Abelian Actions on Pseudo-real Riemann Surfaces

Abstract

A compact Riemann surface is called pseudo-real if it admits orientation-reversing automorphisms but none of them has order two. In this paper, we find necessary and sufficient conditions for the existence of an action on a pseudo-real surface of genus $g\geqslant 2$ of an abelian group containing orientation-reversing automorphisms. Several consequences are obtained, such as the solution of the minimum genus problem for such abelian actions.

1 Introduction

A compact Riemann surface is called pseudo-real if it admits anti-conformal (orientation-reversing) automorphisms, but no anti-conformal automorphism of order 2,  or equivalently, if the surface is reflexible but not definable over the reals. An anti-conformal involution is usually called symmetry, so another term used for such surfaces is asymmetric. Their importance stems from the fact that in the moduli space of compact Riemann surfaces of given genus, pseudo-real surfaces represent the points that have real moduli but are not definable over the reals.

The first results on the existence of pseudo-real surfaces are due to Earle [12] and Shimura [25]. Seppälä [24] showed that the complex algebraic curves of real moduli are coverings of algebraic curves defined over real numbers. This was generalized by Bagiński and Gromadzki [2] who proved, using the language of Riemann surfaces, that a pseudo-real Riemann surface is a cyclic unbranched covering of degree a power of 2 of a compact Riemann surface having a purely imaginary real form. Other examples of pseudo-real surfaces have been found in the family of hyperelliptic ones by Zarrow [30] and Singerman [26]. The groups of automorphisms of these surfaces were determined in [10]. As to the non-hyperelliptic surfaces, it is worth mentioning that Hidalgo in [16] constructed a family of equations for pseudo-real non-hyperelliptic Riemann surfaces of genus 17.

There are no pseudo-real surfaces of genus 0 or 1, since in those cases every reflexible surface admits an anti-conformal automorphism of order 2. On the other hand, it was shown in [6] that there exists at least one pseudo-real surface of genus g for every integer $g > 1$. Groups of automorphisms of pseudo-real surfaces of low genus have been classified in different papers, as in [6, 7] and [1], with a correction to this last in [11]. In [8], the authors study the full groups of automorphisms of pseudo-real Riemann surfaces of genus g that are cyclic p-gonal. In [28], Tyszkowska studies groups of automorphisms of both symmetric and pseudo-real surfaces simultaneously. This paper can be seen as a continuation of [27] and her joint papers with Kozłoswka-Walania [17,18,19], where asymmetric p-hyperelliptic and (q, n)-gonal surfaces, with cyclic automorphism group, are studied.

Every compact Riemann surface S of genus $g\geqslant 2$ can be described as the orbit space ${{\mathcal {H}}}/\Gamma$ of the hyperbolic plane ${{\mathcal {H}}}$ under the action of a torsion-free Fuchsian group $\Gamma .$ Every group G of automorphisms (including orientation-reversing ones) of ${{\mathcal {H}}}/\Gamma$ is isomorphic to the quotient $\Lambda /\Gamma$ where $\Lambda$ is an NEC group. Equivalently, there exist an NEC group $\Lambda$ and an epimorphism $\theta \!:\Lambda \rightarrow G$ whose kernel is $\Gamma .$ In this case we say (for short) that $\theta$ is a smooth epimorphism. A natural question arises: can a given finite group G act as a group of automorphisms of ${{\mathcal {H}}}/\Gamma$? The answer to this question is equivalent to finding the necessary and sufficient conditions on a presentation of the NEC group $\Lambda$ for which there exists a smooth epimorphism from $\Lambda$ onto G.

The case when G is cyclic and all its elements preserve the orientation of S was completely solved by Harvey [15]. As an easy application of this result, Harvey found the maximum order of a cyclic group of automorphisms of a Riemann surface of genus g,  a result already shown by Wiman [29]. This is called the maximum order problem for cyclic groups. Harvey also solved the so-called minimum genus problem for cyclic groups, that is, he computed the minimum genus of a Riemann surface on which a given cyclic group acts. In the last decades, much research has been conducted to extend Harvey’s theorem to other classes of finite groups acting on Riemann surfaces. In particular, the maximum order and the minimum genus problems for different families of finite groups have received a good deal of attention.

However, there are few families of groups for which the above mentioned necessary and sufficient conditions on the presentation of an NEC group have been found. Orientation preserving actions of abelian groups is one of them. The original solution was provided by Breuer in [3], and a refinement given by Rodríguez in [22]. Again, an application of these conditions provides a solution to the maximum order and minimum genus problems for abelian groups. The latter was already solved by Maclachlan in [20], and a more concise proof was given by Rodríguez in [22]. The minimum genus problem for abelian groups acting on other types of surfaces was solved by Gromadzki in [14] for non-orientable surfaces, and by McCullough in [21] for bordered surfaces.

Cyclic groups acting on Riemann surfaces and containing elements which reverse the orientation of the surface were considered by Etayo in [13]. He found the necessary and sufficient conditions on the signature of a proper NEC group for which there exists a smooth epimorphism onto a cyclic group. An application of his results led to finding the largest order of a cyclic group acting on a pseudo-real Riemann surface. Also, the minimum genus problem for cyclic actions on pseudo-real surfaces was considered by Bagiński and Gromadzki in [2], with the right solution given by Conder and Lo in [11]. Non-cyclic abelian actions on pseudo-real surfaces were considered in [5, Section 4], with the focus on actions of the largest possible order.

Following [2], we say that a group A acts essentially on a pseudo-real surface if A contains elements which reverse the orientation of the surface. In this paper, we consider essential abelian actions on pseudo-real surfaces. Given a proper NEC group $\Lambda$ and a finite abelian group A we find necessary and sufficient conditions for the existence of a smooth epimorphism $\theta :\Lambda \rightarrow A$ with symmetry-free image. They are given in terms of the signature of $\Lambda$ and the invariant factors of A. If these conditions are satisfied, then $\mathcal {H}/\ker \theta$ is a pseudo-real Riemann surface on which the abelian group A acts essentially with the prescribed signature of $\Lambda$. This is the main result of this paper, see Theorems 2.1 and 2.2. Several consequences are obtained. The minimum genus problem for essential abelian actions is solved in Sect. 3. When the abelian group has just two invariant factors we find an explicit expression for the minimum genus, Theorem 3.3. In the general case, the solution is given as a minimum to be found among several quantities, Theorem 3.4. In Sect. 4 we fix the order N and let A vary among the abelian groups of order N. We find the minimum genus of a pseudo-real surface on which an abelian group of order N acts essentially. Further, the unique abelian group among those of order N attaining the minimum genus is also determined, Theorem 4.1. Finally, in Sect. 5 we obtain a short proof of three of the bounds computed in [5], Theorem 5.1.

2 Preliminaries

We begin this section with some background about Riemann surfaces and their automorphism groups which can be found in [9]. We then provide some further information about group actions on pseudo-real surfaces, with special emphasis on abelian groups.

2.1 Riemann Surfaces, NEC Groups and Their Signatures

Any compact Riemann surface S of genus $g>1$ can be represented as the orbit space ${{\mathcal {H}}} /\Gamma$ of the upper half-plane ${{\mathcal {H}}}$ under the action of some surface Fuchsian group $\Gamma$, (that is, a torsion-free discrete cocompact subgroup of $\textrm{Aut}^+({{\mathcal {H}}})=\textrm{PSL}(2,\mathbb {R})$, the group of all orientation-preserving isometries of ${{\mathcal {H}}}$). A non-Euclidean crystallographic (NEC, for short) group $\Lambda$ is a discrete cocompact subgroup of $\textrm{Aut}({{\mathcal {H}}})=\textrm{PGL}(2,\mathbb {R})$, the group of all isometries of ${{\mathcal {H}}}$, including those which reverse orientation. A finite group G acts as a group of automorphisms of the surface $S={{\mathcal {H}}} /\Gamma$ if and only if G is isomorphic to the quotient $\Lambda /\Gamma$ for some NEC group $\Lambda$ containing $\Gamma$ as a normal subgroup of index |G|. The canonical epimorphism $\theta \!: \Lambda \rightarrow G$ ($\cong \Lambda /\Gamma$), whose kernel is a surface Fuchsian group, is said to be smooth. The full automorphism group $\textrm{Aut}(S)$ of S is isomorphic to the quotient $\Lambda /\Gamma$ where $\Lambda$ is the normalizer in $\textrm{Aut}({{\mathcal {H}}})$ of the surface group $\Gamma .$

The structure of an NEC group $\Lambda$ is determined by its signature. For the purposes of this paper, it suffices to consider signatures of the form

$$\begin{aligned} (\gamma ;\,-;\,[m_1,\ldots ,m_r];\,\{-\}). \end{aligned}$$

(1.1)

NEC groups with this signature are generated by $\gamma$ glide reflections $d_1,\ldots ,d_\gamma$ and r elliptic elements $x_1,\ldots ,x_r$ which satisfy the defining relations

$$\begin{aligned} x_{i}^{m_{i}} = 1 \quad \text{ for } \ 1 \leqslant i \leqslant r, \ \qquad x_{1}\cdots x_{r}\,d_{1}^{\,2}\cdots d_{\gamma }^{\,2}=1. \end{aligned}$$

We will refer to the last one as the long relation. The integers $m_1,\ldots ,m_r$ are called proper periods. The hyperbolic area of a fundamental region for an NEC group $\Lambda$ with this signature is $2\pi \mu (\Lambda )$ where

$$\begin{aligned} \mu (\Lambda )=\gamma -2 + \sum _{i=1}^{r} \left( 1-\frac{1}{m_i} \right) . \end{aligned}$$

