Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 307–322

APPLICATION OF HOARE’S THEOREM TO SYMMETRIES OF RIEMANN SURFACES

E. Bujalance, A.F. Costa, and D. Singerman

Universidad Nacional de Educacion a Distancia, Departamento de Matemáticas Fundamentales

Facultad de Ciencias, Senda del Rey, s/n, E-28040 Madrid, Spain

Universidad Nacional de Educacion a Distancia, Departamento de Matemáticas Fundamentales

Facultad de Ciencias, Senda del Rey, s/n, E-28040 Madrid, Spain University of Southampton, Department of Mathematics Southampton S09 5NH, United Kingdom

Abstract. Let X be a compact Riemann surface of genus g > 1. A symmetry S of X is an anticonformal involution. We write |S| for the number of connected components of the fixed points set of S. Suppose that X admits two distinct symmetries S_{1} and S_{2} ; then we find a bound for |S_{1}| + |S_{2}| in terms of the genus of X and the order of S_{1}S_{2} . We discuss circumstances in which the bound is attained, showing that this occurs only for hyperelliptic surfaces. In this way we generalize a theorem of S.M. Natanzon.

1. Introduction

Let X be a compact Riemann surface of genus g > 1. A symmetry S of X is an anticonformal involution S: X → X and a Riemann surface that admits a symmetry is called symmetric. By Harnack’s theorem [1, 5, 9], the fixed-point set of S is either empty or consists of k ≤ g + 1 disjoint simple closed curves or, as we shall call them, mirrors. We write |S| for the number of mirrors of S, so that |S| is the number of components of the fixed-point set of S.

Suppose that X admits two distinct symmetries S_{1} , S_{2} . If |S_{1}| = |S_{2}| = g+1 then by a theorem of Natanzon [8], S_{1} and S_{2} commute and X is hyperelliptic. Usually, however, the total number of mirrors of S_{1} and S_{2} is much less than 2g + 2. In Theorem 3 we find a sharp upper bound for |S_{1}| + |S_{2}| in terms of the genus of X and the order of S_{1}S_{2} . In particular, if S_{1} and S_{2} do not commute then |S_{1}| + |S_{2}| ≤ g + 2 (Corollary 3).

Theorem 3 follows from a recent theorem of Hoare [3] on subgroups of NEC groups. Using this theorem we find in Section 5 a graphical technique that allows us to determine |S_{1}| and |S_{2}| in terms of the signature of a certain NEC group. We then find, in Theorem 2, a formula for |S_{1}| + |S_{2}| involving this signature.

1991 Mathematics Subject Classification: Primary 20H10, 30F10, 30F50.

The two first-mentioned authors were partially supported by DGICYT PB 89-201 and Science Plan N SC1-CT91-0716.

Finally, in Section 7, we discuss circumstances in which the bound of Theorem 3 is attained. We show that this is the case only for hyperelliptic surfaces, a result which extends Natanzon’s original theorem.

2. NEC groups

A non-Euclidean crystallographic (NEC) group is a discrete group of isometries of the hyperbolic plane H. As usual, we can choose for our model of H, the upper half-plane with the Poincaré metric. Then every isometry is given by a Möbius or anti-Möbius transformation (the latter being a Möbius transformation composed with z → −z̄). We shall assume that an NEC group has compact quotient space. If Δ is such a group then its algebraic and geometric structure is determined by its signature

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

The quotient space H/Δ is then a surface, possibly with boundary, and in the signature h is the genus of H/Δ, k is the number of its boundary components and + or − is used according to whether the surface is orientable or not. The integers m_{1}, …, m_{r} are the proper periods of Δ and represent the branching over interior points of H/Δ in the natural projection p: H → H/Δ. The k brackets (n_{i1}, …, n_{is_{i}}) are the period cycles and represent the branching over the i^{th} hole. The integers n_{ij} are the link periods. The maximal finite subgroups of Δ are either cyclic of order m_{i} (i = 1, …, r) or dihedral of order 2n_{ij} (i = 1, …, k, j = 1, …, s_{i}) and each period represents a conjugacy class of such subgroups.

Associated to the signature (1) we have a presentation of the group Δ and a formula for the area of a fundamental domain for Δ. If σ(Δ) has a + sign then Δ has generators

x_{1}, …, x_{r}  (elliptic elements)
c_{10}, …, c_{1s_{1}}, …, c_{k0}, …, c_{ks_{k}}  (reflections)
e_{1}, …, e_{k}  (orientation preserving elements)
a_{1}, b_{1}, …, a_{h}, b_{h}  (hyperbolic elements)

and relations

\begin{aligned}x_{i}^{m_{i}} & = 1  (i = 1, …, r) \\ c_{i,j−1}^{2} & = c_{ij}^{2} = (c_{i,j−1}c_{ij})^{n_{ij}} = 1,  i = 1, …, k, j = 1, …, s_{i} \\ e_{i}c_{i0}e_{i}^{−1} & = c_{is_{i}}  (i = 1, …, k) \\ x_{1}x_{2} & ⋯ x_{r}e_{1}e_{2} ⋯ e_{k}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{h}b_{h}a_{h}^{−1}b_{h}^{−1} = 1. \end{aligned}

If σ(Δ) has a minus sign then we just replace the hyperbolic generators a_{i}, b_{i} by glide reflection generators a_{1}, …, a_{h} and replace the last relation by

x_{1}x_{2} ⋯ x_{r}e_{1}e_{2} ⋯ e_{k}a_{1}^{2}a_{2}^{2} ⋯ a_{h}^{2} = 1.

