Automorphism Groups of Symmetric and Pseudo-real Riemann Surfaces

The category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface XC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_C$$\end{document} admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface XC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_C$$\end{document} is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface XC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_C$$\end{document} is called symmetric and the curve C is real.


Introduction
It is known that the moduli space M g of complex algebraic curves of genus g is a quasi-projective variety which can be defined in P n (C) by polynomials with rational coefficients. There is an anti-holomorphic involution ι : M g → M g which maps the class of a complex curve to its conjugate. The fixed points of such a mapping are called complex algebraic curves with real moduli. The defining equations of symmetric Riemann surfaces are over reals, and therefore, they are fixed by ι. Earle [7] proved that not every curve with real moduli is a real curve. Seppälä [22] showed that the non-real complex algebraic curves with real moduli are coverings of real algebraic curves. They are called pseudo-real or asymmetric curves.
The existence of pseudo-real surfaces of every genus g ≥ 2 was proved in [5]. A characterization of automorphism groups of such surfaces and the minimal genus for the cyclic case was given in [2]. The topological types of actions of automorphism groups of pseudo-real surfaces of low genera were described in [4]. The pseudo-real surfaces with cyclic automorphism groups were studied in [9]. The hyperelliptic pseudo-real Riemann surfaces were considered in [3,19] and [6]. The equations for non-hyperelliptic surfaces were found in [10] and [11]. The present paper can be seen as a continuation of papers [12][13][14]24], where asymmetric p-hyperelliptic and (q, n)-gonal surfaces, with cyclic automorphism group, were studied.
Let Aut ± (X) be the group of all conformal and anti-conformal automorphisms of a Riemann surface X of genus g ≥ 2, and let Aut + (X) be the subgroup of all conformal automorphisms. We say that a finite group H acts on the surface X, if it is isomorphic to a subgroup of Aut ± (X). By (X, Y ) H g , we denote a pair of Riemann surfaces X and Y of the same genus g ≥ 2 which have automorphism groups H X ⊆ Aut ± (X) and H Y ⊆ Aut ± (Y ), such that H Y H H X . In the paper, we deal with the so-called (a.s.)-pairs (X, Y ) H g for which X and Y are pseudo-real and symmetric surfaces, respectively. The second chapter of the paper presents the most important facts from theory of non-euclidean crystallograpfic groups, called briefly NEC groups. They are discrete and co-compact subgroups of the group of isometries of the hyperbolic plane H. Fuchsian groups are defined as discrete subgroups of conformal automorphisms of H and they are not always co-compact. However, in this paper, we will assume that they are co-compact and we will treat them as NEC groups. An NEC group not being Fuchsian group will be called a proper NEC group.
According to the Riemann uniformization theorem, a compact Riemann surface X of genus g ≥ 2 can be represented as the orbit space H/Γ for some torsion-free Fuchsian group Γ. Moreover, an automorphism group H ⊆ Aut ± (X) of such a surface can be represented as the quotient group Λ/Γ for a proper NEC group or a Fuchsian group Λ according to whether H contains anti-conformal automorphisms or not. An action of H on X corresponds to a short exact sequence of homomorphisms 1 → Γ → Λ θ → H → 1 and is denoted by (Λ, θ, H).
Let H be a finite group acting on a closed Riemann surface of genus g ≥ 2. Theorem 3.2 of chapter 3 provides a sufficient condition on H to construct an action of H = H Z 2 on another closed Riemann surface X of the same genus as X, such that X/H and X /H are topologically equivalent orbifolds and Z 2 component is generated by a symmetry. This condition is provided by the existence of an automorphism ϕ ∈ Aut(H) of order 2 with certain properties on a set of (geometric) generators of H. It was solved by Singerman for a (k, l, m)-group H generated by two elements h 1 and h 2 of orders k and l, respectively, whose product has order m. He proved in [21] that if such a group has an automorphism ϕ ∈ Aut(H) induced by , then an action of H on a Riemann surface X can be extended to an action of the semidirect product H ρ for a symmetry ρ, such that ρh i ρ = ϕ(h i ) for i = 1, 2. In this paper, we prove that for any (k, l, m)-group H and n ∈ N, there is a symmetric Riemann surface of genus g n with an automorphism group H D 2n , where g n = 1 + 2n|H| or g n = 1 + 2n|H| ( respectively. This result is a particular case of Theorem 3.4 which claims that if a finite group H has a minimal generating set A = {h 1 , . . . , h n } and there is an automorphism ϕ ∈ Aut(H) of order 2, such that for each h i ∈ A, ϕ(h i ) ∈ h j for some h j ∈ A, then for any even q ∈ N, there exists an extension of H by a dihedral group D q = υ, ρ which acts on a symmetric Riemann surface in such a way that ρ is a symmetry and υ is a conformal involution of the surface.
In chapter 4, we study finite group actions on asymmetric Riemann surfaces. Theorem 4.2 determines conditions on generating vectors of the full automorphism group of such a surface. We prove that if A is a generating set of a finite group H and μ A = card(A) − h∈A 1 (h) for a subset A ⊆ A, then for any k ∈ N and ϕ ∈ Aut(H), such that 4k ≡ 0 ( (ϕ)), there is an asymmetric Riemann surface X of genus g = 1+4k|H|μ A for which Aut ± (X) is isomorphic to the semidirect product H ϕ,4k = H δ : δ 4k with respect to the action δhδ −1 = ϕ(h) for h ∈ H. If A = A and ϕ 2 = id H , then for any k ∈ N, there is an (a.s.)-pair (X, Y ) The last chapter is devoted to actions without fixed points. We find the conditions on a generating vector of finite group H for which there exists an (a.s)-pair (X, Y ) H g , such that Y /H is a hyperelliptic Klein surface. Finally, in Theorem 5.5, we prove that if p = g−1 n ( n 2 − 1) for n, g ∈ N, such that g ≥ 3 is odd and n is an even divisor of g − 1, then there exists a symmetric phyperelliptic Riemann surface of genus g being n-sheeted unbranched covering of a hyperelliptic Klein surface of algebraic genus 2(g−1) n + 1. In particular, for any odd g ≥ 3, there exists a symmetric hyperelliptic Riemann surface of genus g with two commuting symmetries.

