Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 3–24

PAIRS OF SYMMETRIES OF RIEMANN SURFACES

Milagros Izquierdo and David Singerman

Mälardalen University, Department of Mathematics S-721 23 Västerås, Sweden; mio@mdh.se University of Southampton, Department of Mathematics Southampton SO17 1BJ, England; ds@maths.soton.ac.uk

Abstract. Let F_{g} be a compact Riemann surface of genus g. A symmetry S of F_{g} is an anticonformal involution acting on F_{g}. The fixed-point set of a symmetry is a collection of disjoint simple closed curves, called the mirrors of the symmetry. The number of mirrors |S| of a symmetry of a surface of genus g can be any integer k with 0 ≤ k ≤ g + 1. However, if a Riemann surface F_{g} admits a symmetry S_{1} with k mirrors then work of Bujalance and Costa [1] and Natanzon [9] on symmetries with g +1 mirrors suggest that there may possibly be restrictions on the number of mirrors of another symmetry S_{2} of F_{g}. In the first three sections of this work we show that the number of such restrictions is few and only occur if one of the symmetries has g + 1 or 0 mirrors. The main result of Sections 1–3 is Theorem 1.1 below. In Section 4 we study a finer classification than the number of mirrors, namely the species of a symmetry. The k mirrors of a symmetry S may or may not separate the surface F_{g} into two non-empty components. If the mirrors do separate, then we say that S has species +k, and if the mirrors do not separate then we say that the species is −k. (See [5].) The species of S determines S up to topological conjugacy. In Section 4 we investigate which pairs of species can occur for two symmetries S_{1} , S_{2} of F_{g}. There are many more restrictions than when we just ask for the number of mirrors.

1. Introduction

Let F_{g} be a compact Riemann surface of genus g ≥ 2. A symmetry S of F_{g} is an anticonformal involution acting on F_{g}. By Harnack’s theorem the fixed points set of S consists of k ≤ g + 1 simple closed curves, called mirrors. The number of mirrors of a symmetry S is denoted by |S|.

Using Hoare’s theorem [6] Bujalance, Costa and Singerman [4] gave a method to calculate the total number of mirrors of two symmetries S_{1} , S_{2} acting on a Riemann surface of genus g. The work there suggests that there may be some restrictions on the possible pairs of integers (|S_{1}|, |S_{2}|) that can occur. In fact, we show that these restrictions are few and in Theorems 2.1, 2.2 and 3.2 we find all pairs (|S_{1}|, |S_{2}|) that can occur. These results can be summarised as follows.

1991 Mathematics Subject Classification: Primary 30F10.

The first author has been partially supported by the Swedish Natural Science Research Council (NFR) and the second author by the EU Human Capital and Mobility project on Computational Conformal Geometry.

Theorem 1.1. Let k_{1} , k_{2} be two integers with 0 ≤ k_{1} ≤ k_{2} ≤ g + 1. Then we can find a Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1} , S_{2} with |S_{1}| = k_{1} , |S_{2}| = k_{2} whenever 1 ≤ k_{1} ≤ k_{2} ≤ g or when

k_{2} = g + 1 with g even and k_{1} = 0, 1 or k_{1} ≡ g + 1 mod 2, or

k_{2} = g + 1 with g odd and k_{1} = 0, 1, 2 or k_{1} ≡ g + 1 mod 2, or
k_{1} = 0 with g even and k_{2} = 0, or k_{2} is odd or
k_{1} = 0 with g odd and 0 ≤ k_{2} ≤ g + 1 arbitrary.

No pairs k_{1}, k_{2} outside of this list can occur.

The results where k_{2} = g+1 follow from work of Natanzon [9] and of Bujalance and Costa [1], (see Theorem 2.1). Further work of Natanzon [11] shows that the restrictions for a surface admitting more than two conjugacy classes of symmetries are likely to be more severe. For example it is shown there that if S_{1} , S_{2} , S_{3} are three non-conjugate symmetries of a surface of genus g then |S_{1}| + |S_{2}| + |S_{3}| ≤ 2g + 4.

1.1. Real algebraic curves. One motivation for this study comes from real algebraic geometry. Whereas a compact Riemann surface corresponds to a complex algebraic curve, a compact symmetric surface corresponds to a real algebraic curve, each conjugacy class of symmetries in Aut(F_{g}), (the group of conformal and anticonformal automorphisms of F_{g}), corresponding to a different real model of the curve. The mirrors of the symmetry correspond to the components of the real curve. Thus, if there are two conjugacy classes of symmetries, S_{1} , S_{2} with |S_{1}| = k_{1} and |S_{2}| = k_{2} then we have exactly two real models for the curve, one with k_{1} components and one with k_{2} components.

1.2. Preliminaries on NEC groups and Riemann surfaces. A Riemann surface of genus g > 1 is the quotient of the hyperbolic plane H by a Fuchsian group, a discrete subgroup of Aut^{+}(H) without elliptic elements. A discrete subgroup of Aut(H) with compact quotient is called an NEC (non-Euclidean crystallographic) group. Given an NEC group Γ the subgroup of Γ consisting of the orientation-preserving elements is called the canonical Fuchsian group of Γ. It is denoted by Γ^{+} . The algebraic structure of an NEC group Γ and the geometric structure of its quotient orbifold H/Γ are determined by the signature of Γ:

(1.1)  s(Γ) = (h; ±; [m_{1}, …, m_{r}]; {(n_{11}, …, n_{1s_{1}}), …, (n_{k1}, …, n_{ks_{k}})}).

The quotient space H/Γ is an orbifold with underlying surface of genus h with r cone points and k mirror lines, each with s_{j} ≥ 0 corner points. The signs + or − correspond to orientable or non-orientable orbifolds respectively. The integers m_{i} are called the proper periods of Γ, they are the orders of the cone points of H/Γ. The k brackets (n_{j1}, …, n_{js_{j}}) are the period cycles of Γ and the integers n_{jh} are the link periods of Γ, the orders of the corner points of H/Γ.

Γ is called the group (or fundamental group) of the orbifold H/Γ.

Associated to the signature (1.1) there is a presentation for Γ with generators

x_{1}, …, x_{r},
e_{1}, …, e_{k},
c_{ij}, 1 ≤ i ≤ k, 0 ≤ j ≤ s_{i},
a_{1}, b_{1}, …, a_{h}, b_{h} if H/Γ is orientable or
a_{1}, …, a_{h}, if H/Γ is non-orientable.

and relators

x_{i}^{m_{i}}, i = 1, …, r,
c_{ij−1}^{2}, c_{ij}^{2}, (c_{ij−1}c_{ij})^{n_{ij}}, i = 1, …, k, j = 0, …, s_{i},
c_{i0}e_{i}^{−1}c_{is_{i}}e_{i},

x_{1}x_{2} ⋯ x_{r}e_{1} ⋯ e_{k}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{h}^{−1}b_{h}^{−1}, if H/Γ is orientable or

x_{1}x_{2} ⋯ x_{r}e_{1} ⋯ e_{k}a_{1}^{2} ⋯ a_{h}^{2}, if H/Γ is non-orientable.

These last two relators are sometimes called the long relators, which give rise to the long relations by putting them equal to 1. In these presentations, the only elements of finite order are the elliptic elements and the reflections. The elliptic elements are conjugates of powers of the x_{i} or c_{ij−1}c_{ij} and the reflections are conjugates of the c_{ij}. The e_{i} generators are orientation preserving. They are called the connecting generators. An NEC group without elliptic elements is called a surface group. If Γ is an NEC group then H/Γ is a Klein surface, i.e., a surface with a dianalytic structure. A Klein surface whose complex double has genus greater than one can be expressed as H/Γ where Γ is an NEC surface group.