The hyperbolic area of a fundamental domain for Δ is given by

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

(see [11]) where ε = 2 if there is a + sign and ε = 1 if there is a − sign. If Λ < Δ is a subgroup of finite index then

(3)  |Δ : Λ| = μ(Λ)/μ(Δ).

For more details about signatures see ([1, 6, 11]).

Now let X be a compact Riemann surface of genus g > 1. Then there is a Fuchsian surface group Γ (i.e. an NEC group of signature (g; +; [ ]{ }) such that X = H/Γ. Let G be a group of automorphisms of X containing a symmetry S. Let Δ be the group generated by all the liftings to H of the elements of G, then Δ is an NEC group and there is a smooth homomorphism θ: Δ → G whose kernel is Γ. (θ is smooth means that θ maps finite subgroups of Δ isomorphically into G.)

Let L = ⟨S⟩, (the group generated by S) and Λ = θ^{−1}(L). We then have

(4)  X/L = H/Γ/Λ/Γ = H/Λ.

For any NEC group Φ, we let Φ^{+} (the canonical Fuchsian group of Φ) denote the subgroup of index one or two consisting of the orientation-preserving elements of Φ. As Γ contains only orientation-preserving transformations and is torsionfree, Λ contains orientation-reversing elements. As |Λ : Γ| = 2, Λ^{+} = Γ so that Λ has no elliptic elements. Therefore Λ has a signature of the form

(5)  (h_{0}; ±; [ ]{( )^{k}})

where the notation signifies that there are k empty period cycles. Thus k is the number of period cycles of Λ which is the number of mirrors of S. Conversely it is clear that if there is an NEC group Λ with signature (5) and containing Γ as subgroup of index two then Λ defines a symmetry on X with exactly k mirrors. This shows that the number of mirrors of a symmetry is an algebraic invariant of an NEC group.

3. Dihedral group actions

We now suppose that X admits two distinct symmetries S_{1} , S_{2} . We let G = ⟨S_{1}, S_{2}⟩ so that G is isomorphic to the dihedral group D_{n} of order 2n. We also let Δ be the lift of G to H, L_{i} = ⟨S_{i}⟩ and Λ_{i} = θ^{−1}(L_{i}), (i = 1, 2), where θ: Δ → G is the homomorphism of Section 2. We have a subgroup diagram

[figure: Δ Λ_{1}  Λ_{2} Γ]

where Λ_{i} has signature (h_{i}; ±; [ ]{( )^{k_{i}}}).

Now given the signature of an NEC group, Hoare’s theorem [3] gives us a procedure for calculating the signature of a subgroup, given the permutation representation of the group on the cosets. We will use the techniques of this theorem to compute k_{1} , k_{2} from the signature of Δ. We find that the algebra in Hoare’s results gives us fairly precise information about k_{1} + k_{2} = |S_{1}| + |S_{2}|.

We shall find the following simple lemma, also used by Hoare, useful.

Lemma 1. Let D_{n} act by right multiplication on the n L_{i} -cosets (i = 1, 2). If n is even then S_{1} fixes exactly two L_{1} -cosets and no L_{2} -cosets while if n is odd then S_{1} fixes exactly one L_{1} -coset and one L_{2} -coset.

Proof. Let Q = S_{1}S_{2} . Then the n L_{1} -cosets are L_{1}Q^{r} (r = 0, …, n−1) and the action of S_{1} is L_{1}Q^{r} → L_{1}Q^{r}S_{1} . If n is even the only fixed cosets correspond to r = 0, n/2 while if n is odd the only fixed cosets correspond to r = 0. If in the action on the L_{2} -cosets, S_{1} fixes L_{2}Q^{r} then Q^{r}S_{1}Q^{−r} = S_{2} and thus Q^{2r+1} = I. Hence there are no fixed cosets if n is even and one if n is odd.

Another way of stating this lemma is as follows:

Lemma 1′. Let D_{n} act by right multiplication on the n L_{1} -cosets. If n is even then S_{1} fixes exactly two L_{1} -cosets and S_{2} fixes no L_{1} -cosets. If n is odd then both S_{1} and S_{2} fix exactly one L_{1} -coset.

We now explain how Hoare’s ideas apply. Let c ∈ Δ be a reflection so that θ maps c to a conjugate of S_{1} or S_{2} in G ≅ D_{n}. The action of c on the Λ_{i} -cosets is the same as the action of θ(c) on the L_{i} -cosets. Suppose that d is another reflection in Δ and that cd has finite order μ. Then cd, in its action on the Λ_{i} -cosets, gives a product of disjoint μ-cycles. (Otherwise some power of cd, not equal to the identity is an elliptic element of Γ which is impossible as Γ is a surface group.) From Lemma 1, each of c and d fixes two cosets or no cosets if μ is even and each fixes 1 coset if μ is odd. By Theorem 1, II(ii) of [3], (or an extension of Lemma 1) each cycle of cd contains either two or none of these fixed cosets.

