Pseudo-real Riemann surfaces and chiral regular maps

Emilio Bujalance Dept. Matemáticas Fund., Fac. Ciencias, UNED Senda del rey, 9, 28040 Madrid, SPAIN eb@mat.uned.es

Partially supported by MTM2005-01637

Marston Conder Department of Mathematics, University of Auckland Private Bag 92019, Auckland, NEW ZEALAND m.conder@auckland.ac.nz

Partially supported by N.Z. Marsden Fund UOA0412

Antonio F. Costa Dept. Matemáticas Fund., Fac. Ciencias, UNED Senda del rey, 9, 28040 Madrid, SPAIN acosta@mat.uned.es

Partially supported by MTM2005-01637

September 22, 2006

1 Introduction

A Riemann surface is called pseudo-real if it admits anticonformal automorphisms but no anticonformal involution. Pseudo-real Riemann surfaces appear in a natural way in the study of the moduli space M_{g}^{K} of Riemann surfaces considered as Klein surfaces. The moduli space M_{g} of Riemann surfaces of genus g is a two-fold branched covering of M_{g}^{K}, and the preimage of the branched locus consists of the Riemann surfaces admitting anticonformal automorphisms — which are either real Riemann surfaces (admitting anticonformal involutions) or pseudo-real Riemann surfaces. We study the latter surfaces in this work. Pseudo-real Riemann surfaces are Riemann surfaces that are equivalent to their conjugate but do not have any anticonformal involution. Note that pseudo-real Riemann surfaces have non-trivial conformal automorphisms, hence the locus of pseudo-real Riemann surfaces in M_{g} is contained in the singular set of the orbifold M_{g} = 𝕋_{g}/Mod_{g}. In Section 3 we study general properties of the automorphism groups of these surfaces and the related uniformizing NEC groups. In particular, we prove that there are pseudo-real Riemann surfaces of every genus g ≥ 2. In Section 4 we study pseudo-real surfaces of genus 2 and 3. In recent years a vast literature has been published about real Riemann surfaces — that is, Riemann surfaces admitting anticonformal involutions — but very little is known about pseudo-real surfaces. A study of hyperelliptic pseudo-real Riemann surfaces was made in [S] and [BC], and pseudo-real surfaces with cyclic automorphism group were investigated in [E]. In Section 5 we study pseudo-real cyclic p-gonal Riemann surfaces, generalizing the results obtained in [BC] for hyperelliptic surfaces. In Section 6 we consider the maximal order of the automorphism group of a pseudo-real Riemann surface (relative to its genus), and we establish a connection between pseudo-real surfaces with maximal automorphism group, and chiral 3-valent regular maps. Finally in Section 7 we show there exist pseudo-real surfaces with automorphism group of maximal order for infinitely many genera, by proving the existence of concrete infinite families of chiral regular maps of type {3, k} for k ≥ 7.

2 Preliminaries on Fuchsian and NEC groups

A non-Euclidean crystallographic group (or NEC group) is a discrete group of isometries of the hyperbolic plane 𝔻. We shall assume that an NEC group has a compact orbit space. If Γ is such a group then its algebraic structure is determined by its signature

(h; ±; [m_{1}, …, m_{r}]; {(n_{11}, …, n_{1s_{1}}), …, (n_{k1}, …, n_{ks_{k}})}). The orbit space 𝔻/Γ is a surface, possibly with boundary. The number h is called the genus of Γ and equals the topological genus of 𝔻/Γ, while k is the number of its boundary components, and the sign is + or − according to whether or not the surface is orientable. The integers m_{i} ≥ 2 are called the proper periods, and represent the branched indices over interior points of 𝔻/Γ in the natural projection π : 𝔻 → 𝔻/Γ. The bracketed expressions (n_{i1}, …, n_{is_{i}}), some or all of which may be empty (with s_{i} = 0), are called the period cycles and represent the branchings over the i^{th} hole in the surface, and the numbers n_{ij} ≥ 2 are the link periods.

Associated with each signature [BEGG] there exists a canonical presentation for the group Γ, and a formula for the hyperbolic area of its fundamental domain. If the signature has sign + then Γ has the following generators:

x_{1}, …, x_{r} (elliptic elements),

c_{10}, …, c_{1s_{1}}, …, c_{k0}, …, c_{ks_{k}} (reflections),
e_{1}, …, e_{k} (boundary transformations),
a_{1}, b_{1}, …, a_{g}, b_{g} (hyperbolic elements);

and these generators satisfy the defining relations x_{i}^{m_{i}} = 1 (for 1 ≤ i ≤ r), c_{ij−1}^{2} = c_{ij}^{2} = (c_{ij−1}c_{ij})^{n_{ij}} = 1, c_{is_{i}} = e_{i}^{−1}c_{i0}e_{i} (for 1 ≤ i ≤ k, 0 ≤ j ≤ s_{i}),

x_{1} … x_{r}e_{1} … e_{k}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} … a_{h}b_{h}a_{h}^{−1}b_{h}^{−1} = 1.