The hyperbolic area of the quotient orbifold is:

(1.2)  μ(Γ) = 2π(εh − 2 + k + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{2} ∑_{i=1}^{k} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} )),

where ε = 2 if there is a + sign and ε = 1 if there is a − sign. If Γ^{∗} is a subgroup of Γ of finite index then the Riemann–Hurwitz formula holds:

(1.3)  |Γ : Γ^{∗}| = \frac{μ(Γ^{∗})}{μ(Γ)}.

Let S be a symmetry acting on a Riemann surface F with group Γ. ThenF/⟨S⟩ is a Klein surface that can be represented as H/Λ where Λ is a surface group with signature

s(Λ) = (h_{0}; ±; [ ]; {( )^{k}}),

and |Λ : Γ| =2. The notation means that there are k empty period cycles in Λ. As H/Λ = F/⟨S⟩, it follows that k is the number of mirrors of S.

We assume that a Riemann surface F with group Γ admits two symmetries S_{1} and S_{2} , with k_{1} and k_{2} mirrors respectively, such that S_{1}S_{2} has order n. Then S_{1} and S_{2} generate a dihedral group D_{n} of order 2n. Let Δ be the NEC group generated by all the liftings to H of the elements of D_{n}. Then there is an epimorphism θ: Δ → D_{n} such that Γ = Ker(θ). Note that Γ is a Fuchsian surface group. If Λ_{1} = θ^{−1}(⟨S_{1}⟩) and Λ_{2} = θ^{−1}(⟨S_{2}⟩), then

s(Λ_{1}) = (h_{1}; ±; [ ]; {( )^{k_{1}}}),  and  s(Λ_{2}) =(h_{2}; ±; [ ]; {( )^{k_{2}}}),

where

|Δ : Λ_{1}| = |Δ : Λ_{2}| = n and |Λ_{1} : Γ| = |Λ_{2} : Γ| = 2

Notation. From now on D_{n} is the group with presentation

(1.4)  ⟨S, Q | S^{2}, Q^{n}, (SQ)^{2}⟩,

where we identify S_{1} with S and S_{2} with SQ.

1.3. Strategy. Our aim is to compute the integers k_{1} , k_{2} , the numbers of mirrors of S_{1} , S_{2} respectively. Now Λ_{1} and Λ_{2} are subgroups of the NEC group Δ and the permutation representation of Δ on the Λ_{i} -cosets is known, being the same as the permutation representation of D_{n} on the ⟨S_{i}⟩-cosets, (i = 1, 2). We can now use Hoare’s theorem [6] which gives an algorithm to compute signatures of subgroups of NEC groups to calculate k_{1} , k_{2} . The use of Hoare’s theorem in this context is explained in detail in [4]. As every reflection of Λ_{i}, (i = 1, 2) is conjugate to a reflection of Δ, the period cycles of Λ are ‘induced’ by the period cycles of Δ. (Topologically, we have an orbifold covering H/Λ_{i} → H/Δ and we have to examine how the holes of H/Δ lift.) In [4] a graphical method was used to facilitate the application of Hoare’s techniques in our context. We have modified this slightly in the description that we now give.

1.4. Hoare diagrams. If θ: Δ → D_{n} is the homomorphism above we associate a Hoare diagram to every period cycle of Δ, the purpose being to be able to read off from the diagram the number of period cycles of Λ_{1} and Λ_{2} and hence the number of mirrors of S_{1} and S_{2} . The diagrams will have vertices and edges of two possible colours which we call blue and red and (as we shall see) the number of mirrors of S_{1} will be the number of blue components and the number of mirrors of S_{2} will be the number of red components. It turns out that the total number of mirrors is found just by adding the number of mirrors coming from each period cycle so we just need do the calculations for groups with one period cycle. Let us assume that Δ has signature

(0; +; [ ]; {(n_{1}, n_{2}, …, n_{s})}),

and presentation

⟨c_{1}, …, c_{s} | c_{1}^{2}, …, c_{s}^{2}, (c_{1}c_{2})^{n_{1}}, …, (c_{s}c_{1})^{n_{s}}⟩.

i.e., Δ is the group generated by reflections in the sides of a hyperbolic polygon with angles π/n_{1}, π/n_{2}, …, π/n_{s}.

Case 1. Here we assume that at least one of the link periods is even. It then follows that n is even. The Hoare diagram of the pair (Δ, θ) is a coloured graph whose vertices are those of a regular s-sided polygon. We label the s vertices with the generating reflections c_{1}, …, c_{s} and colour the vertex c_{j} blue or red if θ(c_{j}) = SQ^{u_{j}} with u_{j} even or odd respectively. The edges of the polygon are labelled with the link periods n_{1}, …, n_{s} with the edge joining the vertices c_{i} and c_{i+1} labelled n_{i}. If we have two consecutive vertices c_{i} and c_{i+1} with the same colour then we colour the edge joining them by that colour if and only if n_{i} is odd. By [4], the number of period cycles of Λ_{1} is the number of blue components and the number of period cycles of Λ_{2} is the number of red components. These numbers are just the numbers of mirrors of S_{1} and S_{2} respectively.

If the above period cycle is just part of a signature then the situation is slightly different in that we have a connecting generator e and the relations

(c_{0}c_{1})^{n_{1}} = (c_{1}c_{2})^{n_{2}} = ⋯ = (c_{s−1}c_{s})^{n_{s}} = ec_{0}e^{−1}c_{s} = 1.

We now consider a polygon with s+1 vertices c_{0}, c_{1}, …, c_{s}. We consider conjugate reflections as representing the same vertex so that ec_{0}e^{−1} represents the same vertex as c_{0} . Then the relation ec_{0}e^{−1}c_{s} = 1 implies that c_{0} and c_{s} are joined by an edge of the same colour as the vertices c_{0} and c_{s}. Note that as c_{0} and c_{s} are conjugate they must have the same colour. Thus the number of components of each colour is the same if the period cycle is on its own or just part of a signature.

Case 2. An empty period cycle. We split this up in two subcases.

2(i) n is even. The generators and relations associated to an empty period cycle are

⟨c, e | c^{2} = ece^{−1}c = 1⟩.

Suppose that θ(c) = S. Then as e and c commute, θ(e) = 1 or Q^{n/2} . In the first case we find that there are two induced period cycles on Λ_{1} and none on Λ_{2} and in the second case we find that there is one induced period cycle on Λ_{1} and none on Λ_{2} . As the empty period cycle case of Hoare’s theorem requires careful interpretation and as there are some misprints in [4], we outline a proof of these results.

Writing L_{i} = ⟨S_{i}⟩ (i = 1, 2), then

D_{n} = L_{1} + L_{1}Q + ⋯ + L_{1}Q^{n−1}.

Let g ∈ Δ obey θ(g) = Q. Then

Δ = Λ_{1} + Λ_{1}g + ⋯ + Λ_{1}g^{n−1}.