Suppose that c fixes a coset Λ_{1}α. Then c_{α} = αcα^{−1} is a reflection in Λ_{i} which is called a reflection induced by c or just an induced reflection. Consider a cycle of cd which contains α (we now identify α and Λ_{i}α). By the above discussion this cycle contains another fixed point β of c, if μ is even, or a fixed point β of d, if μ is odd. In the first case we say that c_{α} and c_{β} are linked and in the second case we say that c_{α} and d_{β} are linked and write c_{α} ∼ c_{β} in the first case and c_{α} ∼ d_{β} in the second case. Once all the links are known we put them together to form chains and then each chain (by Hoare’s results) gives a period cycle in Λ_{i}.

Note. In general c_{α} ∼ c_{β} (or c_{α} ∼ d_{β}) means that c_{α}c_{β} (or c_{α} ∼ d_{β}) has finite order and so gives a link period in a period cycle of Λ_{i}. In our cases the period cycles of the subgroups Λ_{i} are empty so that c_{α} ∼ c_{β} means that (c_{α}c_{β})^{1} = 1 or c_{α} = c_{β} (or c_{α} ∼ d_{β} in the other case).

Before we consider the general situation it might be helpful to look at some examples.

4. Examples

Example 1. Let n = 12 and Δ have signature

(0; +; [ ]; {(2, 4, 4, 6, 3)}),

and presentation

\begin{aligned}⟨c_{i}, (1 ≤ i ≤ 5)|c_{i}^{2} = 1 & (1 ≤ i ≤ 5), \\ & (c_{1}c_{2})^{2} = (c_{2}c_{3})^{4} = (c_{3}c_{4})^{4} = (c_{4}c_{5})^{6} = (c_{5}c_{1})^{3} = 1⟩. \end{aligned}

Consider the following homomorphism

θ: Δ → D_{12} = ⟨S_{1}, Q|S_{1}^{2} = (S_{1}Q)^{2} = Q^{12} = 1⟩;

θ(c_{1}) = S_{1}, θ(c_{2}) = S_{1}Q^{6}, θ(c_{3}) = S_{1}Q^{9}, θ(c_{4}) = S_{1}Q^{6}, θ(c_{5}) = S_{1}Q^{8}.

We have the following decomposition of D_{12} as a union of L_{1} -cosets (L_{1} = ⟨S_{1}⟩):

D_{12} = ⋃_{i=0}^{11} L_{1}Q^{i}

and letting i denote the coset L_{1}Q^{i} we obtain the following permutation representation of D_{12} on the right L_{1} -cosets,

\begin{matrix}c_{1} → & (0) & (6) & (1 & 11) & (2 & 10) & (3 & 9) & (4 & 8) & (5 & 7) \\ c_{2} → & (3) & (9) & (1 & 5) & (2 & 4) & (7 & 11) & (8 & 10) & (0 & 6) \\ c_{3} → & (0 & 9) & (1 & 8) & (2 & 7) & (3 & 6) & (4 & 5) & (10 & 11) \\ c_{4} → & (3) & (9) & (1 & 5) & (2 & 4) & (7 & 11) & (8 & 10) & (0 & 6) \\ c_{5} → & (4) & (10) & (1 & 7) & (2 & 6) & (3 & 5) & (0 & 8) & (11 & 9). \end{matrix}

The induced reflections are

c_{1,0}, c_{1,6}, c_{2,3}, c_{2,9}, c_{4,3}, c_{4,9}, c_{5,4}, c_{5,10}.

We now form the products

\begin{matrix}c_{1}c_{2} → & (0 & 6) & (1 & 7) & (2 & 8) & (3 & 9) & (4 & 10) & (5 & 11) \\ c_{2}c_{3} → & (0 & 3 & 6 & 9) & (1 & 4 & 7 & 10) & (2 & 5 & 8 & 11) \\ c_{3}c_{4} → & (0 & 9 & 6 & 3) & (1 & 10 & 7 & 4) & (2 & 11 & 8 & 5) \\ c_{4}c_{5} → & (1 & 3 & 5 & 7 & 9 & 11) & (2 & 4 & 6 & 8 & 10 & 0) \\ c_{5}c_{1} → & (1 & 5 & 9) & (2 & 6 & 10) & (3 & 7 & 11) & (4 & 8 & 0). \end{matrix}

As fixed points 0, 6 of c_{1} belong to the same cycle of c_{1}c_{2} we have a link c_{1,0} ∼ c_{1,6} . Similarly, we have links c_{2,3} ∼ c_{2,9} , (from c_{2}c_{3} ), c_{4,3} ∼ c_{4,9} (from c_{3}c_{4} ), c_{4,3} ∼ c_{4,9} (from c_{4}c_{5} ), c_{5,4} ∼ c_{5,10} (from c_{4}c_{5} ), c_{5,4} ∼ c_{1,0} (from c_{5}c_{1} ), c_{5,10} ∼ c_{1,6} (from c_{5}c_{1} ). We then get the chains

\begin{aligned}c_{1,0} & ∼ c_{1,6} ∼ c_{5,10} ∼ c_{5,4} ∼ c_{1,0} \\ c_{2,3} & ∼ c_{2,9} ∼ c_{2,3} \\ c_{4,3} & ∼ c_{4,9} ∼ c_{4,3}. \end{aligned}