If the sign is − then we just replace the hyperbolic generators a_{i}, b_{i} by glide reflections d_{1}, …, d_{h}, and the last relation by x_{1} … x_{r}e_{1} … e_{k}d_{1}^{2} … d_{h}^{2} = 1. The hyperbolic area of an arbitrary fundamental region of an NEC group Γ with signature is given by

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 the sign is +, and ε = 1 if the sign is −. Furthermore, any discrete group Λ of isometries of 𝔻 containing Λ as a subgroup of finite index is also an NEC group, and the hyperbolic area of a fundamental region for Λ is given by the Riemann-Hurwitz formula:

[Λ : Γ] = μ(Γ)/μ(Λ).

For any NEC group Λ, let Λ^{+} denote the subgroup of orientation-preserving elements of Λ, called the canonical Fuchsian subgroup of Λ. If Λ^{+} ≠ Λ then Λ^{+} has index 2 in Λ and we say that Λ is a proper NEC group. Let X be a compact Riemann surface of genus g > 1. Then there is a Fuchsian surface group Γ (that is, an NEC group with signature (g; +; [−]; {−}))) such that X = 𝔻/Γ, and the automorphism group Aut(X) of X is isomorphic to Δ/Γ, where Δ is an NEC group containing Γ. We denote by Aut^{+}(X) the group Δ^{+}/Γ of all orientation-preserving automorphisms of X .

3 Pseudo-real Riemann surfaces

Definition 1 An anticonformal involution is an orientation-reversing automorphism of order 2. A Riemann surface is called pseudo-real if it admits anticonformal automorphisms but has no anticonformal involution.

Proposition 2 Let X be a pseudo-real Riemann surface, and let G be the group of conformal and anticonformal automorphisms of X. Then 4 divides the order of G.

Proof. Let g be any anticonformal automorphism of X. If g has order 2^{a}m where m is odd, then g^{m} is an anticonformal automorphism of order 2^{a}, and then a > 1 because X is pseudo-real. Thus G has an element g^{m} of order divisible by 4. □

Theorem 3 Suppose the pseudo-real surface X is conformally equivalent to 𝔻/Γ, where Γ is a surface Fuchsian group, and Γ is normalized by an NEC group Δ such that Δ/Γ ≅ G = Aut(X). Then the signature of Δ has the form (p; −; [m_{1}, ..., m_{r}]), and if Δ^{+} is the canonical Fuchsian subgroup of Δ, then the signature of Δ^{+} is

(p − 1; +; [m_{1}, m_{1}, m_{2}, m_{2}, ..., m_{r}, m_{r}]).

Proof. Since G has anticonformal automorphisms, the signature of Δ is the signature of a proper NEC group — that is, a group with anticonformal transformations. Thus the signature of Δ has the form

(p; −; [m_{1}, ..., m_{r}]) or (p; ±; [m_{1}, ..., m_{r}]; {(n_{i1}, ..., n_{ik_{i}})_{i=1,...,l}}).

Note that in the second case, Δ contains reflections. Now let us consider the monodromy epimorphism

θ : Δ → Δ/Γ ≅ G,

which sends anticonformal transformations to anticonformal automorphisms. If Δ contains reflections, that is, if Δ has signature

(p; ±; [m_{1}, ..., m_{r}]; {(n_{i1}, ..., n_{ik_{i}})_{i=1,...,l}})

where l > 0, then the image by θ of a reflection is an anticonformal involution in G, hence the signature of Δ must be of the form (p; −; [m_{1}, ..., m_{r}]), with no boundary components. The signature of Δ^{+} can now be obtained from the signature of Δ using the Riemann-Hurwitz formula and [BEGG]. □

Theorem 4 For every integer g ≥ 2, there exist pseudo-real surfaces of genus g.

Proof. Let Δ be an NEC group with signature (δ; −; [2, ...^{g+ε}, 2]), where δ = ε = 1 if g is even, or δ = 2 and ε = −1 if g is odd. Let x_{i} (for 1 ≤ i ≤ g + ε) and d_{j} (for 1 ≤ j ≤ δ) be a canonical system of generators of Δ. We may define an epimorphism θ : Δ → ℤ_{4} = ⟨a : a^{4} = 1⟩ by setting

θ(x_{i}) = a^{2} for 1 ≤ i ≤ g + ε, and θ(d_{j}) = a for 1 ≤ j ≤ δ.