We call $\mu (\Lambda )$ the reduced area of $\Lambda .$ If $\Gamma$ is a surface Fuchsian group and the surface ${{\mathcal {H}}}/\Gamma$ has genus g then its reduced area is $\mu (\Gamma ) = 2g-2.$ If $\Delta$ is any subgroup of finite index in $\Lambda$, then $\Delta$ is also an NEC group, and the hyperbolic areas of fundamental regions for $\Delta$ and $\Lambda$ satisfy the Riemann–Hurwitz formula

$$\begin{aligned} \mu (\Delta ) = |\Lambda :\Delta |\,\mu (\Lambda ). \end{aligned}$$

(1.2)

2.2 Group Actions on Pseudo-real Surfaces

Now suppose the surface $S\!=\!{{\mathcal {H}}} /\Gamma$ is pseudo-real. Then, the full group $\textrm{Aut}(S)\!=\!\Lambda /\Gamma$ contains orientation-reversing automorphisms but none of them has order two. Therefore, $\Lambda$ contains no hyperbolic reflection but it contains some glide reflection. This means that the signature of $\Lambda$ is of the form (1.1). Orientation-reversing automorphisms of order two are usually called symmetries, so we will say that $\textrm{Aut}(S)$ is symmetry-free. Observe that, as S admits no symmetry, it also admits no orientation-reversing automorphism of order 2n with n odd. Otherwise, the n-th power of such an automorphism would be a symmetry. So, the order of every element of $\textrm{Aut}(S) \smallsetminus \textrm{Aut}^+(S)$ is divisible by 4. In particular, every involution in $\textrm{Aut}(S)$ preserves orientation, and $|\textrm{Aut}(S)|$ is divisible by 4.

Now, let G be a group that acts on a surface S containing some automorphism that reverses orientation. Following [2], we call such an action essential. The group G may contain no symmetry, but this does not guarantee that the surface S is pseudo-real. In fact, it may happen that $\textrm{Aut}(S)$ strictly contains G with a symmetry in $\textrm{Aut}(S)\smallsetminus G.$ This is a key point in the study of pseudo-real surfaces. A detailed discussion of this topic can be found in [5, Section 2], where the following proposition, concerning abelian groups, is proved.

Proposition 1.1

Let $S=\mathcal {H}/\Gamma$ be a compact Riemann surface, with $\Gamma$ a surface Fuchsian group, and let A be an abelian group of automorphisms of S. If $A=\Lambda /\Gamma$ and the signature of $\Lambda$ is $(1;\,-;\,[m_1,m_2];\,\{-\})$, $(2;\,-;\,[m_1];\,\{-\})$ or $(3;\,-;\,[-];\,\{-\})$, then S is not pseudo-real.

2.3 Further Results for the Abelian Case

Any epimorphism $\theta :\Lambda \rightarrow A$ from an NEC group $\Lambda$ onto an abelian group A factors through its abelianization $\Lambda _{ab}$, that is, $\theta$ induces an epimorphism ${\overline{\theta }}:\Lambda _{ab}\rightarrow A$ such that ${\overline{\theta }}\pi =\theta$ where $\pi :\Lambda \rightarrow \Lambda _{ab}$ is the canonical projection. Conversely, for each epimorphism ${\overline{\theta }}:\Lambda _{ab}\rightarrow A$ the composition $\theta ={\overline{\theta }}\pi :\Lambda \rightarrow A$ is an epimorphism. Therefore, we first focus our attention on the existence of epimorphisms $\Lambda _{ab}\rightarrow A$ between abelian groups. Breuer [3, Lemmas A.1 and A.2] established conditions for the existence of such epimorphisms. For the purposes of this paper we will consider the following particular case. We use multiplicative notation, and denote the cyclic group of order n by $C_n$ and the infinite cyclic group by $C$.

Lemma 1.2

Let q be a prime number and $R, \alpha _1, \ldots ,$ $\alpha _t, \beta _1,\ldots ,$ $\beta _r$ be non-negative integers with $\alpha _i\leqslant \alpha _{i+1}$ and $\beta _i\leqslant \beta _{i+1}$. There is an epimorphism

$$\begin{aligned} C^R\times C_{q^{\beta _1}}\times \cdots \times C_{q^{\beta _r}} \quad \rightarrow \quad C_{q^{\alpha _1}}\times \cdots \times C_{q^{\alpha _t}}, \end{aligned}$$

if and only if the following condition holds: if $R<t$, then for each $i\leqslant t-R$, the elementary divisor $q^{\alpha _i}$ divides at least $t-R+1-i$ elementary divisors $q^{\beta _j}$.

Clearly, the existence of an epimorphism $\Lambda _{ab}\rightarrow A$ from the abelianization $\Lambda _{ab}$ of an NEC group $\Lambda$ onto an abelian finite group A is equivalent to the fulfillment of the conditions in Lemma 1.2 for each prime q dividing the order of A. Our goal now is to describe the structure of $\Lambda _{ab}$ in terms of the signature of $\Lambda$.

We write $m_i=p_1^{\mu _{i1}}\cdots p_s^{\mu _{is}}$ with prime numbers $p_1<\cdots <p_s$ and non-negative integers $\mu _{ij}$ such that $\mu _{1j}+\cdots +\mu _{rj}>0$. For each prime $p_j$, we rearrange the integers $\mu _{1j}, \ldots , \mu _{rj}$ to obtain increasing sequences of integers $\widehat{\mu }_{1j}\leqslant \cdots \leqslant \widehat{\mu }_{rj}$ and define $\widehat{m}_i=p_1^{\widehat{\mu }_{i1}}\cdots p_s^{\widehat{\mu }_{is}}$. Then $\widehat{m}_i\mid \widehat{m}_{i+1}$. Some $\widehat{m}_i$ may be equal to 1. We can obtain the abelianization $\Lambda _{ab}$ by computing the Smith normal form of the relation matrix of the abelianized canonical presentation of $\Lambda$.

Lemma 1.3

The abelianization of an NEC group $\Lambda$ with signature $(\gamma ;\,-;$ $[m_1,\ldots ,m_r];\,\{-\})$ is $\Lambda _{ab} \,\approx \, C^{\gamma -1} \times C_2$ if $r=0$ and $C^{\gamma -1} \times C_{\widehat{m}_{1}} \times \cdots \times C_{\widehat{m}_{r-1}} \times C_{2\widehat{m}_r}$ otherwise.

Proof

See [23, Lemma 3.4] for instance. $\square$

Remark 1.4

Observe that each odd elementary divisor of $\Lambda _{ab}$ is of the form $p^{\widehat{\mu }_{ij}}$ and so it is a prime power divisor of a proper period of $\Lambda$. The same holds true for the even elementary divisors of $\Lambda _{ab}$ except for the largest one. In fact, if $2^{\widehat{\mu }_r}$ is the largest 2-power divisor of any proper period, then the largest even elementary divisor of $\Lambda _{ab}$ is $2^{{\widehat{\mu }_r}+1}$.

We will also need the following inequality, a proof of which can be found in [22]:

$$\begin{aligned} \sum _{i=1}^{r}\left( 1-\frac{1}{\widehat{m}_i}\right) \;\leqslant \; \sum _{i=1}^{r}\left( 1-\frac{1}{m_i}\right) . \end{aligned}$$

(1.3)

A finite abelian group A has a unique invariant factor decomposition $A\approx C_{v_1}\times \cdots \times C_{v_t}$ with invariant factors $v_1,v_2,\ldots ,v_t$ such that $v_i | v_{i+1}$ for $i=1,\ldots , t-1$. Also, A has a unique primary decomposition $A\approx A_{q_1}\times \cdots \times A_{q_s}$ with primes $q_1<\cdots < q_s$, where $A_q = \{x\in A\mid x^{q^n}=1 \hbox { for some n > 0}\}$ is the q-primary component of A (or the q-Sylow subgroup of A) for each prime q dividing the order of A. Each $A_q$ is isomorphic to $C_{q^{\alpha _1(q)}}\times \cdots \times C_{q^{\alpha _t(q)}}$ with $0\leqslant \alpha _1(q)\leqslant \alpha _2(q)\cdots \leqslant \alpha _t(q),$ where the integers $q_j^{\alpha _i(q_j)}$ are the elementary divisors of A.

3 Main Results

Let $\Lambda$ be an NEC group with signature $(\gamma ;\,-;\,[m_1,\ldots ,$ $m_r];\,$ $\{-\})$ where $\gamma +r>3$, and $A\approx C_{v_1}\times \cdots \times C_{v_t}$ an abelian group where $v_i\mid v_{i+1}$ and $4\mid v_t$. In this section, we find the necessary and sufficient conditions on the integers $\gamma ,m_1,\ldots ,m_r,v_1,\ldots ,v_t$ for which there exists a smooth epimorphism $\theta :\Lambda \rightarrow A$ with symmetry-free image. If these conditions are satisfied, then $\mathcal {H}/\ker \theta$ is a pseudo-real Riemann surface on which the abelian group A acts essentially with signature $(\gamma ;\,-\,;\,[m_1,\ldots ,$ $m_r];\,$ $\{-\}).$ Observe that the condition $\gamma +r>3$ is imposed by Proposition 1.1.

We first consider the case when just one invariant factor of A is divisible by four, Theorem 2.1. The general case will be analyzed in Theorem 2.2. We say that an even integer 2k is singly even if k is odd.