As there are three chains, Hoare’s theorem implies that Λ_{1} has three period cycles and hence |S_{1}| = 3. Similarly, we have the action of the generators on the L_{2} - cosets,

\begin{matrix}c_{1} → & (0 & 11) & (1 & 10) & (2 & 9) & (3 & 8) & (4 & 7) & (5 & 6) \\ c_{2} → & (0 & 5) & (1 & 4) & (2 & 3) & (6 & 11) & (7 & 10) & (8 & 9) \\ c_{3} → & (4) & (10) & (0 & 8) & (1 & 7) & (2 & 6) & (3 & 5) & (9 & 11) \\ c_{4} → & (0 & 5) & (1 & 4) & (2 & 3) & (6 & 11) & (7 & 10) & (8 & 9) \\ c_{5} → & (0 & 7) & (1 & 6) & (2 & 5) & (3 & 4) & (8 & 11) & (9 & 10), \end{matrix}

and notice how Lemma 1 applies: e.g. c_{1} fixes 2 L_{1} -cosets and 0 L_{2} -cosets, c_{3} fixes 0 L_{1} -cosets and 2 L_{1} -cosets. The induced reflections are c_{3,4} and c_{3,10} and from the products c_{3}c_{4} and c_{4}c_{5} we find the single chain c_{3,4} ∼ c_{3,10} ∼ c_{3,4} . Thus |S_{2}| = 1. This example suggests that Lemma 1 should restrict the size of |S_{1}| + |S_{2}|. As we shall see in Theorem 1 this is indeed the case, but before then we consider another example which illustrates other aspects of Hoare’s techniques.

Example 2. Let n = 3 and Δ have signature

(0; +; [3]; {(3, 3, 3)}),

and presentation

\begin{aligned}⟨e, x, c_{0}, c_{1}, c_{2}, c_{3}|c_{i}^{2} = 1 & (0 ≤ i ≤ 3), \\ & x^{3} = ex = (c_{0}c_{1})^{3} = (c_{1}c_{2})^{3} = (c_{2}c_{3})^{3} = ec_{0}e^{−1}c_{3} = 1⟩. \end{aligned}

Consider the following homomorphism

θ: Δ → D_{3} = ⟨S_{1}, Q|S_{1}^{2} = (S_{1}Q)^{2} = Q^{3} = 1⟩;
θ(c_{0}) = S_{1}, θ(c_{1}) = S_{1}Q, θ(c_{2}) = S_{1}, θ(c_{3}) = S_{1}Q, θ(e) = Q.

With a similar notation to Example 1,

D_{3} = ⋃_{i=0}^{2} L_{1}Q^{i}

and we have the following permutation representation on the cosets

\begin{matrix}c_{0} → & (0) & (1 & 2) \\ c_{1} → & (0 & 1) & (2) \\ c_{2} → & (0) & (1 & 2) \\ c_{3} → & (0 & 1) & (2) \\ e → & (0 & 1 & 2) \end{matrix}

giving induced reflections c_{0,0} , c_{1,2} , c_{2,0} , c_{3,2} on Λ_{1} . As

\begin{matrix}c_{0}c_{1} → & (0 & 1 & 2), \\ c_{1}c_{2} → & (0 & 2 & 1), \\ c_{2}c_{3} → & (0 & 1 & 2), \\ (ec_{0}e^{−1})c_{3} → & (0) & (1) & (2), \end{matrix}

we get links c_{0,0} ∼ c_{1,2} , c_{1,2} ∼ c_{2,0} , c_{2,0} ∼ c_{3,2} , from the first three products. In the final product c_{0} fixes 0, ec_{0}e^{−1} fixes 2 and c_{3} also fixes 2. By Theorem 1 (II(ii)) and Example 1 of [3], we have a link c_{3,2} ∼ c_{0,0} . This gives the single chain c_{0,0} ∼ c_{1,2} ∼ c_{2,0} ∼ c_{3,2} ∼ c_{0,0} and thus |S_{1}| = 1. As S_{2} is conjugate to S_{1} in D_{3} , |S_{2}| = 1 as well. As we shall see, this case is typical of the cases when n, and hence all the periods of Δ are odd.

5. A graphical technique

We now describe a method that enables us to perform the computations in Section 4 fairly automatically. The examples there show that the number of mirrors depend on the parity of n, the parities of the link periods and the parities of the integers i in S_{1}Q^{i}. We suppose that n is even. If θ(c_{j}) = S_{1}Q^{i} with i even then c_{j} gives a permutation of the cosets with two fixed points α, β; if i is odd then there are no fixed points. If the link period n_{j} is even (recall the relation (c_{j}c_{j+1})^{n_{j}} = 1) then α, β lie in the same cycle of the (permutation induced) by c_{j}c_{j+1} , giving a link c_{j,α} ∼ c_{j,β}. (All suffices are modulo s the length of the period cycle.) If n_{j−1} is even then we get a link c_{j,β} ∼ c_{j,α} giving a chain c_{j,α} ∼ c_{j,β} ∼ c_{j,α}.