We are looking for the number of conjugacy classes of c in Λ_{1} . Every conjugate of c in Δ has the form g^{k}cg^{−k} and g^{k}cg^{−k} ∈ Λ_{1} if and only if Q^{k}SQ^{−k} = S, that is if and only if k = 0 or n/2. Thus Λ contains at most two conjugacy classes represented by c and g^{n/2}cg^{−n/2} . If these are conjugate in Λ_{1} then for some λ ∈ Λ we have

λcλ^{−1} = g^{n/2}cg^{−n/2},

so that λ^{−1}g^{n/2} ∈ centralizer (c). As centralizer (c) is just the group generated by e and c [12 ] we find that g^{n/2} = λe^{k} or g^{n/2} = λe^{k}c. Applying θ we find that Q^{n/2} = 1 or S, which is false so that we have two conjugacy classes of reflections in Λ_{1} and hence there are two induced period cycles in Λ_{1} . In the second case θ(e) = Q^{n/2} and a similar argument shows that there is now only one induced period cycle in Λ_{1} . In both cases it is easy to show that there are no induced period cycles in Λ_{2} .

2(ii) n is odd. A similar method shows that each empty period cycle induces one empty period cycle on Λ_{1} and one on Λ_{2} .

Graphical representation. (i) n even. If θ(e) = 1, and θ(c) = S, or more generally, θ(c) = SQ^{h}, h even, then our graph consists of two disjoint blue vertices; if θ(e) = 1 and θ(c) = QS^{k}, k odd, then our graph just consists of two disjoint red vertices. If θ(e) = Q^{n/2} and θ(c) = SQ^{h}, h even, then our graph consists of one blue vertex; if θ(e) = Q^{n/2} and θ(c) = SQ^{k}, k odd, then our graph consists of one red vertex. (ii) n odd. Here the graph consists of one red and one blue vertex.

Case 3. All link periods n_{1}, …, n_{s} are odd. These are called odd period cycles in [4].

3(i) n is even. Since θ(c_{j−1}c_{j}) has odd order, then θ(c_{j}) = SQ^{u_{j}}, where u_{j} is either even for 1 ≤ j ≤ s or u_{j} odd for 1 ≤ j ≤ s. Therefore an odd period cycle induces empty period cycles either in Λ_{1} or in Λ_{2} , according to the parity of u_{j} in θ(c_{j}) = SQ^{u_{j}}.

Similar arguments as in Case 2, shows that each odd period cycle induces either two empty period cycles in Λ_{1} (respectively Λ_{2} ), if θ(e_{i}) = 1, or one empty period cycle if θ(e_{i}) ≠ 1.

3(ii) n is odd. It is shown that each odd period cycle induces one empty period cycle in Λ_{1} and one in Λ_{2} .

Graphical representation. (i) n even. If θ(e) = 1 and θ(c_{j}) = SQ^{u_{j}}, u_{j} even, then our graph consists of two disjoint blue vertices; if θ(e) = 1 and θ(c_{j}) = QS^{u_{j}}, u_{j} odd, then our graph just consists of two disjoint red vertices. If θ(e) ≠ 1 and θ(c_{j}) = SQ^{h}, h even, then our graph consists of one blue vertex; if θ(e) ≠ 1 and θ(c_{j}) = SQ^{k}, k odd, then our graph consists of one red vertex. (ii) n odd. Here the graph consists of one red and one blue vertex.

As reflections from different period cycles cannot be conjugate the total number of induced period cycles on the Λ_{i} is just the sum of those induced from each period cycle. Thus the Hoare diagram associated to (Δ, θ) is just the disjoint union of the Hoare diagrams for each period cycle. The number of period cycles induced on Λ_{1} which is equal to the number of mirrors of S_{1} is the number of blue components, and the number of period cycles induced on Λ_{2} which is equal to the number of mirrors of S_{2} is the number of red components.

Note. Case 3(i) was missed in [4] which makes Theorem 2(i) of that paper incorrect. The equality there should be replaced by the following inequality.

α + β + 2γ − δ ≤ t ≤ α + 2β + 2γ − δ.

No other result of [4] is affected.

1.5. Examples. For a simple example, let us use the original example of Natanzon of a compact Riemann surface of genus g admitting two symmetries both having g + 1 mirrors. To obtain this we use an NEC group with signature

(0; +; {(2, 2, …, 2)})

where there are 2g + 2 link periods equal to 2 and the generating reflections are c_{1}, …, c_{2g+2} with relations (c_{1}c_{2})^{2} = ⋯ = (c_{2g+1}c_{2g+2})^{2} = (c_{2g+2}c_{1})^{2} = 1 and the homomorphism θ: Δ → D_{2} is defined by the following action on the generators:

c_{1} → S, c_{2} → SQ, c_{3} → S, … c_{2g+2} → SQ.

As there are no odd link periods the Hoare diagram has no edges and so just consists of g + 1 blue vertices and g + 1 red vertices. Thus we see that both S_{1} and S_{2} have g + 1 mirrors as claimed.

For a more complicated example let Δ have signature

(0; +; {(2, 2, 2, 2, 3, 3), (3), ( )})

and a canonical presentation as in 1.2. We define a homomorphism θ: Δ → D_{6} by defining the action on the generators as follows:

\begin{aligned}c_{10} & → S \\ c_{11} & → SQ^{3} \\ c_{12} & → S \\ c_{13} & → SQ^{3} \\ c_{14} & → S \\ c_{15} & → SQ^{4} \\ c_{16} & → SQ^{2} \\ e_{1} & → Q^{4} \end{aligned}  \begin{aligned}c_{20} & → S \\ c_{21} & → SQ^{2} \\ e_{2} & → Q^{2} \end{aligned}  \begin{aligned}c_{30} & → S \\ e_{3} & → 1 \end{aligned}

The Hoare diagram is as follows and from it we see that S_{1} has 5 mirrors and S_{2} has 2 mirrors. Theorem 2 of [4] just tells us that the total number of mirrors of S and SQ is equal to 7.

Figure 2.1

2. Calculation of possible pairs |S_{1}|, |S_{2}|

If F_{g} is a compact Riemann surface admitting a D_{n} action with |S| = g + 1, then Bujalance and Costa [1], gave a complete list of possibilities for the number of mirrors of the paired symmetry SQ. Their result can be stated as follows:

Theorem 2.1. Let D_{n} with presentation (1.4) act as a group of automorphisms of a compact Riemann surface F_{g} of genus g, with S and SQ acting as symmetries, and with |S| = g + 1. If F_{g} is hyperelliptic then |SQ| = g + 1, 0 or 1, if g is even and |SQ| = g + 1, 0, 1 or 2 if g is odd. If F_{g} is non-hyperelliptic then |SQ| = 0 or g + 1 − 2t, 0 ≤ t < \frac{1}{2}(g + 1) if g is even and |SQ| = g + 1 − 2t, 0 ≤ t ≤ \frac{1}{2}(g + 1) if g is odd.

Of course, the existence of surfaces with these pairs of symmetries can be obtained using Hoare diagrams. For example, to get a surface admitting symmetries with g + 1 mirrors and g + 1 − 2t mirrors we use an NEC group Δ with signature

(2.1)  (0; +; [ ]; {(2^{(r)}), ( )^{(t)}})

with r even and 2t + \frac{1}{2}r = g + 1, (where the notation indicates t empty period cycles and r link periods equal to 2), and we map each of the connecting generators e_{i} to 1 and the reflection generators alternately to S and SQ in D_{2} . The Hoare diagram then consists of 2t + \frac{1}{2}r blue vertices and \frac{1}{2}r red vertices.

Now that we have considered the case where one of the symmetries fixes the maximum number of mirrors, we investigate the other possibilities.