Theorem 2.1

Let $\Lambda$ be an NEC group with signature $(\gamma ;\,-;\,[m_1,\ldots ,$ $m_r];\,$ $\{-\})$ with $\gamma +r>3$. Let $A\approx C_{v_1}\times \cdots \times C_{v_t}$ be an abelian group, where $v_i\mid v_{i+1}$, $4\mid v_t$ and $4\not \mid v_i$ if $i<t$. Assume that A has n cyclic factors of singly even order and let $C_{2^{\alpha _1}}\times \cdots \times C_{2^{\alpha _{t}}}$ be the Sylow 2-subgroup of A, $\alpha _i\leqslant \alpha _{i+1}$. Then, there exists a smooth epimorphism $\Lambda \rightarrow A$ with symmetry-free image if and only if the following conditions hold:

• (i) $v_t/m_i$ is an even integer for all i.

• (ii) If $\gamma -1<t$ then $2^{\alpha _{t-\gamma +1}}$ divides $2m_i$ for some i, every odd elementary divisor of $C_{v_{t-\gamma +1}}$ divides at least one proper period, and for $i=1,\ldots ,t-\gamma$ every elementary divisor of $C_{v_i}$ divides at least $t-\gamma +2-i$ proper periods.

• (iii) If $\gamma \!+\!\frac{1}{2}\sum _i v_t/m_i$ is odd then $8\not \mid v_t$, $n\!>\!0$ and either $\gamma$ is odd or there is an odd number $\geqslant 3$ of even proper periods.

Proof

Assume first that there exists a smooth epimorphism $\theta :\Lambda \rightarrow A$ with symmetry-free image.

(i) As $\ker \theta$ is torsion-free, each $\theta (x_i)$ has order $m_i.$ This order has to divide the exponent $v_t$ of A. So $v_t/m_i$ is an integer, and we now show that it is even for all i. Let us write $C_{v_t}=C_{2^{\alpha _t}}\times C_{v'_t}$ with $v'_t$ odd, and let $u_1,\ldots ,u_t$ be generators of $C_{v_1},\ldots ,C_{v_{t-1}},C_{v'_t}$, respectively, which are orientation-preserving since $v_1,\ldots ,v_{t-1},v'_t$ are either odd or singly even. Suppose there exists some proper period $m_{i_0}$ divisible by $2^{\alpha _t}.$ Then, A would be generated by $u_1,\ldots ,u_t,\theta (x_{i_0}),$ and it would not contain orientation-reversing elements, a contradiction. Hence $v_t/m_i$ is even for all i.

(ii) Lemmas 1.2 and 1.3 applied to the epimorphism $\Lambda _{ab}\rightarrow A$ yields that if $\gamma -1<t$ then for each $i\leqslant t-\gamma +1$ every elementary divisor of $C_{v_i}$ divides at least $t-\gamma +2-i$ elementary divisors of $\Lambda _{ab}.$ Since each odd elementary divisor of $\Lambda _{ab}$ is a prime power divisor of a proper period of $\Lambda$, see Remark 1.4, it follows that condition (ii) is true for odd elementary divisors of $C_{v_i}$. The situation with the even ones is the same except that the largest even elementary divisor of $\Lambda _{ab}$ is not a divisor of some $m_i$ but a divisor of $2m_i$ for some i,  see Remark 1.4. The application of Lemma 1.2 to $i=t-\gamma +1$ yields that the elementary divisor $2^{\alpha _{t-\gamma +1}}$ of $C_{v_{t-\gamma +1}}$ divides at least one elementary divisor of $\Lambda _{ab}.$ Hence, $2^{\alpha _{t-\gamma +1}}$ divides $2m_i$ for some i.

(iii) Let $\omega _t$ be a generator of $C_{2^{\alpha _t}}$. To generate $C_{2^{\alpha _t}}$, the exponent of $\omega _t$ in $\theta (d_j)$ must be odd for some j since $v_t/m_i$ is even for all i. We claim that, in fact, the exponent of $\omega _t$ in $\theta (d_j)$ is odd for all j. This is clear if $\gamma =1.$ So assume $\gamma >1$ and that the exponent of $\omega _t$ in, say $\theta (d_1),$ is an odd integer a. Suppose, to get a contradiction, that the exponent of $\omega _t$ in, say $\theta (d_2),$ is an even integer b. Then, there exists an even integer c such that $ac+b\equiv 2^{\alpha _t-1} \pmod {2^{\alpha _t}}$. The orientation-reversing element $\theta (d_1^cd_2)$ satisfies that its square has odd order. This is impossible in a pseudo-real surface. This shows the claim.

Let $e_{it}$ be the exponent of $\omega _t$ in $\theta (x_i).$ It follows from the above claim and the long relation that $\frac{1}{2}\sum _ie_{it}+\gamma$ is even. We now compare the parities of $\frac{1}{2}e_{it}$ and $\frac{1}{2}(v_t/m_i).$ Let $2^{\mu _i}$ be the largest power of 2 dividing $m_i,$ so that $v_t/m_i=2^{\alpha _t-\mu _i}v'_i$ with $v'_i$ odd. Since the $2^{\mu _i}$-th power of $\omega _t^{e_{it}}$ vanishes, we get $e_{it}=2^{\alpha _t-\mu _i}e'_{it}$ for some $e'_{it}.$ Hence, if $v_t/m_i$ is a multiple of 4 then also $e_{it}$ is a multiple of 4. We claim that the converse also holds if $8\mid 2^{\alpha _t}$ or $n=0$. In fact, assume $4\mid e_{it}$ and suppose, to get a contradiction, that $v_t/m_i$ is singly even. Then $m_i$ is even. If $m_i$ is singly even then $\alpha _t=2$ (so $8\not \mid 2^{\alpha _t}$) and $e_{it}\equiv 0\pmod {2^{\alpha _t}}.$ Thus the non-zero components of $\theta (x_i)$ in the Sylow 2-subgroup $(C_2)^n\times C_{2^{\alpha _t}}$ lie in $(C_2)^n.$ But $n=0$ by hypothesis (since $8\not \mid 2^{\alpha _t}$). This gives a contradiction. If $m_i$ is divisible by four then the $(m_i/2)$-th power of $\theta (x_i)$ has null components in $(C_2)^n$ (if any) but not in $C_{2^{\alpha _t}}.$ It follows that $e_{it}=2^{\alpha _t-\mu _i}e'_{it}$ with $e'_{it}$ odd. So if $4\mid e_{it}$ then $4\mid (v_t/m_i).$ This shows our claim. Consequently, if $8\mid 2^{\alpha _t}$ or $n=0$ then $\frac{1}{2}v_t/m_i$ and $\frac{1}{2}e_{it}$ have the same parity and thus $\frac{1}{2}\sum _iv_t/m_i+\gamma$ is even. This shows the first part in (iii).

Finally, if $\gamma +\frac{1}{2}\sum v_t/m_i$ is odd (so $\alpha _t=2$ and $n>0$) and $\gamma$ is even then the component of $\theta (x_1\cdots x_r)= \theta (d_1^2\cdots d_\gamma ^2)^{-1}$ in $(C_2)^n\times C_{2^{\alpha _t}}$ vanishes. This prevents the number of elements $\theta (x_i)$ of (singly) even order from being equal to one. In addition, since the sum $\frac{1}{2}\sum v_t/m_i$ is odd, there is an odd number of odd terms $v_t/(2m_i)$. These correspond to even proper periods. This shows condition (iii).

We prove the sufficiency of the conditions by defining epimorphisms $\theta _{q}:\Lambda \rightarrow A_{q}$ for each prime q in the set $\{q_1,\ldots ,q_\lambda \}$ of prime numbers dividing the order of A, and a smooth epimorphism $\theta :\Lambda \rightarrow A$ as the direct product epimorphism

$$\begin{aligned} \theta : \Lambda \rightarrow A: g \,\mapsto \theta (g) = (\theta _{q_{_1}}(g), \ldots , \theta _{q_{_\lambda }}(g)). \end{aligned}$$

For readability, we write $\mu _i=\mu _i(q)$ in the definition of each homomorphism $\theta _q$. Recall that $q^{\mu _i}$ divides $m_i$ but $q^{\mu _{i}+1}$ does not. Also, we assume that $\mu _i\leqslant \mu _{i+1}$; otherwise, there is a permutation $\tau _q$ of $\{1,\ldots ,r\}$ such that $\mu _{\tau _q(i)}\leqslant \mu _{\tau _q(i+1)}$ and we replace $x_i$ by $x_{\tau _q(i)}$ and $\mu _i$ by $\mu _{\tau _q(i)}$ in the definition of $\theta _q(x_i)$ below. The order of $\theta _q(x_i)$ has to be $q^{\mu _i},$ so that the order of $\theta (x_i)$ is $m_i$. In addition, each $\theta _q$ has to transform the long relation into the identity. We write $\omega _i=\omega _i(q)$ for a generator of $C_{q^{\alpha _i}},$ admitting the possibility $\omega _i=1$ when $\alpha _i=0$ (to lighten notation).

We first define $\theta _2$. Let $\Sigma =\textstyle \sum _i 2^{\alpha _t-\mu _i}$, which is even by (i), and satisfies that $\frac{1}{2}\Sigma$ has the same parity as $\frac{1}{2}\sum _iv_t/m_i$. Note that the following definition for $\theta _2$ prevents $\theta (\Lambda )$ from containing symmetries since the exponent of $\omega _t$ is even in $\theta _2(x_i)$, $i=1,\ldots ,r$, and odd in $\theta _2(d_i)$, $i=1,\ldots ,\gamma$. For the same reason, $\ker {\theta }$ has no orientation-reversing elements.