If n_{j} is odd then the cycle c_{j}c_{j+1} containing α also contains a fixed point γ of c_{j+1} and the cycle containing β contains a fixed point δ of c_{j+1} . We then get links c_{j,α} ∼ c_{j+1,δ} and c_{j,β} ∼ c_{j+1,δ}. If n_{n+1} is odd we get a link c_{j+1,δ} ∼ c_{j+2,ε} and so on (and this argument shows that |S_{1}| = 1 if all the link periods of the period cycle are odd). If n_{j+1} is even we get c_{j+1,δ} ∼ c_{j+1,γ} ∼ c_{j,α} ∼ · · ·. This process can be illustrated graphically as follows. Suppose that the period cycle has length s. Represent each generator c_{j} by a vertex of an s-gon labelled c_{j}. If θ(c_{j}) = S_{1}Q^{i}, colour the vertex black or white according as to whether i is even or odd respectively. If c_{j} and c_{j+1} are two black vertices we join them by a black edge if and only if n_{j} is odd. Then |S_{1}| is the number of black components. To find |S_{2}| we just change the colour of the vertices (as S_{2}Q^{i} = S_{1}Q^{i+1} , changing the parity of the index) and still join black vertices by a black edge if n_{j} is odd.

Example. From Example 1 of Section 4 we obtain the following pictures.

\begin{aligned}& & c_{1} \\ c_{5} & & & • c_{2} \\ & c_{4}• & & ∘ c_{3} \\ & & |S_{1}| = 3 \end{aligned} ; [figure: •; •]  \begin{aligned}& ∘ c_{1} \\ c_{5}∘ & & ∘ c_{2} \\ c_{4}∘ & & • c_{3} \\ & |S_{2}| = 1 \end{aligned}

From these graphical ideas we get a simple numerical result.

Notation. We let t = |S_{1}|+|S_{2}| the total number of fixed curves of S_{1} and S_{2} .

Theorem 1. With the previous notation, let n be even and suppose that Δ has a single period cycle and at least one even period. Then t is equal to the number of even link periods in the period cycle.

Proof. Suppose that the period cycle has length s. In terms of the above graphs we need to show that if the total number of black edges in one graph is u (this is the number of odd link periods) then the number of black components in both graphs is s − u. This is by induction on u. If u = 0 then the total number of black components is the number of black vertices in both graphs which is the number of vertices in one graph, namely s. Suppose the result is true if there are u black edges. Now increase the number of black edges by one. A black edge cannot join vertices of different colours for if θ(c_{j}) = SQ^{i_{1}} and θ(c_{j+1}) = SQ^{i_{2}} then θ(c_{j}c_{j+1}) = Q^{i_{2}−i_{1}} . As n is even and c_{j}c_{j+1} has odd period, i_{2} ≡ i_{1} mod 2. Thus if we introduce another black edge it joins two black vertices in one of the graphs so that the total number of black components is (s − u) − 1 = s − (u + 1), and the proof follows.

We now extend this result to the case when Δ is a general NEC group. Let us define a non-empty period cycle to be odd if it only contains odd link periods. Then, as we saw near the beginning of Section 5, an odd period cycle induces just one mirror on U/Λ_{1} and one on U/Λ_{2} and so contributes 2 to t = |S_{1}| + |S_{2}|. Now consider an empty period cycle. The corresponding generators are c_{i} and e_{i}. These obey the relation e_{i}c_{i}e_{i}^{−1} = c_{i}. With the previous notation let θ(c_{i}) = S_{1} . As the centralizer of S_{1} in G is {1} if n is odd and {1, Q^{n/2}} if n is even, θ(e_{i}) = 1 if n is odd while θ(e_{i}) = 1 or Q^{n/2} if n is even. In the case when n is even the permutation on the L_{1} -cosets induced by c_{i} is

(0)(\frac{n}{2})(1 n − 1)(2 n − 2) …

and the permutation induced by e_{i} is either

(0)(1) … (n − 1) if θ(e_{i}) = 1

or

(0 \frac{n}{2})(1 \frac{n}{2} + 1) … if θ(e_{i}) = Q^{n/2}.

The induced reflections are c_{i,0} and c_{i,n/2} . If θ(e_{2}) = 1 we get two chains c_{i,0} ∼ c_{i,0} and c_{i,n/2} ∼ c_{i,n/2} and if θ(e_{i}) = Q^{n/2} we get a single chain c_{i,0} ∼ c_{i,n/2} ∼ c_{i,0} . (Compare Example 2 in Section 4.) Thus if e_{i} ∈ Γ then every empty period cycle gives rise to two mirrors of S_{1} (and none of S_{2} by Lemma 1) and so contributes 2 to t, whilst if e ∈ Γ then every empty period cycle gives rise to one mirror of S_{1} , and none of S_{2} , contributing 1 to t.

Notation. We let α be the number of even link periods of Δ, β the number of odd period cycles of Δ, γ the number of empty period cycles of Δ and δ the number of e_{i} generators associated to the empty period cycles which do not belong to Γ.

We proved the following result.

Theorem 2. With the above notation:

(i) if n is even then t = α + 2β + 2γ − δ, (ii) if n is odd then t = 2β + 2γ.

Note. If n is even then we usually let θ(e_{i}) = 1 so that δ = 0. If n is odd then |S_{1}| = |S_{2}| = β + γ.

6. The main application

Theorem 3. Let S_{1}S_{2} have order n.

(i) If n is even then t ≤ (4g/n) + 2. (ii) If n is odd then t ≤ ((2g − 2)/n) + 4.

