 2 if δ 1 = δ 2 = 0 or to Z 2n ⊕ Z 2 otherwise.
Case 2.14. Here n = 6 and v = (xρ k1 , it follows that γ 1 ≡ r 1 + r 2 + r 3 (2) and γ 2 ≡ s 1 + s 2 + s 3 (2). Using the pairs Φ k,l,m,n we can show that v is equivalent to (x,x −1 y, y −1 ρ u1+u3 . The numbers u j are integers only for a surface of genus g 11 if one of elements θ(x 2 ) p2 or θ(x 3 ) p3 is the only t-involution among the elements θ(x i ) pi with p i = 1. As in the case 2.5, we can show that both possibilities provide the equivalent actions, however this time the automorphism ψ 2,3 of the group Λ is involved.    on a genus one surface given in [4], we conclude that the corresponding group G has one of the presentations listed in Table 3.5. According to these cases we will refer to an action of the group G as to (1.  Proof. In this case, the group Λ has the signature σ(Λ) = (1;2, r ...,2). Let x 1 ,...,x r ,a 1 ,b 1 be its canonical generators and let x and y denote θ(a 1 ) and θ(b 1 ), respectively. Then

10)
and ν denotes a permutation of the set {p, q}.
Proof. By inspecting Table 3.1 and the formula (3.2) for g in range g 12 ≤ g ≤ g 6 , we conclude that the actions (1.1)-(1.4) are possible only on a surface of genus g 9 if both elements x and y in the presentation of G have the orders 2 and exactly one of elements θ(x i ) pi is the t-involution for some i with 2| G|/ p i = 8.