If $\gamma +\frac{1}{2}\sum _iv_t/m_i$ is even and $\gamma <t$ then we define

$$\begin{aligned} \begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-t+\gamma -1, \\&\theta _2(x_i) = \omega _{i-r+t-\gamma +1}\omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=r-t+\gamma ,\ldots ,r-1, \\&\theta _2(x_r) = \omega _{1}^{-1}\cdots \omega _{t-\gamma }^{-1}\omega _{t}^{2^{\alpha _{t}-\mu _r}}, \\&\theta _2(d_1) = \omega _{t-\gamma +1}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{1-\gamma -\Sigma /2},\\&\theta _2(d_i) = \omega _{t-\gamma +i}\omega _{t}, \quad i=2,\cdots ,\gamma -1, \\&\theta _2(d_\gamma ) = \omega _{t}\, \quad \text{ if } \gamma >1. \end{aligned} \end{aligned}$$

(2.1)

It is easy to check that $\theta _2$ is onto and that it preserves the long relation. The order of $\omega _t^{2^{\alpha _t-\mu _i}}$ is $2^{\mu _i}$. So, $\theta _2(x_i)$ has order $2^{\mu _i}$ for $i=1,\ldots , r-t+\gamma -1.$ If $\mu _i\geqslant 1$ then the same happens to $\theta _2(x_i)$ for $i\geqslant r-t+\gamma$ because $\omega _i^2=1$ for $i<t.$ It remains to show that if $\mu _i=0$ then $\theta _2(x_i)=1$ for $i\geqslant r-t+\gamma .$ If $\mu _r=0$ (so $m_r$ is odd) then $\theta _2(x_r)= \omega _{1}^{-1}\cdots \omega _{t-\gamma }^{-1}$. Suppose this element is not trivial. Then 2 is an elementary divisor of $C_{2^{\alpha _i}}$ for some $i\in \{1,\ldots ,t-\gamma \}$. Since $\gamma -1<t,$ we may apply condition (ii) to get that 2 divides at least $t-\gamma +2-i\geqslant 2$ proper periods. This contradicts that $m_r$ is odd. The proof that if $\mu _i=0$ for $i\ne r$ then $\theta _2(x_i)=1$ is similar.

If $\gamma +\frac{1}{2}\sum _iv_t/m_i$ is even and $\gamma \geqslant t$ then we define

$$\begin{aligned} \begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}} \quad \text{ for } \text{ all } \ i, \\ {}&\theta _2(d_1) = \omega _1^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{1-\gamma -\Sigma /2},\\ {}&\theta _2(d_i) = \omega _i\omega _t \quad \text{ if } \ 1<i<t, \\ {}&\theta _2(d_i) = \omega _{t} \quad \text{ for } \ i\geqslant t. \end{aligned} \end{aligned}$$

(2.2)

It is straightforward to check that $\theta _2$ is onto, preserves the long relation and that $\theta _2(x_i)$ has order $2^{\mu _i}$ for all i.

If $\gamma +\frac{1}{2}\sum _iv_t/m_i$ is odd, then $8\not \mid v_t$ (so $\alpha _t=2$) and $n>0$ by (iii). Observe that each proper period is odd or singly even by (i). In addition, if $\gamma$ is odd then $\frac{1}{2}\sum _iv_t/m_i$ is even and so there is an even number of even proper periods. Moreover, there are some even proper periods (at least two) since otherwise $\theta (d_1\cdots d_{\gamma })^2=\theta (x_1\cdots x_r)^{-1}$ would have odd order, which is impossible in a pseudo-real surface because $\theta (d_1\cdots d_{\gamma })$ reverses the orientation. On the other hand, if $\gamma$ is even, then $\frac{1}{2}\sum _iv_t/m_i$ is odd and there is an odd number of even proper periods—at least three by (iii). In particular, $r\geqslant 2$ for any value of $\gamma .$ If $\gamma <t$ then we define $\theta _2$ as in (2.1) but we redefine the images of $x_{r-1},x_r$ and $d_1$ in the following way:

$$\begin{aligned}&\theta _2(x_{r-1}) = \omega _{t-\gamma }\omega _t^{2}, \quad \theta _2(x_r) = \omega _{1}^{-1}\cdots \omega _{t-\gamma }^{-1} \quad \text{ if } \ \gamma \leqslant n, \\ {}&\theta _2(x_{r-1}) = \omega _{t-1}\omega _t^{2}, \quad \theta _2(x_r) = \omega _{t-1}^{-1} \quad \text{ if } \ \gamma >n,\\ {}&\theta _2(d_1) = \omega _{t-\gamma +1}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{2-\gamma -\Sigma /2}. \end{aligned}$$

By definition of n, the elements $\omega _1,\ldots ,\omega _{t-n-1}$ are trivial whilst $\omega _{t-n},\ldots ,\omega _{t-1}$ have order two.

If $\gamma \geqslant t$, then we define $\theta _2$ as in (2.2) but we redefine the images of $x_{r-1},x_r$ and $d_1$ as

$$\begin{aligned} \theta _2(x_{r-1}) = \omega _{t-1}\omega _t^{2}, \qquad \theta _2(x_r) = \omega _{t-1}^{-1},\qquad \theta _2(d_1) = \omega _{t-\gamma +1}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{2-\gamma -\Sigma /2}. \end{aligned}$$

It is easy to see that in both cases $\theta _2$ is onto, preserves the long relation and $\theta _2(x_i)$ has order $2^{\mu _i}.$

Now, let $q\ne 2$ be a prime number dividing the order of A, $C_{q^{\alpha _1}}\times \cdots \times C_{q^{\alpha _{t}}}$ the Sylow q-subgroup of A, $\alpha _i\leqslant \alpha _{i+1}$, $\omega _i$ a generator of $C_{q^{\alpha _i}}$ and $\Sigma \equiv \textstyle \sum _i q^{\alpha _t-\mu _i}\pmod {q^{\alpha _t}}$, $0\leqslant \Sigma <q^{\alpha _t}$. We also write $s=-\Sigma /2-1$ if $\Sigma$ is even and $s=(q^{\alpha _t}-\Sigma )/2-1$ otherwise, so that $\omega _t^{2\,s+2}=\omega _t^{-\Sigma }.$ If $\gamma \leqslant t$ then we define:

$$\begin{aligned}&\theta _q(x_i) = \omega _{t}^{q^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-t+\gamma -1, \\ {}&\theta _q(x_i) = \omega _{i-r+t-\gamma +1}\omega _{t}^{q^{\alpha _{t}-\mu _i}}, \quad i=r-t+\gamma ,\ldots ,r-1, \quad \text{ if } \gamma<t,\\ {}&\theta _q(x_r) = \omega _{1}^{-1}\cdots \omega _{t-\gamma }^{-1}\omega _{t-\gamma +1}^{2}\omega _{t}^{q^{\alpha _{t}-\mu _r}}, \quad \text{ if }\ \gamma <t,\\ {}&\theta _q(x_r) = \omega _{1}^{2}\omega _{t}^{q^{\alpha _{t}-\mu _r}}, \quad \text{ if } \ \gamma =t,\\ {}&\theta _q(d_1) = \omega _{t-\gamma +1}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{s},\\ {}&\theta _q(d_i) = \omega _{t-\gamma +i}, \quad i=2,\cdots ,\gamma . \end{aligned}$$

If $\gamma >t$, then we define:

$$\begin{aligned}&\theta _q(x_i) = \omega _{t}^{q^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r, \\&\theta _q(d_1) = \omega _{1}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{s},\\&\theta _q(d_i) = \omega _{i-1}, \quad i=2,\cdots ,t+1,\\&\theta _q(d_i) = 1, \quad i>t+1. \end{aligned}$$

For either value of $\gamma$, the proof that $\theta _q$ is onto, preserves the long relation and that $\theta _q(x_i)$ has order $q^{\mu _i}$, can be done in a similar way as in the case $q=2.$ $\square$

We now consider the case when more than one invariant factor of A is divisible by four. Some parts of the proof of Theorem 2.2 are similar to the proof of Theorem 2.1 and will be omitted.

Theorem 2.2

Let $\Lambda$ be an NEC group with signature $(\gamma ;\,-;\,[m_1,\ldots ,$ $m_r];\,$ $\{-\})$ with $\gamma +r>3$, and $A\approx C_{v_1}\times \cdots \times C_{v_t}$ an abelian group, where $v_i\mid v_{i+1}$ and $4\mid v_t$. Assume that A has n cyclic factors of singly even order, $m>1$ of order multiple of 4 and let $C_{2^{\alpha _1}}\times \cdots \times C_{2^{\alpha _{t}}}$ be the Sylow 2-subgroup of A, $\alpha _i\leqslant \alpha _{i+1}$. Then, there exists a smooth epimorphism $\Lambda \rightarrow A$ with symmetry-free image if and only if the following conditions hold:

• (i) $m_i$ divides $v_t$ for all i; if $r=1$, then $v_t/m_1$ is even.

• (ii) If $\gamma -1<t$, then $2^{\alpha _{t-\gamma +1}}$ divides $2m_i$ for some i, odd elementary divisors of $C_{v_{t-\gamma +1}}$ divide some proper period and every elementary divisor of $C_{v_i}$ divides at least $t-\gamma +2-i$ proper periods for $i=1,\ldots ,t-\gamma$.

• (iii) If $\gamma$ is odd, then $2^{\alpha _{t-m+1}}$ divides $2m_i$ for some i.