Then X = 𝔻/ ker θ is a Riemann surface such that Δ/ ker θ ≅ ℤ_{4} is a group of automorphisms of X. Choosing Δ to be maximal (see [BCF]) ensures that the group Δ/ ker θ will be the full automorphism group of X, and this will contain anticonformal automorphisms but only one involution, namely a^{2}, and that involution is conformal. Hence X is pseudo-real. □

4 Pseudo-real surfaces of genus 2 and 3

Theorem 5 Let X be a pseudo-real Riemann surface of genus 2. Then Aut(X) is isomorphic to ℤ_{4}, and if Aut(X) ≅ Δ/Γ where X = 𝔻/Γ, then Δ has signature (1; −; [2, 2, 2]).

Proof. If G = Δ/Γ then G^{+} = Δ^{+}/Γ is the conformal automorphism group of a Riemann surface of genus 2. By Theorem 3 and [Br, Table 4], we know that the only possibilities for G^{+} and the signature s(Δ^{+}) of Δ^{+} are the following (where n^{r} denotes n, ...^{r}, n):

G^{+} ≅ ℤ_{2} and s(Δ^{+}) = (0; +; [2^{6}]),
or G^{+} ≅ ℤ_{2} and s(Δ^{+}) = (1; +; [2^{2}]),
or G^{+} ≅ ℤ_{4} and s(Δ^{+}) = (0; +; [2^{2}, 4^{2}]).

The second and third of these three cases are ruled out by the analysis undertaken in [BC], so we are left with only the first case. Then, since Δ must contain Δ^{+} as a subgroup of index two, the signature of Δ must be (1; −; [2, 2, 2]), and since G^{+} ≅ ℤ_{2}, also G ≅ ℤ_{4}. □

We can easily construct such a pseudo-real Riemann surface X of genus 2, for example as in the proof of Theorem 4 (with δ = ε = 1 and g = 2).

Theorem 6 Let X be a pseudo-real Riemann surface of genus 3, with automorphism group G ≅ Δ/Γ, where X = 𝔻/Γ. Then there are three possible cases, all of which are realisable:

(a) G ≅ ℤ_{4}, and Δ has signature (2; −; [2, 2]), or
(b) G ≅ D_{4}, and Δ has signature (1; −; [2, 2, 2]), or
(c) G ≅ ℤ_{4} ⊕ ℤ_{2}, and Δ has signature (1; −; [2, 2, 2]).

Proof. The index 2 subgroup G^{+} ≅ Δ^{+}/Γ of G ≅ Δ/Γ is the automorphism group of a Riemann surface of genus 3. By Theorem 3 and [Br, Table 4], and using the results of [BC] and [BCC], we find the only possibilities for G^{+} and the signature of Δ^{+} are the following (where 2^{r} denotes 2, ...^{r}, 2):

G^{+} ≅ ℤ_{2}  and s(Δ^{+}) = (0; +; [2^{8}]),
or G^{+} ≅ ℤ_{2}  and s(Δ^{+}) = (1; +; [2^{4}]),
or G^{+} ≅ ℤ_{2} ⊕ ℤ_{2} and s(Δ^{+}) = (0; +; [2^{6}]).

In the first case G must be ℤ_{4} and Δ must have signature (1; −; [2^{4}]), but then there is no epimorphism θ : Δ → ℤ_{4} with an appropriate kernel Γ, so this case is ruled out. From the other two cases, we deduce that the only possibilities for G, G^{+} and Δ are the following:

G ≅ ℤ_{4},  G^{+} ≅ ℤ_{2}  and s(Δ) = (2; −; [2, 2]),
or G ≅ D_{4},  G^{+} ≅ ℤ_{2} ⊕ ℤ_{2} and s(Δ) = (1; −; [2, 2, 2]),
or G ≅ ℤ_{4} ⊕ ℤ_{2}, G^{+} ≅ ℤ_{2} ⊕ ℤ_{2} and s(Δ) = (1; −; [2, 2, 2]).

In each case it is easy to establish an epimorphism θ : Δ → G where Δ is a maximal NEC group with the corresponding signature, so that G = Δ/ ker θ is the automorphism group of a pseudo-real Riemann surface of genus 3. □

5 Pseudo-real cyclic p-gonal Riemann surfaces

Definition 7 A cyclic p-gonal Riemann surface is a Riemann surface X that admits an automorphism h of order p such that the quotient surface X/⟨h⟩ has genus 0.