Non-Euclidean Crystallographic Groups
We shall use the combinatorial approach, based on theory of non-Euclidean crystallographic groups (in short NEC groups) which are discrete and cocompact subgroups of the group of isometries of the hyperbolic plane H, including those which reverse orientation. If such a subgroup contains only orientation preserving isometries, then it is called a Fuchsian group. Macbeath and Wilkie [15], [25] associated with every NEC group Λ a signature σ(Λ), which determines its algebraic structure. It has the form The orbit space H/Λ is a surface of topological genus h, having u+v boundary components, and being orientable or not according to the sign being + or  The group with the signature (1) has the presentation given by the following generators and relations: Any generators of an NEC group satisfying the above relations are called canonical generators.
The hyperbolic area μ(Λ) of any fundamental region of an NEC group Λ with the signature (1) is given by where ε = 2 if the sign is + and ε = 1 otherwise. An abstract group with the presentation (2) can be realized as an NEC group if and only if (3) is positive.
If Γ is a finite index subgroup of an NEC group Λ, then it is an NEC group itself and there is a Hurwitz-Riemann formula, which says that A Fuchsian group can be regarded as an NEC group with the signature An NEC group Λ with the signature (1) has the so-called canonical Fuchsian subgroup Λ + consisting of all conformal automorphisms in Λ. Singerman [18] proved that Λ + has the signature (εh + u + v − 1; m 1 , m 1 , . . . , m t , m t , n 11 , . . . , n 1s1 , . . . , n u1 , . . . , n usu ). (4)