Proof. (i) Let Δ have signature (1). Then by [11], Δ^{+} is a Fuchsian group of signature

(h^{+}; +; [m_{1}, m_{1}, …, m_{r}, m_{r}, n_{11}, …, n_{1s_{1}}, …, n_{ks_{k}}]{ })

where h^{+} = εh + k − 1, ε being defined in (2). As there is an epimorphism from Δ^{+} onto C_{n} we know by Harvey’s theorem [2] that

(i) If m is the least common multiple (lcm) of the periods of Δ^{+} then m|n and also these periods obey the lcm condition which says that the lcm of all the periods with one period deleted is also equal to m.

(ii) If h^{+} = 0, then m = n. (iii) The number of periods of Δ^{+} cannot equal 1 and if h^{+} = 0 their number if

≥ 3. (iv) If 2|m then the number of periods divisible by the maximum power of 2

dividing m is even.

Besides the notation α, β, γ introduced earlier we also let η equal the number of period cycles with at least one even link period. Then k = β + γ + η. Our proof is just to apply the area and index formulae (2) and (3) but we need to break up the calculation into several cases.

(a) r ≥ 1. From (2) and (3)

\begin{aligned}2g − 2 & ≥ 2n(−2 + k + ∑_{i=1}^{n}(1 − \frac{1}{m_{1}} ) + \frac{1}{2} ∑_{i=1}^{k} ∑_{j=1}^{s_{i}}(1 − \frac{1}{n_{ij}} )) \\ & ≥ 2n(−2 + β + γ + η + \frac{1}{2} + \frac{α}{4}). \end{aligned}

If α ≠ 0, then η ≥ 1 so that

2g − 2 ≥ 2n(β + γ + \frac{α}{4} − \frac{1}{2}) ≥ 2n(\frac{t}{4} − \frac{1}{2}).

Hence t ≤ 4(g − 1)/n + 2 < (4g/n) + 2. If α = 0 then η = 0 and now

2g − 2 ≥ 2n(β + γ − \frac{3}{2}) ≥ 2n(\frac{t − 3}{2}).

Therefore

t ≤ \frac{2(g − 1)}{n} + 3 < \frac{4g}{n} + 2,

the final inequality being true unless n ≥ 2g + 2; but in this case it follows from the proof of Theorem 4 of [12] that Δ^{+} is a triangle group or has signature (0; +; [2, 2, m, m]{ }) with m odd. As Δ^{+} has at least one proper period, Δ has at most two even link periods so that t ≤ 2, by Theorem 2.

We can now assume that r = 0.

(b) α = 0. We first assume that Δ has no link periods and so has signature (h; ±; [ ]{( )^{k}}) with εh − 2 + k ≥ 1 as μ(Δ) > 0. Then t = 2k by Theorem 2. Also 2g − 2 = 2n(εh − 2 + k) ≥ 2n giving n ≤ g − 1. Hence 2g − 2 ≥ 2n((t/2) − 2) giving

t ≤ \frac{2(g − 1)}{n} + 4 < \frac{4g}{n} + 2

as n < g + 1.

If Δ has link periods then they are all odd. If there are β_{0} odd link periods then β_{0} ≥ 2 by (i) of Harvey’s theorem, and clearly β_{0} ≥ β. Also, by (iii) of Harvey’s theorem h^{+} > 0 as m is odd and n is even. Therefore μ(Δ^{+}) ≥ 2h^{+} − 2 + \frac{2}{3}β_{0} ≥ \frac{4}{3} . As 2g − 2 = nη(Δ^{+}), n < \frac{3}{2}(g − 1). Now

\begin{aligned}2g − 2 & = 2n(εh − 2 + k + \frac{1}{2} ∑_{i=1}^{k} ∑_{j=1}^{s_{i}}(1 − \frac{1}{n_{ij}} )) \\ & ≥ 2n(k − 2 + \frac{2}{3}) = 2n(\frac{t}{2} − \frac{4}{3}). \end{aligned}

Therefore t ≤ (2(g − 1)/n) + 8/3 < (4g/n) + 2 as n < \frac{3}{2}(g + 1).

Thus from now on we can assume that r = 0, α ≥ 1 and thus η ≥ 1.

(c) We now assume that h^{+} > 0. Then

\begin{aligned}2g − 2 & ≥ 2n(εh + β + γ + η − 2 + \frac{α}{4}) \\ & = 2n(εh + \frac{t}{4} + \frac{β + γ}{2} + η − 2). \end{aligned}

If β = γ = 0, then h^{+} = εh+η−1 ≥ 1 so that 2g−2 ≥ 2nt/4 and t ≤ (4g−4)/n. If β + γ > 0, then εh + η + (β + γ)/2 ≥ \frac{3}{2} , and 2g − 2 ≥ 2n((t/4) − \frac{1}{2}) giving

t ≤ \frac{4(g − 1)}{n} + 2 < \frac{4g}{n} + 2.

(d) We now assume that h^{+} = 0 so that k = 1. If the unique period cycle is odd or empty then t = 2 so that we can assume that it contains even link periods.