Theorem 2.2. Let k_{1} and k_{2} be two integers with 1 ≤ k_{1} ≤ g, 1 ≤ k_{2} ≤ g. Then there exists a compact Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1} , S_{2} , with |S_{1}| = k_{1} and |S_{2}| = k_{2} .

Proof. As above, we find an action of D_{2} on a surface of genus g with S_{1} = S and S_{2} = SQ. We first assume that k_{1} ≡ k_{2} mod 2 and that k_{1} ≥ k_{2} . Let Δ be a group with signature

(2.2)  (0; +; [2^{g−k_{1}+1}]; {(2^{(2k_{2})}), ( )^{(k_{1}−k_{2})/2}}).

Consider the homomorphism θ: Δ → D_{2} that takes every elliptic generator x_{i} to Q, the reflection generators in the period cycle (2^{2k_{2}}) alternately to S and SQ, the connecting generators e_{i} of the empty period cycles to 1 and the reflection generators of the empty period cycles to S. We choose the image of the connecting generator e_{1} of the non-empty period cycle to be Q or 1 so as to make the long relation

x_{1}x_{2} ⋯ x_{g−k_{1}+1}e_{1} ⋯ e_{(k_{1}−k_{2})/2+1} = 1

consistent with the homomorphism θ. In the Hoare diagram the period cycle (2^{2k_{2}}) contributes k_{2} blue vertices and k_{2} red vertices while the empty period cycles contribute k_{1} − k_{2} blue vertices, giving in total k_{1} blue vertices and k_{2} red vertices, as desired. Also, the genus of the kernel of θ is g, so the corresponding quotient surface has genus g and the symmetries S_{1} and S_{2} have k_{1} and k_{2} mirrors respectively.

If k_{1} ≡ k_{2} + 1 mod 2 then we use an NEC group with signature

(2.3)  (0; +; [2^{(g−k_{1})}]; {(2^{(2k_{2})}), ( )^{(k_{1}−k_{2}+1)/2}}).

The homomorphism θ is defined as before except that θ maps exactly one of the connecting generators of an empty period cycle to Q and all the others to 1. Thus the empty period cycles contribute 2(k_{1} − k_{2} − 1)/2+1 = k_{1} −k_{2} red vertices and now the calculation goes exactly as before.

3. Surfaces admitting a fixed-point free symmetry

We now investigate the case where the compact Riemann surface F_{g} admits a pair of symmetries S_{1} , S_{2} with |S_{1}| = 0. For a group element A we let o(A) denote the order of A.

Theorem 3.1. If F_{g} admits a pair of symmetries S_{1} , S_{2} with |S_{1}| = 0, and if o(S_{1}S_{2}) is divisible by 4, then g is odd.

Proof. Suppose that o(S_{1}S_{2}) = 4a, for some positive integer a. Then T = (S_{2}S_{1})^{a−1}S_{2}S_{1}S_{2}(S_{1}S_{2})^{a−1}

commutes with S_{1} , is conjugate to S_{1} and is distinct from S_{1} . Let A = ⟨S_{1}, T ⟩, a group generated by two commuting fixed-point free symmetries. Let Δ_{1} be the lift of A to H. As S_{1} and T act without fixed points, Δ_{1} cannot have period cycles and so has signature (h; −; [2^{(r)}]; { }). In the homomorphism from Δ_{1} to A the elliptic generators of Δ_{1} must map to S_{1}S_{2} = Q so that r is even. The Riemann–Hurwitz formula gives

g − 1 = 2h − 4 + r

and so g is odd.

Theorem 3.2. If g is even and if F_{g} admits a fixed-point free symmetry S_{1} and a symmetry S_{2} with non-empty fixed-point set, then o(S_{1}S_{2}) ≡ 2 mod 4 and |S_{2}| is odd.

Proof. By Theorem 3.1, o(S_{1}S_{2}) is not divisible by 4. The order is not odd for then S_{1} and S_{2} would be conjugate which is impossible as S_{2} has non-empty fixed-point set. Hence o(S_{1}S_{2}) ≡ 2 mod 4. If o(S_{1}S_{2}) = 4a + 2 then letting

T = (S_{2}S_{1})^{a}S_{2}(S_{1}S_{2})^{a}

we see that T is conjugate to S_{2} and o(S_{1}T) = 2. Let B =⟨S_{1}, T ⟩ and Δ_{2} be the lift of B to H. We first note that Δ_{2} cannot have non-empty period cycles. For otherwise these period cycles would consist of link periods equal to 2. In the homomorphism θ from Δ_{2} to B the reflection generators would map alternately to S_{1} and T. The Hoare diagram would then have both blue and red vertices contradicting the hypothesis that |S_{1}| = 0. Thus the signature of Δ_{2} must have the form (h; ±; [2^{r}]; {( )^{s}}) and θ maps each reflection generator c_{i} (i = 1, …, s) of Δ_{2} to T, as S_{1} acts freely. Let e_{1}, …, e_{s} be the canonical generators of Δ_{2} that commute with the c_{i} (the connecting generators). Suppose that θ maps e_{1}, …, e_{s} to Q (=S_{1}S_{2} ) and e_{u+1}, …, e_{s} to 1. Then the Hoare diagram consists of 2(s−u)+u red vertices and no blue vertices, so that |T | = 2s−u. As θ(x_{j}) = Q (j = 1, …, r), then applying θ to the ‘long relation’ shows that r + u is even. The Riemann–Hurwitz formula gives

4g − 1 = 4εh + 2s − 4 + r

(where ε = 2 or 1 depending on whether Δ_{2} is orientable or not) and as g is even, r is odd so that u is odd and thus |S_{2}| = |T | is odd.

Theorem 3.2 puts some restrictions on the possible pairs (0, |S_{2}|) describing the number of fixed curves of a pair of symmetries S_{1} , S_{2} . We now show by constructing examples that these are the only restrictions. In the following g denotes the genus of a surface admitting a fixed-point free symmetry S_{1} , and m denotes the number of mirrors of another symmetry S_{2} .

Case 1. g odd, m odd, m ≤ g. Let Δ be an NEC group with signature

(3.1)  (0; +; [4, 2^{(g−m)/2)}]; {(2^{(m)})}).

We can construct a homomorphism θ from Δ to D_{4} by mapping the elliptic generator of order 4 to Q, the elliptic generators of order 2 to Q^{2} , the reflection generators alternately to SQ and SQ^{3} and the connecting generator e_{1} to Q or Q^{−1} . The kernel of θ is a Fuchsian surface group Γ. The Hoare diagram has m isolated red vertices and no blue vertices, so that S_{1} and S_{2} are symmetries of H/Γ with |S_{1}| = 0 and |S_{2}| = m as claimed.

Case 2. g odd, m even, 1 < m < g. We now let Δ have signature

(3.2)  (0; +; [2^{g+1−m}]; {( )^{m/2}})

and construct a homomorphism ϕ: Δ → D_{2} that takes the elliptic generators to Q the reflection generators to SQ and the connecting generators e_{i} to 1. As above we see that the kernel of ϕ is a Fuchsian surface group of genus g and that the corresponding Riemann surface has a pair S_{1} , S_{2} of symmetries with |S_{1}| = 0, |S_{2}| = m.

Case 3. g even, m odd. We now let Δ have signature

(3.3)  (0; +; [2^{g+2−m}]; {( )^{(m+1)/2}})

and construct a homomorphism ψ: Δ → D_{2} that takes the elliptic generators to Q, the reflection generators to SQ, the connecting generators e_{1}, …, e_{(m−1)/2} to 1 and e_{(m+1)/2} to Q. The kernel of ψ is a Fuchsian surface group of genus g and the corresponding Riemann surface has a pair of symmetries S_{1} , S_{2} with |S_{1}| = 0, |S_{2}| = m.