Riemann Surfaces
Any compact Riemann surface X of genus h > 1 is uniformized by a torsionfree Fuchsian group Γ with the signature (h; −) called a surface Fuchsian group. An automorphism group G of X is isomorphic to a quotient group Λ/Γ for some NEC group Λ normalizing Γ. If there does not exist another NEC group containing Λ properly, then Λ is called a maximal NEC group and G = Λ/Γ is the full automorphism group of X. The detailed exposition of maximality can be found in [8].
A signature σ is called maximal, if for every NEC group Λ with a signature σ containing an NEC group Λ with the signature σ and having the same Teichmüller dimension, the equality Λ = Λ holds. For any maximal signature σ, there exists a maximal NEC group with the signature σ.
In the paper, we will use Macbeath's formula [16] on the total number of fixed points of a conformal automorphism of a Riemann surface.
is the normalizer of h in H and the sum is taken over indices i for which h is conjugate to a power of θ(x i ).

Klein Surfaces
A Klein surface is a compact topological surface equipped with a dianalytic structure. A Riemann surface can be seen as an oriented Klein surface without boundary. For a given Klein surface X, Alling and Greenleaf [1] constructed certain double cover X + being a Riemann surface. The algebraic genus of X is defined as the genus of X + , and for a surface of topological genus g having k boundary components, it is equal to d = αg + k − 1, where α = 2 if X is orientable and α = 1 otherwise. Preston [17] proved that any compact Klein surface X of algebraic genus d ≥ 2 can be represented as H/Γ for the so-called surface NEC group Γ. If X has topological genus g and k boundary components, then σ(Γ) = (g; ±; It is said that a Klein surface X is p-hyperellitic, if it has a an involution δ for which the orbit space X/ ρ has algebraic genus p. The automorphism δ is called a p-hyperelliptic involution. If p = 0, then the surface X is called hyperelliptic.

On Symmetric Generating Vectors
A finite group H acts on a Riemann surface X of genus g ≥ 2 if and only if there is a short exact sequence of homomorphisms where Λ is an NEC group and Γ is a surface Fuchsian group isomorphic to the fundamental group of X. The action corresponding to this sequence is denoted by (Λ, θ, H). We say that H acts with the signature σ(Λ) and that θ is a smooth epimorphism. In this chapter, we will consider actions of a finite group H with a signature where the sign is + for γ = 0 and it is − otherwise. Let (h) denote the order of an element h ∈ H. Then, we have the following.
and no composition of generators containing an odd number of elements h i is trivial. The surface X has genus Proof. Suppose that there is an action (Λ, θ, H) for which Λ has the signature (6). Then, Λ is generated by glide reflections d 1 , . . . , d γ and elliptic elements x i , . . . , x r which satisfy the relations Any product of these generators containing an odd number of elements h i is nontrivial, because otherwise Γ = kerθ would have an anti-conformal automorphism. In particular, all elements h i have even orders. The hyperbolic area (3) of a fundamental region of Λ is positive what implies that satisfying the conditions (7) and any composition of these elements containing an odd number of h i is nontrivial, then there is an action (Λ, θ, H) on the Riemann surface X = H/kerθ, where Λ is an NEC group with the signature (6), such that m i = (b i ) for 1 ≤ i ≤ r, and θ : Λ → H is an epimorphism induced by By the Hurwitz-Riemann formula, the surface X has genus (8). An ordered sequence T r,γ = [b 1 , . . . , b r , h 1 , . . . h γ ] of generators of a finite group H which satisfies the conditions of Lemma 3.1 is called a generating vector of H. We say that this vector is essential or unessential according to whether γ > 0 or γ = 0, respectively. In the first case H has anti-conformal automorphism, and in the second case, it has only conformal automorphisms. We will also say that the vector T r,γ is exceptional, if the group H has not a symmetry, by which we mean an anti-conformal automorphism of order 2. Clearly, each unessential vector T r,γ is exceptional. In order to an essential vector T r,γ was exceptional, no composition of its elements containing an odd number of h i can have order 2, because otherwise H would have a symmetry. In particular, no h i can be an involution and so (h i ) ≡ 0 (4) for 1 ≤ i ≤ γ.
Throughout the paper, we use the following notation. The symbol m (n) denotes m mod n for m, n ∈ N. If ϕ is an automorphism of a finite group H and q ∈ N is an integer, such that q ( (ϕ)) = 0, then H ϕ,q denotes the semidirect product of H by a cyclic group Z q = δ with respect to the action In a particular case, when ϕ 2 = id H and ϕ fixes a central element z ∈ Z(H) of even order, then there is a normal subgroup z −1 δ 2 in H ϕ,q with q = 2 (z) and the quotient group H ϕ,q / z −1 δ 2 will be denoted by H ϕ/z .
We will say that a generating vector T r,γ = [b 1 , . . . , b r , h 1 , . . . , h γ ] of a finite group H is symmetric, if there is an integer t in the range 1 ≤ t ≤ r 2 for which the condition (10) is satisfied and the assignment (11) induces an automorphism η ∈ Aut(H). We will denote such a vector by T t,η r,γ .