• (iv) $2^{\alpha _{t}}$ divides either no proper period or at least two proper periods; if $2^{\alpha _{t}}$ divides an odd number of proper periods, then $\alpha _{t-1}=\alpha _{t}$ and $m>2$.

Proof

Assume first that $\theta :\Lambda \rightarrow A$ is a smooth epimorphism with symmetry-free image.

(i) The order of $\theta (x_i)$ is $m_i$ and so it divides the exponent $v_t$ of A. If $r=1$ and $v_t/m_1$ is odd, then the long relation is not preserved.

(ii) These conditions follow from Lemma 1.2 applied to the epimorphism $\Lambda _{ab}\rightarrow A$.

(iii) Suppose that $2^{\alpha _{t-m+1}}\not \mid 2m_i$ for all i. Let $\omega _i$ be a generator of $C_{v_i}$. If $v_i$ is singly even (i.e., $\alpha _i=1$, $C_{2^{\alpha _i}}=C_2$), then the component of $\theta (x_1\cdots x_r)$ in $C_{2^{\alpha _i}}$ is trivial by the long relation since the component of $\theta (d_1^2\cdots d_\gamma ^2)=\theta (d_1\cdots d_\gamma )^2$ in $C_{2^{\alpha _i}}$ is trivial, and thus the exponent of $\omega _i$ in $\theta (x_1\cdots x_r)$ is even. Since $2^{\alpha _{t-m+1}}\not \mid 2m_i$ for all i, the exponent of $\omega _j$ in $\theta (x_i)$ is multiple of 4 for $j\geqslant t-m+1$. Therefore, the exponents of $\omega _1,\ldots ,\omega _t$ in $\theta (x_1\cdots x_r)$ are $a_1, \ldots , a_{t-m-n}$, $2a_{t-m-n+1}, \ldots , 2a_{t-m}$, $4a_{t-m+1}, \ldots , 4a_t$, respectively, for some integers $a_1,\ldots ,a_t$. Let $h\in \Lambda$ be such that $\theta (h)= (\omega _{t-m+1}^{a_{t-m+1}}\cdots \omega _t^{a_t})^2$, hence h is orientation-preserving. As $\gamma$ is odd (by hypothesis) $(hd_1\cdots d_\gamma )^{v_t/2^{\alpha _t}}$ is orientation-reversing and, by the long relation, $\theta ((hd_1\cdots d_\gamma )^{v_t/2^{\alpha _t}})\in (C_2)^n$, so that either $(hd_1\cdots d_\gamma )^{v_t/2^{\alpha _t}}\in \ker \theta$ or $\theta ((hd_1\cdots d_\gamma )^{v_t/2^{\alpha _t}})$ is an orientation-reversing involution. Therefore, $\mathcal {H}/\ker \theta$ would not be a pseudo-real Riemann surface, a contradiction. Hence $2^{\alpha _{t-m+1}}$ divides $2m_i$ for some i.

(iv) The long relation is not preserved if either $2^{\alpha _t}$ divides only one proper period or $\alpha _{t-1}<\alpha _t$ and $2^{\alpha _t}$ divides an odd number of proper periods. If $m=2$, $\alpha _{t-1}=\alpha _t$ and $2^{\alpha _t}$ divides an odd number of proper periods, then the long relation would not be preserved, some orientation-reversing element would belong to $\ker {\theta }$ or $\theta (\Lambda )$ would contain a symmetry.

We prove the sufficiency of the conditions (i)–(iv) by defining epimorphisms $\theta _{q}$ as in Theorem 2.1 for $q\ne 2$, and $\theta _2$ as follows. Let $\omega _i$ be a generator of $C_{2^{\alpha _i}}$ and $\Sigma =\textstyle \sum _i 2^{\alpha _t-\mu _i}$, so that $\Sigma$ is odd if and only if $\sum _iv_t/m_i$ is odd. Note that, if $\Sigma$ is odd, then $2^{\alpha _t}$ divides an odd number of proper periods; recall condition (iv) in that case.

We observe, as in the proof of Theorem 2.1, that neither $\theta (\Lambda )$ has symmetries nor $\ker {\theta }$ has orientation-reversing elements since the exponent of $\omega _{t-m+1}$ is even in $\theta (x_i)$, $i=1,\ldots ,r$, and odd in $\theta (d_i)$, $i=1,\ldots ,\gamma$ (in some cases, we should consider $\omega _{t-\gamma +1}$, $\omega _{t-\gamma }$ or $\omega _{t-\gamma -1}$ instead of $\omega _{t-m+1}$; in any case, the considered generator has order multiple of 4).

(a) $2^{\alpha _t}\not \mid m_i$.

If $m<\gamma \leqslant t$:

$$\begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-t+\gamma -1, \\&\theta _2(x_i) = \omega _{i-r+t-\gamma +1}\omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=r-t+\gamma ,\ldots ,r-1, \\&\theta _2(x_r) = \omega _{1}^{-1}\cdots \omega _{t-\gamma }^{-1}\omega _{t-\gamma +1}^{2}\omega _{t}^{2^{\alpha _{t}-\mu _r}} \quad \text {if } \gamma \text { is even}, \\&\theta _2(x_r) = \omega _{1}^{-1}\cdots \omega _{t-\gamma }^{-1}\omega _{t-\gamma +1}^{2}\omega _{t-m+1}^{-2}\omega _{t}^{2^{\alpha _{t}-\mu _r}} \quad \text {if } \gamma \text { is odd},\\&\theta _2(d_1) = \omega _{t-\gamma +1}^{-1}\cdots \omega _{t-m}^{-1}\omega _{t-m+1}^{1-\gamma }\omega _{t-m+2}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{-1-\Sigma /2}, \quad \text {if } \gamma \text { is even}, \\&\theta _2(d_1) = \omega _{t-\gamma +1}^{-1}\cdots \omega _{t-m}^{-1}\omega _{t-m+1}^{2-\gamma }\omega _{t-m+2}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{-1-\Sigma /2}, \quad \text {if } \gamma \text { is odd}, \\&\theta _2(d_i) = \omega _{t-\gamma +i}\omega _{t-m+1}, \quad i>1, i\ne \gamma -m+1, \\&\theta _2(d_{\gamma -m+1}) = \omega _{t-m+1}. \end{aligned}$$

If $\gamma >t$:

$$\begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-1, \\&\theta _2(x_r) = \omega _{t}^{2^{\alpha _{t}-\mu _r}} \quad \text {if } \gamma \text { is even}, \\&\theta _2(x_r) = \omega _{t-m+1}^{-2}\omega _{t}^{2^{\alpha _{t}-\mu _r}} \quad \text {if } \gamma \text { is odd},\\&\theta _2(d_1) = \omega _{1}^{-1}\cdots \omega _{t-m}^{-1}\omega _{t-m+1}^{1-\gamma }\omega _{t-m+2}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{-1-\Sigma /2}, \quad \text {if } \gamma \text { is even}, \\&\theta _2(d_1) = \omega _{1}^{-1}\cdots \omega _{t-m}^{-1}\omega _{t-m+1}^{2-\gamma }\omega _{t-m+2}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{-1-\Sigma /2}, \quad \text {if } \gamma \text { is odd}, \\&\theta _2(d_i) = \omega _{i}\omega _{t-m+1}, \quad i=2,\ldots ,t, i\ne t-m+1, \\&\theta _2(d_{i}) = \omega _{t-m+1}, \quad i>t \text { or } i=t-m+1. \end{aligned}$$

(Here, we replace $\omega _{i}$ by $\omega _{i-1}$ in the definition of $\theta _2(d_i)$ if also $n=0$.)

If $\gamma \leqslant m$:

$$\begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-t+\gamma -1, \\&\theta _2(x_i) = \omega _{i-r+t-\gamma +1}\omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=r-t+\gamma ,\ldots ,r-1, \\&\theta _2(x_r) = \omega _{1}^{-1}\cdots \omega _{t-\gamma }^{-1}\omega _{t-\gamma +1}^{2}\omega _{t}^{2^{\alpha _{t}-\mu _r}}, \\&\theta _2(d_1) = \omega _{t-\gamma +1}^{-1}\cdots \omega _{t-1}^{-1}\omega _{t}^{-1-\Sigma /2}, \\&\theta _2(d_i) = \omega _{t-\gamma +i}, \quad i=2,\ldots , \gamma . \end{aligned}$$

(b) $2^{\alpha _{t}}|m_i$ for some i and $\Sigma$ is even.