Thus Δ^{+} has signature (0; +; [n_{1}, …, n_{t}, ν_{1}, …, ν_{τ}]) where the n_{i} are even and the ν_{i} are odd. Then

\begin{aligned}\frac{2g − 2}{n} & = −2 + ∑_{i=1}^{t}(1 − \frac{1}{n_{i}} ) + ∑_{i=1}^{τ}(1 − \frac{1}{ν_{i}} ) \\ & ≥ −2 + \frac{t}{2} + ∑_{i=1}^{τ}(1 − \frac{1}{ν_{i}} ). \end{aligned}

Now if τ > 0 then τ ≥ 2 by Harvey’s theorem, part (i). Therefore

\frac{2g − 2}{n} ≥ −2 + \frac{t}{2} + \frac{4}{3}

and thus

t ≤ \frac{4g − 4}{n} + \frac{4}{3} < \frac{4g}{n} + 2.

Thus we can assume that τ = 0, i.e. that all the link periods in the period cycle are even. We assume that u of them are equal to 2 and v are greater than 2. We denote the latter by n_{1}, …, n_{v}. Then

\begin{aligned}2g − 2 & = n(−2 + \frac{u}{2} + ∑_{i=1}^{v}(1 − \frac{1}{n_{i}} )) ≥ n(−2 + \frac{u}{2} + \frac{3v}{4} ) \\ & = n(−2 + \frac{u + v}{2} + \frac{v}{4} ) ≥ n(−1 + \frac{t}{2}) \end{aligned}

if v ≥ 4, by Theorem 2. Now t ≤ (4(g − 1)/n) + 2 < (4g/n) + 2. If v = 3, then {n_{1}, n_{2}, n_{3}} ≠ {4, 4, 4}, {4, 4, 6} or {4, 6, 6} by conditions (i) and (iv) of Harvey’s theorem and thus ∑_{1}^{3} 1/n_{i} ≤ \frac{1}{2} , so that

2g − 2 = n(1 + \frac{u}{2} − ∑_{1}^{3} \frac{1}{n_{i}} ) ≥ n(\frac{u + 1}{2}).

Thus

t = u + 3 ≤ \frac{4(g − 1)}{n} + 2 < \frac{4g}{n} + 2.

If v = 2 then by the lcm condition Δ^{+} has signature (0; +; [2, 2, …, n, n]; { }) and as n is even t = (4g/n) + 2. We cannot have v = 1 by the lcm condition and if v = 0 then n = 2 and t = 2g + 2 = (4g/2) + 2.

(ii) This is much easier! We have

2g − 2 ≥ 2n(−2 + k) = 2n(−2 + β + γ) = 2n(\frac{t}{2} − 2)

by Theorem 2. Thus t ≤ (2(g − 1)/n) + 4.

Notes. The bound in Theorem 3 (i) is only obtained if and only if Δ has signature (0; +; [ ]; {2, …, 2, n, n}) with n even. A similar bound, but with topological hypotheses, has been found by Natanzon in [9].

7. Attainment of the bounds

We show that the bounds in Theorem 3 are attained and then study some of the Riemann surfaces which attain these bounds. First we prove the following lemma which is of independent interest.

Lemma 2. Let Λ be a Fuchsian group of signature (0; +; [m_{1}, …, m_{r}]; { }) and suppose that there exists a homomorphism φ: Λ → C_{n}. If Λ^{∗} is an NEC group of signature (0; +; [ ]{(m_{1}, …, m_{r}); }) with (Λ^{∗})^{+} = Λ then φ extends to a homomorphism φ^{∗}: Λ^{∗} → D_{n}. This extension is essentially unique.

Proof. Suppose that Λ has presentation ⟨x_{1}, …, x_{r} | x_{1}^{m_{1}} = ⋯ = x_{r}^{m_{r}} = x_{1}x_{2} ⋯ x_{r} = 1⟩ and suppose that C_{n} is generated by q and that φ(x_{i}) = q^{σ_{i}}. Then n | ∑_{i=1}^{r} σ_{i}. Let

D_{n} = ⟨y, q | y^{2} = q^{n} = (yq)^{2} = 1⟩

and

Λ^{∗} = ⟨c_{1}, …, c_{r} | c_{1}^{2} = ⋯ = c_{r}^{2} = (c_{1}c_{2})^{m_{1}} = ⋯ = (c_{r−1}c_{r})^{m_{r−1}} = (c_{r}c_{1})^{m_{r}} = 1⟩,

with c_{i}c_{i+1} = x_{i}.

If we let φ^{∗}(c_{1}) = y then we must have φ^{∗}(c_{2}) = yq^{σ_{1}} , φ^{∗}(c_{3}) = yq^{σ_{1}+σ_{2}} , …, φ^{∗}(c_{r}) = yq^{σ_{1}+···+σ_{r−1}} . φ^{∗}: Λ^{∗} → D_{n} is a homomorphism that extends φ, and it is unique once the image of c_{1} is known. As the image of c_{1} must be a reflection any two extensions differ by an automorphism of D_{n}.

Theorem 4. (i) The bound in Theorem 3 (i) is attained for every even n and every g > 1 such that n|4g.

(ii) The bound in Theorem 3 (ii) is attained for every odd n and every g > 1 such that n|g − 1.