Case 4. g odd, m = 0. This is achieved using an NEC group with signature (1; −; [2^{g+1}]; { }) and considering a homomorphism onto D_{2} , taking the elliptic generators to Q and the glide reflection generator to S.

Case 5. g even, m = 0. This is achieved using an NEC group with signature (1; −; [3^{(g+2)/2}]; { }) and considering a homomorphism onto D_{3} .

Theorems 2.1, 2.2 and 3.2 together with the above examples prove Theorem 1.1 announced in the introduction.

4. Computing pairs of species

In this section we consider a finer classification by taking into account whether the mirrors of S separate or do not separate the surface.

4.1. Separating symmetries. If S is a symmetry of F_{g} then either the fixed point set of S separates F_{g} into two homeomorphic components or the fixed point set of S does not separate. Thus we may have separating or non-separating symmetries. If |S| = k and S is separating (respectively non-separating) then we say that S has species +k (respectively −k), (see [5]). The species of a symmetry determines the symmetry up to topological conjugacy. In [7] an algorithmic method was described that can be used to determine whether a symmetry is separating. We briefly recall this method.

Let Γ be the Fuchsian surface group that uniformizes F_{g} and let G be a group of automorphisms of F_{g} that contains a symmetry S. Let Δ be the lift of G to the upper half-plane H and θ: Δ → G be the canonical epimorphism with kernel Γ. Let Λ = θ^{−1}(⟨S⟩) and consider the Schreier coset graph 𝒮 (Δ, Λ). If c_{i} is a reflection in Δ that fixes a coset then this corresponds to a loop in 𝒮 (Δ, Λ). We let 𝒮̂ = 𝒮̂ (Δ, Λ) be the Schreier graph with all loops corresponding to reflection generators deleted. Each edge of 𝒮̂ is labelled by a generator of Δ so every path corresponds to an element of Δ, namely the products of the labels of the edges. In [7] it is shown that S is a separating symmetry if and only if every closed path in 𝒮̂ (Δ, Λ) corresponds to an orientation preserving element. Thus we only need one closed path corresponding to an orientation-reversing element to imply that S is non-separating. We would thus expect S to be separating to impose strong restrictions on the signature of Δ and hence on the quotient orbifold F_{g}/G. For example it is well known that if S is separating then

(4.1)  |S| ≡ g + 1 mod 2.

This follows by an easy Euler characteristic argument as the quotient F_{g}/⟨S⟩ is orientable or see [5].

We now revert to our special case G = D_{n} = ⟨S, Q|S^{2} = Q^{n} = (SQ)^{2} = 1⟩. The Schreier graph 𝒮 is isomorphic to the Schreier graph of D_{n} with respect

to the subgroup ⟨S⟩. As D_{n} = ⟨S⟩∪⟨S⟩Q∪⋯∪⟨S⟩Q^{n−1} we can label the vertices of the graph by the elements of Z_{n} and the edges by the elements of D_{n}. Note that an edge labelled Q^{i} will join the vertex r to the vertex r + i and SQ^{j} will join i to j − i.

Notation. We let Y denote the canonical set of generators of Δ and let Y^{+} , Y^{−} denote the subsets consisting of the orientation-preserving and orientationreversing generators respectively. We may assume that S ∈ θ(Y^{−}).

Lemma 4.1. Assume that S separates. Then if n ≡ 2 mod 4, θ(Y^{+}) ⊆ {1, Q^{n/2}}; otherwise, θ(Y^{+}) = 1.

Proof. If Q^{i} ∈ θ(Y^{+}), (i ≠ n/2), then an orientation-reversing path lies in 𝒮 as in Figure 4.1.

Figure 4.1

Thus if S separates and Q^{i} ∈ θ(Y^{+}) then i = 0 or n/2. If n is odd then i = 0; if 4|n then we can find the following orientation-reversing path with k = n/4.

Figure 4.2

Lemma 4.2. Assume that S separates. If n is odd then θ(Y^{−}) = {S, SQ^{i}} for some integer i with (i, n) = 1.

Proof. If SQ^{i}, SQ^{j} ∈ θ(Y^{−}) then because n is odd, we can find k such that i−j ≡ k mod n and we can now find an orientation-reversing path as in Figure 4.3.

Figure 4.3

By Lemma 4.1, θ(Y^{+}) = {1}, so θ(Y) = {S, SQ^{i}}. As θ is an epimorphism (i, n) = 1.

As there is an automorphism of D_{n} that fixes S and maps Q to Q^{r}, for any r co-prime to n, we may assume that i = 1 in Lemma 4.2.

Lemma 4.3. Assume that S separates. If n is even and if S, SQ^{i}, SQ^{j} ∈ θ(Y^{−}), with i ≢ j mod n then i, j have different parities.

Proof. If i ≡ j mod 2 then we can find k such that i − j ≡ 2k mod n and then we can find an orientation-reversing path as in Lemma 4.2.

We may assume that i is odd and j is even.

Lemma 4.4. Assume that S separates. If n is even and S, SQ^{i}, SQ^{j} ∈ θ(Y^{−}) with i ≢ j mod n, then j ≡ 2i mod n.

Proof. We consider the following pentagon in the Schreier graph 𝒮 .

Figure 4.4

As S separates we cannot have this orientation-reversing path so that one of the edges must be a loop. We first consider the possibility that the edge labelled S is a loop. In this case, i − j ≡ j − i mod n or i − j ≡ \frac{1}{2}n mod n. By Lemma 4.3, n ≡ 2 mod 4. Now θ(Y^{+}) contains at most 1 and Q^{n/2} and θ(Y^{−}) contains at most S, SQ^{i}, SQ^{i+(n/2)} with i odd. As θ(Y) generates D_{n} we must have (i, n) = 1 and so we can apply an automorphism of D_{n} and assume that i = 1 and so j = 1 + \frac{1}{2}n. Now SQ^{j}: 1 → \frac{1}{2}n and so we can find the following path of odd length in 𝒮 :

1 → −1  (by S)
−1 → 2 (by SQ)
2 → −2  (by S)
−2 →  3  (by SQ)
.  .  .
.  .  .
\frac{1}{2}n − 1 → 1 − \frac{1}{2}n  (by S)
1 − \frac{1}{2}n → \frac{1}{2}n  (by SQ)
\frac{1}{2}n →  1  (by SQ^{j}).

This is a contradiction so that the edge labelled S cannot be a loop. Hence the edge labelled SQ^{j} joining i to j − i is a loop and then i ≡ j − i mod n or j ≡ 2i mod n.

Now if n is odd and S separates then as all symmetries in D_{n} are conjugate SQ must necessarily separate so that both symmetries of the pair are separating symmetries. Now suppose that n is even and that both symmetries of the pair separate. By Lemma 4.4 we may suppose, after applying an automorphism of D_{n} that θ(Y^{−}) ⊆ {S, SQ, SQ^{2}}.

Lemma 4.5. If n is even and if both S and SQ separate then θ(Y^{−}) = {S, SQ}.