If $\gamma \geqslant t$:

$$\begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-1, \\&\theta _2(x_r) = \omega _{1}^{-2}\cdots \omega _{t-m+1}^{-2g}\cdots \omega _{t-1}^{-2}\omega _{t}^{1-\Sigma }, \\&\theta _2(d_i) = \omega _{t-m+1}\omega _{t-\gamma -1+i} \quad \text { if } i<t \text { and } i\ne \gamma -m+2, \\&\theta _2(d_{i}) = \omega _{t-m+1} \quad \text { if } i=\gamma -m+2\text { or }i\geqslant t. \end{aligned}$$

If $\gamma <t$:

$$\begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-t+\gamma -1, \\ {}&\theta _2(x_i) = \omega _{i-r+t-\gamma +1}\omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=r-t+\gamma ,\ldots ,r-2, \\ {}&\theta _2(x_{r-1}) = \omega _{t}, \\ {}&\theta _2(x_r) = {\left\{ \begin{array}{ll}\omega _{1}^{-1}\cdots \omega _{t-\gamma -1}^{-1}\omega _{t-\gamma }^{-2}\cdots \omega _{t-1}^{-2}\omega _{t}^{1-\Sigma } &{}{}\quad \text{ if } \gamma<m, \\ \omega _{1}^{-1}\cdots \omega _{t-\gamma -1}^{-1}\omega _{t-\gamma }^{-2}\cdots \omega _{t-m+1}^{-2g}\cdots \omega _{t-1}^{-2}\omega _{t}^{1-\Sigma } &{}{}\quad \text{ if } \gamma \geqslant m, \end{array}\right. }\\ {}&\theta _2(d_i) = {\left\{ \begin{array}{ll}\omega _{t-\gamma -1+i} &{}{}\quad \text{ if } \gamma <m \ \ \text{ or } \ \ i=\gamma -m+2,\\ \omega _{t-m+1}\omega _{t-\gamma -1+i} &{}{}\quad \text{ otherwise },\end{array}\right. } \quad i=1,\ldots , \gamma . \end{aligned}$$

(c) $2^{\alpha _{t}}|m_i$ for some i and $\Sigma$ is odd.

If $\gamma \geqslant t-1$:

$$\begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-2, \\ {}&\theta _2(x_{r-1}) = \omega _{t-1}, \\ {}&\theta _2(x_r) = \omega _{1}^{-2}\cdots \omega _{t-m+1}^{-2g}\cdots \omega _{t-2}^{-2}\omega _{t-1}^{-1}\omega _{t}^{2-\Sigma }, \\ {}&\theta _2(d_i) = \omega _{t-m+1}\omega _{t-\gamma -1+i} \quad \text{ if } i<t-1 \text{ and } i\ne \gamma -m+2, \\ {}&\theta _2(d_{i}) = \omega _{t-m+1} \quad \text{ if } i=\gamma -m+2 \text{ or }\ i\geqslant t-1. \end{aligned}$$

If $\gamma <t-1$:

$$\begin{aligned}&\theta _2(x_i) = \omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=1,\ldots , r-t+\gamma -1, \\ {}&\theta _2(x_i) = \omega _{i-r+t-\gamma +1}\omega _{t}^{2^{\alpha _{t}-\mu _i}}, \quad i=r-t+\gamma ,\ldots ,r-3, \\ {}&\theta _2(x_{r-2}) = \omega _{t}, \\ {}&\theta _2(x_{r-1}) = \omega _{t-1}, \\ {}&\theta _2(x_r) = {\left\{ \begin{array}{ll} \omega _{1}^{-1}\cdots \omega _{t-\gamma -2}^{-1}\omega _{t-\gamma -1}^{-2}\cdots \omega _{t-2}^{-2}\omega _{t-1}^{-1}\omega _{t}^{2-\Sigma } &{}{}\quad \text{ if } \gamma<m, \\ \omega _{1}^{-1}\cdots \omega _{t-\gamma -2}^{-1}\omega _{t-\gamma -1}^{-2}\cdots \omega _{t-m+1}^{-2g}\cdots \omega _{t-2}^{-2}\omega _{t-1}^{-1}\omega _{t}^{2-\Sigma } &{}{}\quad \text{ if } \gamma \geqslant m, \end{array}\right. }\\ {}&\theta _2(d_i) = {\left\{ \begin{array}{ll}\omega _{t-\gamma -2+i} &{}{}\quad \text{ if } \gamma <m \ \text{ or } \ i=\gamma -m+3,\\ \omega _{t-m+1}\omega _{t-\gamma -2+i} &{}{}\quad \text{ otherwise. } \end{array}\right. } \quad i=1,\ldots , \gamma . \end{aligned}$$

$\square$

4 Minimum Genus

As an application of Theorems 2.1 and 2.2, in this section, we find the minimum genus $>1$ of a pseudo-real Riemann surface on which a given abelian group A acts essentially as a group of automorphisms. We denote this value by $g^*(A).$ Recall that the order of A is divisible by 4. The case when A is cyclic was solved by Conder and Lo in [11]. Here, we provide a shorter proof based on our previous results.

Theorem 3.1

Let $v_1$ be an integer such that $4\mid v_1$. The minimum genus of a pseudo-real surface on which the cyclic group $C_{v_1}$ acts essentially is $g^*(C_{v_1})=v_1/2$.

Proof

An easy application of Theorem 2.1 shows that the signature $(1;-;[2,2,2n];\{-\})$ satisfies the conditions for the existence of an action of the cyclic group $C_{4n}$ on a pseudo-real Riemann surface. Its genus, by the Riemann–Hurwitz formula, equals $g^*=2n.$ The reduced area of this signature is $1-\frac{1}{2n}.$ The unique signatures of the form (1.1) with $\gamma +r>3$ having reduced area $<1$ are $(1;-;[2,3,3];\{-\})$, $(1;-;[2,3,4];\{-\})$, $(1;-;[2,3,5];\{-\})$ and $(1;-;[2,2,m_3];\{-\})$. Theorem 2.1 shows that $C_{4n}$ does act with the first three signatures if $n=3,$ $n=6$ and $n=15$, respectively. Since their reduced area is also $1-\frac{1}{2n}$, they provide additional actions of the same group $C_{4n}$ on genus 2n for these particular values of n. Theorem 2.1 shows that $C_{4n}$ does not act with the last signature unless $m_3=2n.$ In fact, it has to be $m_3=2n/k$, by condition (i), and k odd, by condition (iii). If $k>1$ and p is a prime divisor of k then the elementary divisor $p^\alpha$ of $C_{4n}$ does not divide, at least, one proper period, contradicting condition (ii). Therefore, $C_{4n}$ does not act with signature of reduced area smaller than $1-\frac{1}{2n}$. $\square$

Remark 3.2

Observe that if $n\ne 3,6,15$ then the signature with which $C_{4n}$ acts attaining the bound is unique, namely, $(1;-;[2,2,2n];\{-\})$. The epimorphism is also unique, and the surfaces are hyperelliptic, see [4].

Theorem 3.3

Let $v_1$ and $v_2$ be integers such that $v_1\mid v_2$ and $4\mid v_2$. The minimum genus of a pseudo-real surface on which $C_{v_1}\times C_{v_2}$ acts essentially is $g^*(C_{v_1}\times C_{v_2}) = 1+\frac{3}{4}v_1v_2-R$, where

Proof

The following signatures fulfill the conditions of Theorem 2.1 or Theorem  2.2 assuring the existence of an essential action of $C_{v_1}\times C_{v_2}$ on a pseudo-real surface of genus $1+\frac{3}{4}v_1v_2-R$ with R as above:

$$\begin{aligned}&(2;\,-\,;\,[2,v_1/2];\, \{-\}){} & {} \text {if } 8\mid v_1 \text { or } v_1=4,\\&(2;\,-\,;\,[2,v_1/4];\, \{-\}){} & {} \text {if } v_1/4 \text { is odd and } v_1\ne 4,\\&(1;\,-\,;\,[2,v_1/2,v_2/2];\, \{-\}){} & {} \text {if } v_1/2 \text { is odd and } v_1\ne 2,\\&(1;\,-\,;\,[2,v_1,v_2/4];\, \{-\}){} & {} \text {if } 8\not \mid v_2 \text { and either } v_1=2 \text { or } v_1 \text { is odd,}\\&(1;\,-\,;\,[2,v_1,v_2/2];\, \{-\}){} & {} \text {otherwise.} \end{aligned}$$

We have to prove that the reduced area of any other signature fulfilling the conditions of Theorem  2.1 or Theorem 2.2 is greater than or equal to that of the above signatures. The reduced area of each of these is smaller than 3/2. So we focus on signatures of the form (1.1) with $\gamma +r>3$ for which the reduced area is smaller than 3/2. They satisfy $(\gamma ,r)=(2,2),$ (1, 3) or (1, 4),  as is easy to see.

We deal just with the first case, $8\mid v_1$ or $v_1=4$, the other cases can be proven likewise. The reduced area of $(2;-;[2,v_1/2];\{-\})$ is $\frac{3}{2}-\frac{2}{v_1},$ so we will prove that any signature satisfying the conditions of Theorem 2.2 (the one applicable for these values of $v_1$) has reduced area $\geqslant \frac{3}{2}-\frac{2}{v_1}.$

Let $(2;\,-;\,[m_1, m_2];\, \{-\})$ be a signature fulfilling the conditions of Theorem 2.2. Then each prime power divisor of $v_1/2$ divides $m_1$ or $m_2$ by condition (ii). So $v_1/2$ divides ${{\,\textrm{lcm}\,}}(m_1,m_2)$. We have to prove that $\frac{1}{m_1}+\frac{1}{m_2}\leqslant \frac{1}{2}+\frac{2}{v_1}.$ This is clear if $v_1=4,$ so we assume $8\mid v_1.$ Without lost of generality we may assume that $m_1\leqslant m_2$. If $m_1\geqslant 4$, then $\frac{1}{m_1}+\frac{1}{m_2} \leqslant \frac{1}{2}$ and we are done. Assume $m_1=3$. If $m_2\geqslant 6$ then $\frac{1}{m_1}+\frac{1}{m_2}\leqslant \frac{1}{3}+\frac{1}{6}=\frac{1}{2}$. Assume $m_2\in \{3,4,5\}.$ As $\frac{v_1}{2}\mid {{\,\textrm{lcm}\,}}(3,m_2)$ it must be $m_2=4$. So $\frac{v_1}{2}=4$ or 12. Both values of $v_1$ satisfy $\frac{1}{m_1}+\frac{1}{m_2}\leqslant \frac{1}{2}+\frac{2}{v_1}.$ Finally, if $m_1=2$ then $v_1/2$ divides $m_2$ and thus $\frac{1}{m_1}+\frac{1}{m_2} \leqslant \frac{1}{2}+\frac{2}{v_1}$.