Theorem 8 Let X be a pseudo-real cyclic p-gonal Riemann surface of genus g ≥ (p − 1)^{2}, where p is prime. Let G be the automorphism group of X, and let H = ⟨h⟩ ≅ ℤ_{p} be the subgroup of G generated by an automorphism h of p-gonality. Let Δ and Γ be NEC groups such that X ≃ 𝔻/Γ and Γ ◁ Δ with Δ/Γ ≅ G. Then g is even, and one of the following two cases holds:

(a) G ≅ ℤ_{n} ⊕ H, where 4 divides n and the first factor ℤ_{n} is generated

by an anticonformal automorphism, and the NEC group Δ has

signature (1; −; [p, ...^{\frac{2(g+p−1)}{n(p−1)}}, p, \frac{n}{2}]); or

(b) G ≅ ℤ_{pn}, where 4 divides n, and G is generated by an anticonformal

automorphism, and Δ has signature (1; −; [p, ...^{\frac{2g}{n(p−1)}}, p, \frac{n}{2}p]).

Proof. Let Λ be the Fuchsian group of genus 0 such that Γ < Λ < Δ, with X/H ≃ 𝔻/Λ and Λ/Γ ≅ H. The signatures of Λ and Δ have the form

(0; +; [p, ...^{q}, p]) and (g′; −; [m_{1}, ..., m_{r}])

respectively. Let n = [Δ : Λ]. By [A], we know that if the genus of X satisfies g ≥ (p − 1)^{2}, then there is a unique p-gonal covering X → ℂ̂ = X/H. Hence Λ ◁ Δ, and so

[m_{1}, ..., m_{r}] = [ps_{1}, ...., ps_{t}, m_{t+1}, ..., m_{r}] and q = \frac{n}{s_{1}} + ... + \frac{n}{s_{t}}.

Applying the Riemann-Hurwitz formula, we find

−2 + q(1 − \frac{1}{p}) = n(g′ − 2 + t ; ∑_{i=1}(1 − \frac{1}{s_{i}p}) + r ; ∑_{i=t+1}(1 − \frac{1}{m_{i}})). (*)

Since the genus of Λ is 0, the genus of Δ must be 1, and hence the formula (*) is equivalent to

2 − n + n t ; ∑_{i=1}(1 − \frac{1}{s_{i}}) + n r ; ∑_{i=t+1}(1 − \frac{1}{m_{i}}) = 0.

From this formula it is easy to deduce that the only possible signatures for Δ are the following:

(1; −; [p, ...^{\frac{2(g+p−1)}{n(p−1)}}, p, \frac{n}{2}]) and (1; −; [p, ...^{\frac{2g}{n(p−1)}}, p, \frac{n}{2}p]).

Now suppose Δ has one of the above signatures, and define l = \frac{2(g+p−1)}{n(p−1)} for the first signature and l = \frac{2g}{n(p−1)} for the second. We will consider the epimorphism θ : Δ → Δ/Λ. Let d, x_{1}, ..., x_{l}, x_{l+1} be the generators of a canonical presentation of Δ. From the form of the signature of Λ, we see that θ(x_{1}) = ... = θ(x_{l}) = 1, and that θ(x_{l+1}) = b is an element of order \frac{n}{2}. Also the two elements a = θ(d) and b = θ(x_{l+1}) generate the image Δ/Λ. Next, from the relation

d^{2}x_{1}...x_{l}x_{l+1} = 1

we find that a^{2}b^{−1} = 1, and so Δ/Λ is cyclic of order n. Since the subgroup H generated by the p-gonal automorphism is unique, it is central in G and hence Δ/Γ is a central extension of a cyclic group of order n. But then since G/Z(G) is cyclic, G is abelian (by an easy theorem from group theory). Thus G is isomorphic to either ℤ_{n} ⊕ H or ℤ_{pn}, and the rest follows. □

6 The maximal order of the automorphism group of a pseudo-real Riemann surface

Theorem 9 If X is a pseudo-real Riemann surface of genus g with automorphism group G, then |G| ≤ 12(g − 1). Moreover, if |G| = 12(g − 1) and G ≅ Δ/Γ where X ≃ 𝔻/Γ, then the signature of Δ is (1; −; [2, 3]).

Proof. By Theorem 3, the NEC group Δ has signature (p; −; [m_{1}, ..., m_{r}]), and then from the Riemann-Hurwitz formula, we find

2g − 2 = |G| (p − 2 + r ; ∑_{i=1}(1 − \frac{1}{m_{i}})).