Corollary 3.3. Any finite group acts on infinite many symmetric Riemann surfaces.
Proof. Clearly, this statement is true for the trivial group. Let {g 1 , . . . , g n } be a set of nonidentity generators of a finite nontrivial group H. By repeating one of elements g i two or three times if necessary, we can assume that n ≥ 3.
Then, H has an unessential symmetric generating vector . . , g t , g −1 t . . . , g −1 1 ] for η = id H and t = n, and according to Theorem 3.2, there is a Riemann surface of genus g = 1 The group H acts on infinite many Riemann surfaces of different genera, because in the construction of generating vector, we can repeat generators of H an infinite number of times. A = {a 1 , . . . , a n } be a minimal generating set of a finite noncyclic group H, and suppose that there is an automorphism ϕ ∈ Aut(H) of order 2, such that for each a i ∈ A, ϕ(a i ) ∈ a j for some a j ∈ A. Then for any even q ∈ N, there is a Riemann surface with an automorphism group H D q , where D q is a dihedral group generated by a conformal involution υ and a symmetry ρ, such that ρhρ = h and υhυ = ϕ(h) for h ∈ H.

Theorem 3.4. Let
Proof. Let m i = (a i ) for i = 1, . . . , n, and let α ij be an integer, such that what implies that α 2 ii ≡ 1 (m i ). Thus, for g i = a i , we have ϕ(g i ) = g −1 i or ϕ(g i ) = g i . Now, assume that i = j. Since ϕ(a i ) = a αij j , it follows that m i divides m j . If ϕ(a j ) = a α jk k for k = i, then a i = ϕ 2 (a i ) ∈ a k , against the assumption that A is the minimal generating set. Thus, ϕ(a and g j = a j . Therefore, without lost of generality, we can assume that H has a generating set {g 1 , . . . , g n }, such that for some integers s, p, u ≥ 0 with 2s + p + u = n. For an even q ∈ N, let H be the semidirect product H ϕ,q = H δ : δ q = 1 with respect to the action δhδ = ϕ(h) for h ∈ H. Then, for γ = u +2− u (2) and r = 2(s + p), there is a generating set {b 1 , . . . , b r , h 1 , . . . , h γ } of H which satisfies the conditions (7). It can be chosen as follows: Any product of listed generators containing an odd number of h i has the form hδ l for some h ∈ H and an odd integer l. Therefore, it cannot be trivial, because otherwise 1 = δ l ∈ H, a contradiction. Thus, T r,γ = [b 1 , . . . , b r , h 1 , . . . , h γ ] is a generating vector of H , such that the condition (10) is satisfied for t = s+p. Let η : H → H be the mapping induced by (11). Then, η(h i ) = h −1 i for 1 ≤ i ≤ γ. In particular, η(δ) = δ −1 and Since η(b i ) = b −1 t+1−i , it follows that η(g 2s+i ) = g 2s+i for i = 1, . . . , p and η(g 2i ) = g 2i for i = 1, . . . , s. By the last equalities, we get what implies that η(δg i δ −1 ) = η(ϕ(g i )) for i = 1, . . . , n. Therefore, η is an automorphism of the group H and vector T r,γ can be interpreted as a symmetric generating vector T s+p,η r,γ . According to Theorem 3.2, there is a Riemann surface with an automorphism group H η,2 = H ρ for a symmetry ρ and the last group is isomorphic to H D q for a dihedral group of order 2q generated by conformal involution υ = δρ and symmetry ρ. Theorem 3.4 and preserves a central element z ∈ H of order 2, then there is a symmetric Riemann surface Y such that H ϕ,2 = H υ :