Proof. (i) Let Δ be an NEC group of signature (0; +; [ ]{(2^{(4g/n)}, n, n); }) (4g/n periods equal to 2). Then Δ^{+} has signature (0; +; [2^{(4g/n)}, n, n]; { }) and as n is even there is a homomorphism θ: Δ^{+} → C_{n} = ⟨Q⟩. This takes the involution generators of Δ^{+} to Q^{n/2} and the generators of order n to Q and Q^{−1} . By Lemma 2, θ extends to a homomorphism θ^{∗}: Δ → D_{n}. If Γ is the kernel of θ^{∗} then X = H/Γ is the surface we require.

(ii) We let Δ_{1} have signature (0; +; [ ]{( )((g − 1)/n) + 2}). There is clearly a smooth homomorphism θ_{1}: Δ_{1} → D_{n} and if Γ_{1} is the kernel of θ_{1} then X_{1} = H/Γ_{1} as the required surface.

Theorem 5. Let X be a Riemann surface for which the bound of Theorem 3 (i) is attained. Then X is hyperelliptic with (S_{1}S_{2})^{n/2} being the hyperelliptic involution.

Proof. The proof of Theorem 3 (i) shows that if X = H/Γ is a surface for which the bound is attained then Γ must be a normal subgroup of an NEC group Δ of signature {0; +; [ ]{(2^{(4g/n)}n, n)}} and Theorem 4 shows that Γ must be the kernel of θ^{+}: Δ^{+} → C_{n}. By [10] we calculate that (θ^{+})^{−1}(⟨(S_{1}S_{2})^{n/2}⟩) has signature (0; +; [2^{(2g+2)}]; { }) so by [7], (S_{1}S_{2})^{n/2} is the hyperelliptic involution.

Finally, we investigate the number of mirrors of each class of symmetries of the surfaces for which the bound of Theorem 3 (i) is attained. If Δ, Δ^{+} are as in the proof of Theorem 4 (i), then with the notation of the proof of Lemma 2, the homomorphism θ^{+}: Δ^{+} → C_{n} is given by x_{1} → Q^{n/2}, …, x_{4g/n} → Q^{n/2} , x_{(4g/n)+1} → Q, x_{(4g/n)+2} → Q^{−1} , (this is unique up to automorphisms of C_{n}) and then the homomorphism θ: Δ → D_{n} is given, in the case 4g/n even, by c_{1} → S_{1} , c_{2} → S_{1}Q^{n/2} , c_{3} → S_{1} , …, c_{4g/n} → S_{1}/Q^{n/2} , c_{(4g/n)+1} → S_{1} , c_{(4g/n)+2} → S_{1}Q and in the case 4g/n odd by

\begin{aligned}c_{1} & → S_{1}, c_{2} → S_{1}Q^{n/2}, …, c_{4g/n} → S_{1}, \\ c_{(4g/n)+1} & → S_{1}Q^{n/2}, c_{(4g/n)+2} → S_{1}Q^{n/2+1}. \end{aligned}

As there are no odd link periods, the graphs of Section 5 contain no black edges and the number of fixed curves of S_{1} is the number of black vertices, i.e. the number of even exponents of Q. We immediately obtain

Theorem 6. If 4|n then the surfaces X for which the bounds of Theorem 3 (i) are attained admit two classes of symmetries, one with (4g/n)+1 mirrors and the other with 1 mirror. If n ≡ 2 mod 4 then both classes of symmetries have (2g/n) + 1 mirrors.

We consider some corollaries concerning “M -symmetries” [8]. These are symmetries with g + 1 mirrors, the maximum possible number by Harnack’s bound, and they are the symmetries to which most attention has been paid in the literature. From Theorems 3 (i) and 5 we obtain Natanzon’s theorem [8].

Corollary 1. If a Riemann surface X admits two M -symmetries S_{1} , S_{2} then S_{1}S_{2} has order 2. Also, X is hyperelliptic and S_{1}S_{2} is the hyperelliptic involution.

Also from Theorem 3 we deduce

Corollary 2. If X admits an M -symmetry S_{1} and another symmetry S_{2} then S_{1}S_{2} has order 2 or 4.

Corollary 3. If S_{1} and S_{2} do not commute then |S_{1}| + |S_{2}| ≤ g + 2 and this bound is attained for every g ≥ 2.

Proof. The order of S_{1}S_{2} is not equal to 2. We see from Theorem 3 that |S_{1}| + |S_{2}| ≤ g + 2 if n is even and, by Theorem 4, this bound is attained for every g with n = 4. If n is odd then |S_{1}| + |S_{2}| ≤ \frac{2}{3}(g − 1) + 4 and this bound is attained only if 3|g − 1. Thus for n odd, |S_{1}| + |S_{2}| ≤ g + 2 with equality only for g = 4, n = 3.

Another important question about a symmetry S of a Riemann surface X is to decide whether its mirrors separate X, (e.g. see [1]). This is equivalent to whether X/⟨S⟩ is orientable or not, or the NEC group Λ of (5) has orientable quotient space or not. The techniques of [4] allow us to do this. We note that the mirrors of an M -symmetry always separate. We do not go into the details here but it can be shown that in Theorem 6 that when n > 4, the mirrors of all symmetries do not separate. If n = 4, then one class of symmetries consists of M -symmetries and by the above remark the mirrors of each such symmetry do separate. The other class of symmetries have only one mirror which can be shown to be non-separating.

322 E. Bujalance, A.F. Costa, and D. Singerman

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Received 25 February 1992