Proof. Suppose that θ(Y^{−}) = {S, SQ, SQ^{2}}. Let Λ_{1} = θ^{−1}(⟨SQ⟩) and form the Schreier coset graph 𝒮_{1} = 𝒮 (Δ, Λ_{1}) and delete the reflection loops, as before, to form 𝒮̂_{1} . Again, SQ is non-separating if there is an orientation-reversing path in 𝒮_{1} . The cosets are now ⟨SQ⟩, ⟨SQ⟩Q, …, ⟨SQ⟩Q^{n−1} , which we denote by 0, 1, …, n − 1. We have (0)SQ^{2} = 1, 1(SQ) = n − 1 and (n − 1)S = 0, so we have an orientation reversing triangle.

These lemmas tell us about the possible link periods and proper periods when there are separating symmetries in the dihedral group.

Theorem 4.1. If G is a group of automorphisms generated by a pair of symmetries and if Δ is the lift of G to the upper half-plane H, then if one of the symmetries separates,

(a) (i) if n is odd then Δ has no proper periods and all link periods are equal to n, (a) (ii) if n is even then Δ has no proper periods if 4|n and all proper periods

are equal to 2, if n ≡ 2 mod 4, and in both cases, all link periods are equal to n or \frac{1}{2}n.

(a) (iii) However, if Δ contains a proper period equal to 2 then Δ can have no link periods equal to n. If both of the symmetries separate then

(b) Δ has no proper periods and all link periods are equal to n.

Proof. a(i). Let θ: Δ → D_{n} be the canonical homomorphism. If Δ contains an elliptic element x then by Lemma 4.1, θ(x) = 1 and then x ∈ ker θ which contradicts ker θ being a surface group. Now assume that c_{k} and c_{k+1} are reflections in Δ with c_{k}c_{k+1} having finite order m. By Lemma 4.2 we may assume that θ(c_{k}) = S, and θ(c_{k+1}) = SQ^{i} with (i, n) = 1. As S(SQ^{i}) has order n and ker θ is a surface group, m = n. The first part of a(ii) also follows from Lemma 4.1 and the second part from Lemma 4.4, or more particularly the remark at the end of the proof that θ(Y^{−}) ⊆ {S, SQ, SQ^{2}}. For a(iii) we consider the cosets of ⟨S⟩ and label the coset ⟨S⟩Q^{r} by r. We join 0 to \frac{1}{2}n by a path of length n − 1 alternately labelled S and SQ. This path goes from 0 to 1, 1 to n − 1, n − 1 to 2, 2 to n − 2 etc. Once we have arrived at \frac{1}{2}n we go back to 0, by a path labelled Q^{n/2} . The resulting closed path is an orientation-reversing path, which shows that if Δ contains a period equal to 2 and link periods equal to n then S is non-separating. To prove (b) we may assume, by Lemma 4.1, that n ≡ 2 mod 4. By Lemma 4.5 S, SQ ∈ θ(Y^{−}). We consider the cosets of ⟨SQ⟩ and label the coset ⟨SQ⟩Q^{r} by r. We can find r such that 2r ≡ \frac{1}{2}n − 1 mod n and then (r)S = −r − 1. Now if there is an elliptic period, then by Lemma 4.1, it must be equal to 2, and its image in G is Q^{n/2} . We then have

(r)Q^{n/2} = r + \frac{1}{2}n ≡ −r − 1 mod n

and so we can find an orientation-reversing closed path in the Schreier graph. The fact that all link periods are equal to n follows directly from Lemma 4.5.

4.2. Signatures. We now have enough information to determine the possible signatures of Δ given that one or both symmetries of the pair separates.

Theorem 4.2. If n is odd and one (and hence both) symmetries of the pair separates then the signature of Δ has the form

(4.2)  (h; +; [ ]; {(n, …, n), …, (n, …, n), ( ), …, ( )}).

where each non-empty period-cycle has even length.

Proof. By Theorem 4.1 above we only need prove that the period cycles have even length and that we have a positive sign in the signature. The first follows as the homomorphism θ must be of the form c_{i0} → S, c_{i1} → SQ, c_{i2} → S, …, c_{is_{i}} → S, with c_{i0} and c_{is_{i}} conjugate in Δ and the second because a glide reflection generator must give an orientation-reversing loop in both the Schreier graphs.

Theorem 4.3. If n is divisible by 4 and if one of the symmetries separates then Δ has a signature of the form

(4.3) (h; ±; [ ]; \{(…, \frac{1}{2}n, n, n, …, n, \frac{1}{2}n, …) … (\frac{1}{2}n, …, \frac{1}{2}n) …, ( ), …, ( )\}),

where there are always an even number of link periods equal to n between two link periods equal to \frac{1}{2}n, and the period cycles containing only link periods equal to \frac{1}{2}n have even length.

Proof. The only statement that we have not proved concerns the even number of link periods equal to n between two link periods equal to \frac{1}{2}n. To see this we normalize the homomorphism so that the images of θ are S, SQ, SQ^{2} . We only get link periods equal to n when SQ is an image of θ. The images of the neighbouring reflections are then S and SQ^{2} showing that the link periods equal to n occur in pairs.

Theorem 4.4. If n ≡ 2 mod 4 and if one of the symmetries separates then Δ has signature either of the form

(i) as given in Theorem 4.3 or (ii) (h; ±; [2, …, 2]; \{(\frac{1}{2}n, …, \frac{1}{2}n), …, (\frac{1}{2}n, …, \frac{1}{2}n), ( ), …, ( )\}) where each period cycle has even length.

Proof. This follows from the above Lemmas and Theorem 4.1.

Theorem 4.5. If both symmetries of the pair separate, then Δ has signature of the form

(h; +; [ ]; {(n, …, n), …, (n, …, n), ( ), …, ( )}).

Proof. This follows from part (b) of Theorem 4.1.

Remarks. 1. Each of the above theorems tells us about the possible quotient orbifolds by the D_{n} action given that one of the symmetries separates. We have seen for example, that there can only be cone points of order 2 (and this occurs only in the case that n ≡ 2 mod 4) and the corner points can only have orders n and \frac{1}{2}n.

2. The converse of the above theorems will be true as long as the images of the hyperbolic and glide reflection generators obey the restrictions of Lemmas 4.1–4.4. In particular the converse always holds if the above NEC groups Δ have genus 0.

4.3. The existence of Riemann surfaces admitting pairs of symmetries with given species. We are now in a position to tackle the question: Let k_{1} , k_{2} , be two integers with −g ≤ k_{1} ≤ k_{2} ≤ g + 1. Does there exist a Riemann surface X of genus g admitting a pair of symmetries S_{1} , S_{2} with sp(S_{1}) = k_{1} , sp(S_{2}) = k_{2} ? (Here we use the above convention that a positive species refers to a separating symmetry, and a non-positive species to a non-separating symmetry.) We shall see that most pairs of species do exist but that there are some interesting exceptions. We divide our investigation into a number of cases.

Case 1. k_{1} ≤ −1, k_{2} ≤ −1.

Theorem 4.7. If −g ≤ k_{1} ≤ k_{2} ≤ −1 then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1}, S_{2} with sp(S_{1}) = k_{1} , sp(S_{2}) = k_{2} .

Proof. In Theorem 2.2 we constructed symmetries with |S_{1}| = |k_{1}|, |S_{2}| = |k_{2}|. Signature (2.2) must correspond to a negative species by Theorems 4.4, 4.5 and (2.3) gives a negative species as the connecting generator maps to Q.

Case 2. k_{1} ≤ 0, k_{2} = 0.

Theorem 4.8. If g is odd and if −g ≤ k_{1} ≤ 0 then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1} , S_{2} with sp(S_{1}) = k_{1} , sp(S_{2}) = 0. If g is even and if k_{1} = 0 or k_{1} ≤ g is odd then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1} , S_{2} with sp(S_{1}) = k_{1} , sp(S_{2}) = 0.