Let $(1;\,-;\,[m_1, m_2, m_3];\, \{-\})$ be a signature fulfilling the conditions of Theorem 2.2. Then every elementary divisor of $C_{v_1}$ divides at least two proper periods. We have to show that $\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\leqslant \frac{1}{2}+\frac{2}{v_1}.$ We may assume that $m_1\leqslant m_2\leqslant m_3$. If $m_1\geqslant 6$ then $\sum _i\frac{1}{m_i}\leqslant \frac{3}{m_1}\leqslant \frac{1}{2}$ and we are done. If $m_1=2$ then every elementary divisor of $C_{v_1}$ divides $m_2$ and $m_3$. So $v_1$ divides $m_2$ and $m_3$ and therefore $\sum _i\frac{1}{m_i}=\frac{1}{2}+\frac{1}{m_2}+\frac{1}{m_3}\leqslant \frac{1}{2}+\frac{2}{v_1}$. So we assume $m_1\in \{3,4,5\}$. If $v_1=4$ then $\sum _i\frac{1}{m_i}\leqslant 1 = \frac{1}{2}+\frac{2}{v_1}$ and we are done. If $8\mid v_1$ then 8 also divides $m_2$ and $m_3$. Moreover, every elementary divisor $>5$ of $C_{v_1}$ (if any) divides $m_2$ and $m_3.$ It follows that if 3 and 5 are not elementary divisors of $C_{v_1}$ then $v_1$ divides $m_2$ and $m_3$ and so $\sum _i\frac{1}{m_i}\leqslant \frac{1}{m_1}+\frac{2}{v_1}.$ If 3 is an elementary divisor of $C_{v_1}$ then 3 divides at least one $m_i$ with $i\ne 1.$ So either $\sum _i\frac{1}{m_i} \leqslant \frac{1}{3}+\frac{1}{24}+\frac{1}{8} =\frac{1}{2}$ or $\sum _i\frac{1}{m_i}\leqslant \frac{1}{3}+\frac{1}{24}+\frac{1}{24}<\frac{1}{2}.$ If 5 is an elementary divisor of $C_{v_1}$ then 5 divides at least one $m_i$ with $i\ne 1.$ So either $\sum _i\frac{1}{m_i} \leqslant \frac{1}{5}+\frac{1}{40}+\frac{1}{8} <\frac{1}{2}$ or $\sum _i\frac{1}{m_i} \leqslant \frac{1}{3}+\frac{1}{40}+\frac{1}{40} <\frac{1}{2}$.

Finally, let $(1;\,-;\,[m_1, m_2, m_3,m_4];\, \{-\})$ be a signature fulfilling the conditions of Theorem 2.2. Suppose, to get a contradiction, that it has reduced area $<\frac{3}{2}-\frac{2}{v_1}.$ In particular, $\sum _i \frac{1}{m_i}>\frac{3}{2}$. It is straightforward to see that unique sets of proper periods that satisfy this inequality are $[m_1,m_2,m_3,m_4]$ = [2, 2, 2, m] (for any $m\geqslant 2)$, [2, 2, 3, 3],  [2, 2, 3, 4] and [2, 2, 3, 5]. However, the corresponding signatures do not fulfill the conditions of Theorem 2.2 because the elementary divisor $2^{\alpha _1}$ of $C_{v_1}$ does not divide at least two proper periods. $\square$

Theorem 3.4

Let $A\approx C_{v_1}\times \cdots \times C_{v_t}$ be an abelian group such that $t>2$, $v_i\mid v_{i+1}$ and $4\mid v_t$. Assume that A has n cyclic factors of singly even order and m of order multiple of 4. The minimum genus of a pseudo-real surface on which A acts essentially is

$$\begin{aligned} g^*(A)=1+\frac{|A|}{2}\cdot \underset{i\in I}{\min }\left\{ t-1-\frac{1}{v_1}-\cdots -\frac{1}{v_{i-1}}-\frac{\delta _i}{v_i}\right\} , \end{aligned}$$

where $I=\{t-m-n+1, t-m+1, t-m+2,\ldots ,t\}$ if $n>0$ and $m+n$ is even and $I=\{t-m+1,\ldots ,t\}$ otherwise, $\delta _i=2$ if $i=1$ or $\frac{v_{i}}{v_{i-1}}$ is even and $\delta _i=1$ otherwise.

Proof

For each $i=1,\ldots ,t$ we consider a signature of the form $(t-i+1;\,-;\,[m_1,\ldots , m_i];\, \{-\})$, whose reduced area is $\mu _i=t-1-\frac{1}{m_1}-\cdots -\frac{1}{m_i}$. Other signatures $(\gamma ;\,-;\,[m_1,\ldots , m_r];\, \{-\})$ with $\gamma >t$ provide greater values of the reduced area since $\gamma -2\geqslant t-1$ in that case.

Assume first that $m<t.$ For each $i\in \{1,\ldots ,t-m\}$, we consider the signature $S_i=(t-i+1;\,-\,;\,[v_1,\ldots , v_{i-1}, 2^av_{i}];$ $\{-\})$ where $a=-1$ if $i=t-m-n+1$ and $m+n$ is even, $a=0$ if $i\ne t-m-n+1$ and $t-i+1$ is even, $a=\alpha _{t-m+1}-1$ if $t-i+1$ and $v_i$ are odd and $a=\alpha _{t-m+1}-2$ otherwise.

Signature $S_i$ provides a smaller reduced area than that of any other signature $(t-i+1;\,-;\,[m_1,\ldots , m_r];\, \{-\})$ fulfilling the conditions of Theorems 2.1 and 2.2.

For $i\in \{t-m+1,\ldots ,t\}$, we consider the following signatures:

$$\begin{aligned} S_i = {\left\{ \begin{array}{ll} (t-i+1;\,-;\,[v_1,\ldots , v_{i-1}, v_{i}/2];\, \{-\}) &{}\quad \text {if } i=1 \text { or } \frac{v_{i}}{v_{i-1}} \text {is even,}\\ (t-i+1;\,-;\,[v_1,\ldots , v_{i}];\, \{-\})&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

(Recall condition (ii) of Theorem 2.2.) The corresponding reduced area is $\mu _i = t-1-\frac{1}{v_1}-\cdots -\frac{1}{v_{i-1}}-\frac{\delta _i}{v_i}$, where $\delta _i=2$ if $i=1$ or $\frac{v_{i}}{v_{i-1}}$ is even, and $\delta _i=1$ otherwise.

If $i\in \{1,\ldots , t-m\}$ and either $i\ne t-m-n+1$ or $m+n$ is odd, then clearly $\mu _i\geqslant \mu _{t-m+1}$ and thus we dismiss these values of i.

Given $i\in \{t-m+1,\ldots ,t\}$, the reduced area of any other signature $(\gamma ;\,-;\,[m_1,\ldots , m_r];\, \{-\})$, with $\gamma =t-i+1$, fulfilling the conditions of Theorem 2.2 is greater than or equal to $\mu _i$ since, by condition (ii), $r\geqslant i$ and any prime power in any proper period of $S_i$ also appears in some proper period of such signature—recall (1.3) as well. $\square$

Remark 3.5

We can discard some values in the set I of Theorem 3.4. If $i\in \{t-m+1,\ldots ,t-1\}$, then $\mu _i$ can be the smallest reduced area only if either $i=1$ or $\frac{v_{i-1}}{v_{i-2}}$ is odd and $\frac{v_i}{v_{i-1}}$ is even (i.e., $\delta _{i-1}=1$ and $\delta _i=2$). Indeed,

• (a) if $i<t$, $\delta _{i-1}=1$ and $\delta _i=1$, then $\mu _i=\mu _{i+1}+\delta _{i+1}/v_{i+1}$, and thus $\mu _i\geqslant \mu _{i+1}$, and,

• (b) if $\delta _{i-1}=2$, then $-\dfrac{\delta _{i-1}}{v_{i-1}}\leqslant -\dfrac{1}{v_{i-1}}-\dfrac{\delta _i}{v_i}$ and thus $\mu _{i-1}\leqslant \mu _i$ since $\delta _iv_{i-1}|v_i$ (either if $\delta _i=1$ or 2) so that $\frac{\delta _i}{v_i}\leqslant \frac{1}{v_{i-1}}$.

It also follows from (b) that $\mu _{t-m+1}\leqslant \mu _{t-m+2}$ and $\mu _t$ can be the smallest reduced area only if $\frac{v_{t-1}}{v_{t-2}}$ is odd (i.e., $\delta _{t-1}=1$).

Remark 3.6

The following examples show that the minimum genus can be attained by different values of $i\in I$.

• (a) $C_2\times C_{4a}\times \overset{t-1}{\cdots }\times C_{4a}$, $t>2$ and $a>(t-1)/2$. The smallest reduced area is $\mu _1=t-2$.

• (b) $C_4\times \overset{j-1}{\cdots }\times C_4\times C_8\times C_{8a}\times \overset{t-j}{\cdots }\times C_{8a}$, $t>2$, $j\in \{1,\ldots ,t\}$, $j\ne 2$, and $a>t-j$. The smallest reduced area is $\mu _j=t-\frac{j-6}{4}$.