Corollary 3.5. If an automorphism ϕ ∈ Aut(H) satisfies the assumptions of
Proof. For any central element z ∈ Z(H) of order 2, such that ϕ(z) = z, there is a normal subgroup δ 2 z −1 of H ϕ,4 = H δ : δ 4 = 1 . The quotient group H ϕ/z = H ϕ,4 / δ 2 z −1 has a symmetric generating vector T s+p,η r,γ defined by (19). Thus, according to Theorem 3.2, there is a Riemann surface Y with an automorphism group G = H ϕ/z ρ , where ρ is a symmetry, such that ρδρ = δ −1 and ρhρ = h for h ∈ H. For υ = δρ, the subgroup H υ < G consists of all conformal automorphisms of G.

On Asymmetric Generating Vectors
A Riemann surface of genus g ≥ 2 is called asymmetric or pseudo-real, if it has anti-conformal automorphisms, but none of them is a symmetry. The next result comes from [2]; however, its proof is different from the original. The genus g of the surface X is given by (8).
Proof. If (Λ, θ, H) is a full action on an asymmetric Riemann surface, then the group Λ must have glide reflections but no reflection, and therefore, it has the signature   = [b 1 , . . . , b r , h 1 , . . . , h γ ] divisible by 4, and any composition of elements of vector T r,γ containing an odd number of h i neither is trivial nor has order 2. The signature (22) with m i = (b i ) for 1 ≤ i ≤ r appears on the lists of non-maximal signatures for (γ, r) = (1, 2), (3, 0), (2, 1), and it is maximal for all remaining pairs (γ, r). There exists a maximal NEC group Λ with any maximal signature. Let θ be an epimorphism induced by (9) from a maximal NEC group Λ with the signature (22) onto H. Then, Γ = kerθ is a surface Fuchsian group, and by the Hurwitz-Riemann formula, the genus of the Riemann surface X = H/Γ is equal to (8).
Since Λ is a maximal NEC group, it follows that H Λ/Γ is the full automorphism group of X, because otherwise Λ would be contained properly in another NEC group what contradicts the maximality of Λ. The surface X has anti-conformal automorphisms h i and it has no symmetry, because no composition of elements of an exceptional vector containing an odd number of h i has order 2. Therefore, X is asymmetric.
A signature (22) with (γ, r) ∈ {(1, 2), (3, 0), (2, 1)} is non-maximal and any NEC group Λ with such a signature is a subgroup of another NEC group Λ . There are five such cases on the lists of non-maximal signatures All listed above pairs of signatures have been considered in the proof of Theorem 3.2. Indeed, in cases (1) − (4), the signatures σ(Λ) and σ(Λ ) have the forms (12) and (13), respectively. In case (5), the group Λ contains a subgroup Λ with the signature (0; +; [2, m] (12) and (13), respectively. According to Theorem 3.2, if H has a symmetric vector T r,γ , then any action of this group with a nonmaximal signature σ(Λ) on a Riemann surface X extends to an action of the group H ρ with the signature σ(Λ ) for some symmetry ρ, where in case (5), we should take σ(Λ ) instead of σ(Λ ). It means that H is not the full automorphism group and the surface X is symmetric. Consequently, vector T r,γ cannot be symmetric for any action of H with a non-maximal signature on an asymmetric Riemann surface.