Proof. If g is odd and k_{1} is even then let Δ be an NEC group of signature

(4.5)  (1; −; [2^{(g+1−|k_{1}|}]; {( )^{|k_{1}|/2}}).

We construct a homomorphism from Δ to D_{2} by mapping the glide reflection and reflection generators to S_{1} , the elliptic generators to S_{1}S_{2} = Q and the connecting e-generators to the identity. This is a homomorphism as g + 1 − k_{1} is even and so the long relation is preserved. As the glide reflection generator maps to S_{1} the Schreier graph 𝒮 (Δ, Λ_{1}), where Λ_{1} is inverse image of ⟨S_{1}⟩, has orientation reversing loops not just coming from reflections so that S_{1} is non-separating. As the connecting generators map to the identity each empty period cycle contributes two mirrors so the species of S_{1} is k_{1} as claimed. The Riemann–Hurwitz formula gives the genus of the Riemann surface as g. If g is odd and k_{1} is odd we use the signature (3.1) to construct the symmetries. By forming the Schreier graph we see that the symmetry with non-empty fixed-point set does not separate. In this case the product of the symmetries has order 4, and by comparison with the above case where g is odd it is easy to see that we cannot achieve this when the product of the symmetries has order 2. If g is even and k_{1} is odd then we let Δ be an NEC group of signature

(1; −; [2^{g−|k_{1}|}]; {( )^{(|k_{1}|+1)}/2})

and construct a homomorphism from Δ to D_{2} , again sending the glide-reflection and reflection generators to S_{1} and the elliptic generators to S_{1}S_{2} . This time, however we map one of the connecting generators to S_{1}S_{2} = Q and the others to the identity. As g − k_{1} is odd this is a homomorphism as the long relation is preserved. The species is calculated as before. If k_{1} = 0 we refer to Case 5 in Section 3 where we find a commuting pair of symmetries of a surface of odd genus both with species 0.

Case 3. k_{1} = 0, k_{2} > 0.

Theorem 4.9. If g is odd and if 1 < k_{2} ≤ g + 1, with k_{2} even then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1}, S_{2} with sp(S_{1}) = 0, sp(S_{2}) = k_{2} . If g is even and k_{2} is odd (1 ≤ k_{2} ≤ g + 1), then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1} , S_{2} with sp(S_{1}) = 0, sp(S_{2}) = k_{2} .

Proof. We use Cases 2, 3 of Section 3 and verify that in each case that S_{2} is a separating symmetry.

Note. By Theorem 3.2 there does not exist a Riemann surface of even genus admitting a symmetry of zero species and another one with a non-zero even species.

Case 4. k_{1} > 0, k_{2} > 0. Note that by (4.1) we must have k_{1} ≡ k_{2} ≡ g + 1 mod 2.

Theorem 4.10. (i) If 1 < k_{1} ≤ k_{2} ≤ g + 1 then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries S_{1} , S_{2} with sp(S_{1}) = k_{1} , sp(S_{2}) = k_{2} . (ii) If g is even, 1 ≤ k_{2} ≤ g + 1, and k_{2} ≡ g + 1 mod 4 then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries of species S_{1} , S_{2} with sp(S_{1}) = +1, sp(S_{2}) = k_{2} .

Proof. Let Δ be an NEC group with signature

(4.7)  (h; +; [ ]; {(2^{2t}), ( )^{u_{1}+u_{2}}})

Let θ: Δ → D_{2} = ⟨S_{1}, S_{2}⟩ be defined by mapping the reflections of the nonempty period cycle alternately to S_{1} and S_{2} , the reflections of u_{1} of the period cycles to S_{1} the periods of u_{2} of the period cycles to S_{2} and the connecting generators to the identity. Then letting k_{i} = |S_{i}|, (i = 1, 2) we see that

k_{i} = t + 2u_{i}

and the Riemann–Hurwitz formula gives

g + 1 = 4h + 2u_{1} + 2u_{2} + t.

If k_{2} ≡ g + 1 mod 4 we let u_{1} = 0, u_{2} = \frac{1}{2}(k_{2} − k_{1}), t = k_{1} , h = \frac{1}{4}(g + 1 − k_{2}) to give sp(S_{1}) = +k_{1} , sp(S_{2}) = +k_{2} . We get part (ii) by putting t = 1.

If k_{2} ≡ g−1 mod 4 and k_{1} > 1 we let u_{1} = 1, u_{2} = \frac{1}{2}(k_{2} − k_{1})+1, t = k_{1}−2, h = \frac{1}{4}(g − 1 − k_{2}) to give sp(S_{1}) = +1, sp(S_{2}) = +k_{2} .

Theorem 4.9 does not give us information when k_{1} = 1, k_{2} ≡ g − 1 mod 4. If k_{1} = 1 we must have t = 1, u_{1} = 0 so the Riemann–Hurwitz formula implies (when n = 2) that

g + 1 = 4h + 2u_{2} + 1 = 4h + k_{2}

and so necessarily k_{2} ≡ g + 1 mod 4. Thus we cannot have a pair of commuting separating symmetries with one of the symmetries having just one mirror and the other having k_{2} ≡ g − 1 mod 4 mirrors. We now investigate the case where we have a pair S_{1} , S_{2} of separating symmetries with S_{1}S_{2} having even order n > 2. By Theorem 4.5 the only way we can do this is via a homomorphism from an NEC group of signature

(4.8)  (h; +; [ ]; {(n, …, n), …, (n, …, n), ( )^{u_{1}+u_{2}}})

onto D_{n} = ⟨S_{1}, S_{2}⟩, where the reflections of the non-empty period cycles map alternately to S_{1} and S_{2} and the length of each period cycle is even, the reflections of u_{1} of the empty period cycles map to S_{1} and the reflections of u_{2} of the empty period cycles map to S_{2} and the connecting generators map to the identity. If there are q non-empty period cycles and 2t link-periods equal to n then the Riemann–Hurwitz formula gives

2g − 2 = n(4h + 2q + 2u_{1} + 2u_{2} − 4 + 2t) − 2t.

However, k_{1} = 1 and so t + 2u_{1} = 1 giving t = 1 and thus q = 1 and u_{1} = 0. We then obtain g = n(2h + u_{2}) and k_{2} = 2u_{2} + 1 = (2g/n) − 4h + 1. Thus k_{2} ≤ (2g/n) + 1 and k_{2} ≡ (2g/n) + 1 mod 4. On the other hand we can choose u_{2} and Δ with signature (4.8) to give this result and thus we have

Theorem 4.11. If k_{2} ≡ g − 1 mod 4 then there exists a Riemann surface F_{g} of genus g admitting a pair of symmetries of species 1 and k_{2} > 0 if and only if

k_{2} ≡ (\frac{2g}{n} + 1) mod 4  and  k_{2} ≤ \frac{2g}{n} + 1,

where n > 2 is an integer dividing 2g.

For example if k_{2} ≡ g − 1 mod 4 then there is no Riemann surface of genus g that admits a pair of symmetries of species 1 and k_{2} where k_{2} > \frac{1}{2}g + 1.

Case 5. k_{1} < 0, k_{2} > 0. We first deal with the case when we have commuting symmetries. As k_{2} > 0 we must have k_{2} ≡ g + 1 mod 2. What we now find is that k_{1} ≡ g + 1 mod 2 also.