5 Least Minimum Genus of Abelian Groups of the Same Order

For a fixed value of N there are several abelian groups of the same order N. In this section we find the minimum genus $g^*(N)$ of a pseudo-real surface on which an abelian group of order N acts essentially. The minimum is attained by a unique group, and we determine such a group. We again recall that N is divisible by 4.

Theorem 4.1

If an abelian group of order N acts essentially on a pseudo-real compact Riemann surface of genus $g>1$, then $g\geqslant g^*(N)$, where

$$\begin{aligned} g^*(N) = {\left\{ \begin{array}{ll}16 &{}\quad \text {if }N=36,\\ \dfrac{N}{2} &{}\quad \text {if } N\ne 36 \text { and } 8\not \mid N,\\ \dfrac{N}{2}-3 &{}\quad \text {if } 8|N.\end{array}\right. } \end{aligned}$$

The minimum genus is attained uniquely by $C_3\times C_{12}$, $C_N$ and either $C_2\times C_{N/2}$ if $16\not \mid N$ or $C_2\times C_2\times C_{N/4}$ if $16\mid N$, respectively.

Proof

The minimum genus of $C_N$ is N/2 by Theorem 3.1. We now analyze which abelian groups of order N have minimum genus smaller than N/2. The minimum genus of $C_{v_1}\times \cdots \times C_{v_t}$ is greater than $N/2=v_1\cdots v_t/2$ if $t>3$; for, note that the quantities $\mu _i$ defined in the proof of Theorem 3.4 verify $\mu _i\geqslant t-1-\frac{1}{2}-\overset{t}{\cdots }-\frac{1}{2}=\frac{t}{2}-1\geqslant 1$. Assume now that $t=3$. The minimum genus of $C_{2}\times C_{2}\times C_{v_3}$ equals $2v_3-3=N/2-3,$ as is easy to see using Theorem 3.4. Observe that $v_3=N/4$ so $16\mid N$ in this case. The minimum genus of $C_{v_1}\times C_{v_2}\times C_{v_3}$ with $(v_1,v_2)\ne (2,2)$ is greater than $N/2=v_1v_2v_3/2$. In fact, in this case we have $\mu _1=2-\frac{\delta _1}{v_1}\geqslant 1$, $\mu _2=2-\frac{1}{v_1}-\frac{\delta _2}{v_2}\geqslant 1$, and $\mu _3=2-\frac{1}{v_1}-\frac{1}{v_2}-\frac{\delta _3}{v_3}$ with $4\mid v_3$; if $v_1>2$, then $\mu _3\geqslant 2-5/6>1$; if $v_1=2$, $v_2\geqslant 4$, then $\delta _3/v_3\leqslant 1/4$ and thus $\mu _3\geqslant 2-1/2-1/5-1/4>1$. Assume now that $t=2.$ Theorem 3.3 yields that the minimum genus of $C_{v_1}\times C_{v_2}$ is smaller than $N/2=v_1v_2/2$ in three cases:

• (i) if $v_1=3$ and $v_2=12$ (that is, $N=36$) then $g^*(C_{v_1}\times C_{v_2})=N/2-2$,

• (ii) if $v_1=2$ and $8\not \mid v_2$ then $g^*(C_{v_1}\times C_{v_2})=N/2-3$,

• (iii) if $v_1=2$ and $8\mid v_2$ then $g^*(C_{v_1}\times C_{v_2})=N/2-1.$

Therefore, we conclude that

• (a) if $16\mid N$ then $g^*(N)=N/2-3$, attained by $C_2\times C_2\times C_{N/4}$,

• (b) if $8\mid N$ and $16\not \mid N$ then $g^*(N)=N/2-3$, attained by $C_2\times C_{N/2}$,

• (c) if $8\not \mid N$ and $N\ne 36$ then $g^*(N)=N/2$, attained by $C_N$, and

• (d) if $N=36$ then $g^*(N)=16,$ attained by $C_3\times C_{12}$.

Finally, we show that the minimum genus is attained by a unique abelian group in each case. In case (a), the minimum genus is attained when $t=3$ since $g^*(C_N)=N/2$ and either $g^*(C_{v_1}\times C_{v_2})>N/2$ or $g^*(C_{v_1}\times C_{v_2})=N/2-1$ by Theorem 3.3 because $16\mid N$. For $t=3$ it is $g^*(C_{v_1}\times C_{v_2}\times C_{v_3})>N/2$ if $(v_1,v_2)\ne (2,2)$, as shown above. In case (b), $g^*(C_{v_1}\times \cdots \times C_{v_t})>N/2$ if $t>2$. In case (c), $v_i$ is odd if $i<t$ and $4\mid v_t$ so that $g^*(C_{v_1}\times \cdots \times C_{v_t})=1+N/2 (t-1-1/v_1-\cdots -1/v_{t-1}-2/v_t) \geqslant 1+N/2(t-1-(t-1)/3-2/12)>N/2$ if $t>2$ by Theorem 3.4; if $t=2$ then Theorem 3.3 yields $g^*(C_{v_1}\times C_{v_2})=1+N/2+N(1/4-1/2v_1-2/v_2)>N/2$ since $v_2\geqslant 24$ if $v_1=3$ and $v_2\geqslant 20$ if $v_1\geqslant 5$ (recall that $v_1$ is odd, $4v_1\mid v_2$ and $v_1v_2>36$). In case (d), the unique abelian groups of order $N=36$ that act essentially on a pseudo-real surface are $C_{36},$ with $g^*(C_{36})=18,$ and $C_3\times C_{12},$ with $g^*(C_3\times C_{12})=16.$ $\square$

6 Maximum Order

The maximum order of an abelian group acting essentially on a pseudo-real Riemann surface of genus $g>1$ was computed in [5, Theorems 4.1, 4.3 and 4.5]. Here we provide a shorter proof based on our previous results.

Theorem 5.1

The maximum order of an abelian group that acts essentially on some pseudo-real Riemann surface of genus $g>1$ is 36 if $g=16$, 2g if g is even and $g\ne 16$, $2\,g+6$ if $g\equiv 1\pmod {4}$, and $2\,g+2$ if $g\equiv 3\pmod {4}$.

Proof

If $g=16$ then Theorem 4.1 yields that the maximum order is 36 since $C_3\times C_{12}$ acts on genus 16 and, if $N>36$ and $4\mid N$, then $g^*(N)\geqslant N/2-3\geqslant 17>16$.

Assume now that g is even and $g\ne 16.$ Theorem 3.1 yields that the cyclic group $C_{2g}$ acts on genus g. Let $N>2g$ with $4\mid N$ be the order of an abelian group A. If $N\geqslant 2g+8$ then Theorem 4.1 yields $g^*(N)\geqslant N/2-3>g,$ so A does not act on genus g. Assume $N=2\,g+4.$ If $g\equiv 0\pmod {4}$ then $8\not \mid 2g+4$ and so $g^*(2g+4)=g+2>g$ by Theorem 4.1. So A does not act on genus g. Assume now that $g\equiv 2\pmod {4}$. Riemann–Hurwitz formula (1.2) and Proposition 1.1 yield that the signatures with which a group of order $2g+4$ may act on genus g are $(1;-;[m_1,m_2,m_3];\{-\})$ with $\{m_1,m_2,m_3\}=\{2,3,3\}$ if $g=16$, $\{2,3,4\}$ if $g=34$, $\{2,3,5\}$ if $g=88$, and $\{2,2,(g+2)/3\}$ if $g\equiv 1\pmod {3}$. The first and the third signatures are ruled out because $g\not \equiv 2\pmod {4}$. In the second case, the unique abelian groups of order $N=2g+4=72$ with an element of order 4 are $C_{72}$ and $C_2\times C_{36}$, $C_{3}\times C_{24}$ and $C_6\times C_{12}.$ Theorem 3.1 rules out $C_{72}$, Theorem 2.1 (i) rules out $C_{2}\times C_{36}$ and $C_6\times C_{12}$, and Theorem 2.1 (ii) rules out $C_{3}\times C_{24}$. Finally, if an abelian group A of order $2g+4$ acts with signature $(1;-;[2,2,(g+2)/3];\{-\})$ where $g\equiv 1\pmod 3,$ then its exponent $v_{t}$ is an even multiple of $(g+2)/3$ by Theorem 2.1 (i). Since A cannot be cyclic by Theorem 3.1, we get $v_t=2(g+2)/3$ and $A=C_3\times C_{2(g+2)/3}$ with $(g+2)/3$ divisible by 3. But the elementary divisor of $C_3$ does not divide, at least, two proper periods, contrary to Theorem 2.1 (ii).

Assume now that $g\equiv 1\pmod 4$. Then $N=2g+6$ is a multiple of 8 and Theorem 4.1 yields $g^*(N)=N/2-3=g$. If $N>2g+6$ is a multiple of 4 then $g^*(N)\geqslant N/2-3\geqslant (2g+10)/2-3>g$, again by Theorem 4.1. So, no abelian group of order $N>2g+6$ acts on genus $g\equiv 1\pmod 4.$

Finally, if $g\equiv 3\pmod 4$ then $C_2\times C_{g+1}$ acts with signature $(1;-;[2,2,(g+1)/2];\{-\})$ on genus g by Theorem 2.1. Now, Theorem 4.1 yields that for $N=2g+6$, it is $g^*(N)=N/2>g$ (because $8\not \mid 2g+6),$ and for $N>2g+6$ and multiple of 4, it is $g^*(N)\geqslant N/2-3 \geqslant (2g+10)/2-3 > g.$ So no abelian group of order $N> 2g+2$ acts on genus g. $\square$