Theorem 4.12. If −g ≤ k_{1} < 0 < k_{2} ≤ g then there exists a Riemann surface F_{g} of genus g admitting a commuting pair of symmetries S_{1} , S_{2} with sp(S_{1}) = k_{1} , sp(S_{2}) = k_{2} if and only if k_{1} ≡ k_{2} ≡ g + 1 mod 2.

Proof. To prove the existence of such a pair of symmetries we use an NEC group Δ of signature

(\frac{1}{2}(g + 1 − k_{2}); −; [ ]; \{(2^{2|k_{1}|)}, ( )^{(k_{2}−|k_{1}|)/2}\})

and we define a homomorphism from Δ to D_{2} which maps the glide reflection generators to S_{1} , the reflection generators of the non-empty period cycle alternately to S_{1} and S_{2} , and the generators of the empty period cycles to S_{2} . On the other hand, it follows from Theorem 4.4 that the only possible signatures for Δ are (h; ±; [2^{r}]; {( )^{u}}) or (h; ±; [ ]; {(2^{s_{1}}), …, (2^{s_{k}})}). In the first signature both symmetries do not separate, for the second for one of the symmetries to separate we must have all the connecting generators mapping to the identity. In this case every empty period cycle contributes two mirrors while the non-empty period cycles contribute the same number of mirrors to both symmetries. Thus k_{1} ≡ k_{2} mod 2.

Theorem 4.11 puts a restriction on the possible pairs k_{1} , k_{2} when k_{1} < 0 < k_{2} for commuting symmetries S_{1} , S_{2} that is, when S_{1}S_{2} has order n = 2. We now see what other pairs k_{1} , k_{2} are possible if S_{1}S_{2} has order n > 2. We shall see that not only are restrictions needed but, unlike the other cases it is not that easy to find, in a uniform way, what pairs are possible for a given genus. The restrictions come about because of the inequality in the next theorem.

Theorem 4.13. Let S_{1} , S_{2} be a pair of symmetries of a Riemann surface F_{g} of genus g > 1 with S_{1}S_{2} of even order n > 2. If sp(S_{i}) = k_{i} and if k_{1} < 0 < k_{2} then

\frac{1}{2}n|k_{1}| + (\frac{1}{2}n − 1)k_{2} ≤ g − 1 + n.

Proof. By Theorems 4.3 and 4.4 we find that the lifted NEC group Δ has a signature of the form

(p; ±; [ ]; \{(… \frac{1}{2}n, n, …, n, \frac{1}{2}n, …), …, (\frac{1}{2}n, …, \frac{1}{2}n), …, ( )^{u_{1}+u_{2}}\})

with an even number of link periods equal to n between link periods equal to \frac{1}{2}n, and where for i = 1, 2, u_{i} of the empty period cycles correspond to reflections that map to S_{i}. Suppose that there are 2s link periods equal to n and r link periods equal to \frac{1}{2}n. By Lemma 4.1 we need the connecting generators to map to the identity so by Section 1.4 we see that each empty period cycle contributes two mirrors and then by constructing the appropriate Hoare diagram we find that

\begin{aligned}k_{2} & = r + s + 2u_{2}, \\ k_{1} & = s + 2u_{1}. \end{aligned}

If the total number of period cycles is k ≥ u_{1} + u_{2} + 1 then the Riemann–Hurwitz formula now gives

2g − 2 = 2n(εp − 2 + k + s(1 − \frac{1}{n}) + \frac{r}{2}(1 − \frac{2}{n}))

≥ n(2u_{1} + 2u_{2} − 2 + 2s(1 − \frac{1}{n}) + r(1 − \frac{2}{n}))

≥ n(2h_{1} + 2h_{2} + 2u + v) − 2u − 2v − 2h_{1} − 2n ≥ (n − 2)(s + r + 2u_{1}) + n(s + 2u_{2}) − 2n ≥ (n − 2)k_{2} + n|k_{1}| − 2n and the result follows.

Corollary 4.14. With the notation of Theorem 4.12, if S_{1}S_{2} has order n ≥ 4 then

2|k_{1}| + k_{2} ≤ g + 3.

Proof. We can write the inequality of Theorem 4.12 in the form

2|k_{1}| + k_{2} + (\frac{1}{2}n − 2)(|k_{1}| + k_{2}) ≤ g + n − 1.

As |k_{1}| ≥ 1, |k_{2}| ≥ 1, the result follows.

Corollary 4.15. If a Riemann surface F_{g} of genus g admits a non-separating symmetry S_{1} with more than \frac{1}{2}g + 1 mirrors and if S_{2} is another symmetry not commuting with S_{1} then S_{2} must be non-separating.

5. Summary

The results 4.7–4.12 give necessary and sufficient conditions for the existence of a pair S_{1} , S_{2} of symmetries of species k_{1} , k_{2} . However, 4.13 and 4.14 only give necessary conditions on the pair k_{1} , k_{2} . For example, there is no pair of symmetries of a Riemann surface of genus 2 of species −2, +1. Such a pair cannot exist for commuting symmetries by 4.12, and otherwise we have to solve the equations r + s + 2u_{2} = 1, s + 2u_{1} = 2. The only solution is s = u_{2} = 0, r = u_{1} = 1 and then the corresponding NEC group admits no surface-kernel homomorphism onto D_{n} for n > 2.

A way of summarising our results is to say that a symmetric Riemann surface F_{g} of genus g admits (k_{1}, k_{2})_{n} if it admits a pair of symmetries S_{1} , S_{2} of species k_{1} , k_{2} repectively with S_{1}S_{2} having order n. Then we have shown:

If k_{1} ≤ k_{2} ≤ −1 then (k_{1}, k_{2})_{2} exists for all g (Theorem 4.7).

If k_{1} ≤ −1 is even then (k_{1}, 0)_{2} exists if g is odd (Theorem 4.8).

If k_{1} ≤ −1 is odd then (k_{1}, 0)_{4} exists if g is odd and (0, 0)_{2} exists (Theorem 4.9).

If 1 < k_{1} ≤ k_{2} ≤ g + 1 then (k_{1}, k_{2})_{2} exists. If 1 ≤ k_{2} ≤ g + 1, g even and k_{2} ≡ g + 1 mod 4 then (1, k_{2})_{2} exists (Theorem 4.10).

If k_{2} ≥ 1, k_{2} ≡ g − 1 mod 4 and k_{2} ≡ ((2g/n) + 1) mod 4, k_{2} ≤ ((2g/n) + 1), n > 2 then (1, k_{2})_{n} exists (Theorem 4.11).

If k_{1}k_{2} ≥ 0 and if there is a symmetric Riemann surface admitting (k_{1}, k_{2})_{n} for some n, then it appears in the above list with the least positive value of n.

If −g < k_{1} < 0 < k_{2} < g then (k_{1}, k_{2})_{2} exists if and only if k_{1} ≡ k_{2} ≡ g + 1 mod 2 (Theorem 4.12).

If −g ≤ k_{1} < 0 < k_{2} < g, n > 2 is even, and (k_{1}, k_{2})_{n} exists then it follows that \frac{1}{2}n|k_{1}| + (\frac{1}{2}n − 1)k_{2} ≤ g − 1 + n (Theorem 4.13). This implies that 2|k_{1}| + k_{2} ≤ g + 3 (Corollary 4.14).

We would like to thank Emilio Bujalance and Antonio Costa for pointing out an error in a previous version of this paper.

24 Milagros Izquierdo and David Singerman

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Received 3 May 1995