The minimum positive value of the bracketed expression on the right-hand side is \frac{1}{6}, which is attained when p = 1, r = 2, m_{1} = 2 and m_{2} = 3, and in that case |G| = 12(g − 1). □

We are interested in the construction of pseudo-real Riemann surfaces with automorphism group of maximal order. In order to find such a pseudo-real surface of genus g with maximal symmetry, we need to find:

(a) a maximal NEC group Δ with signature (1; −; [2, 3]),
(b) a finite group G of order 12(g − 1), and
(c) an epimorphism θ : Δ → G such that ker θ is a Fuchsian surface

group of genus g. Here we remark that the monodromy epimorphism θ is determined by the image of the canonical generators. If we have a group G of order 12(g − 1) and a monodromy epimorphism θ : Δ → G, then the group Δ is maximal (see [S] and [BEGG]) unless there is another NEC group Δ′ with signature (0; +; [2], {(2, 3)}) containing Δ and an epimorphism θ′ : Δ′ → G′, where G′ is an index two extension of G and θ′|_{Δ} = θ.

Proposition 10 Let Δ be an NEC group with signature (1; −; [2, 3]), let d, x_{1} and x_{2} be the generators of a canonical presentation for Δ, and let θ : Δ → G be an epimorphism such that θ(d_{1}) = a and θ(x_{1}) = b. Then θ can be extended to an epimorphism θ′ : Δ′ → G′, where Δ′ is an NEC group containing Δ as a subgroup of index 2 and G′ is a group containing G as a subgroup of index 2, if and only if G admits an automorphism of order 2 such that α(a) = a^{−1} and α(b) = b^{−1}.

Proof. If G admits such an automorphism α, then we can construct the semidirect product G′ = G ⋊_{α} ℤ_{2}, which is generated by G = ⟨a, b⟩ and an involution c, conjugation by which induces the automorphism α on G. Also we can let Δ′ be an NEC group with signature (0; +; [2], {(2, 3)}) and having canonical generators x′_{1} c′_{1}, c′_{2}, c′_{3}, and then define an epimorphism θ′ : Δ′ → G′ = G ⋊_{α} ℤ_{2} by setting

θ′(x′_{1}) = ac, θ′(c′_{1}) = c, θ′(c′_{2}) = cb, and θ′(c′_{3}) = a^{2}c.

Conversely, if such an extension θ′ : Δ′ → G′ of θ exists, then by [S] and [BEGG], Δ′ must have signature (0; +; [2], {(2, 3)}) with canonical generators x′_{1} c′_{1}, c′_{2}, c′_{3}, and without loss of generality the embedding of Δ in Δ′ is given by

d_{1} ↦ x′_{1}c′_{1}, x_{1} ↦ c′_{1}c′_{2}, x_{2} ↦ c′_{2}c′_{3};

hence if c is the involution θ′(c′_{1}), then

cac = θ′(c′_{1}d_{1}c′_{1}) = θ′(c′_{1}x′_{1}) = θ(d_{1})^{−1} = a^{−1}

and

cbc = θ′(c′_{1}x_{1}c′_{1}) = θ′(c′_{2}c′_{1}) = θ(x_{1})^{−1} = b^{−1},

so conjugation by c gives the required automorphism. □

The last proposition and a theorem to follow provide a link with the theory of 3-valent regular maps.

Definition 11 An orientably-regular map M is a 2-cell embedding of a connected graph into an orientable surface, such that the group Aut^{+}(M) of all orientation-preserving automorphisms of the surface that preserve the embedding has a single orbit on the arcs (directed edges) of the graph. The map is called reflexible if there exist orientation-reversing automorphisms that preserve the embedding, and otherwise it is said to be chiral.

More details may be found in [CD], where all orientably-regular maps of genus 2 to 15 were determined. If M is an orientably-regular map of type {m, n} (with vertices of valence m and faces of size n), then Aut^{+}(M) is generated by two elements R and S satisfying R^{m} = S^{n} = (RS)^{2} = 1, and M is reflexible if and only if there is an automorphism τ of G = ⟨R, S⟩ such that τ(R) = R^{−1} and τ(S) = S^{−1} (or equivalently, an automorphism inverting any one of the pairs (R, S), (R, RS) or (S, RS)).

Theorem 12 For each chiral regular map M of type {3, n}, where n is odd, if M has automorphism group G, then there exists a pseudo-real Riemann surface X with automorphism group of maximal order and isomorphic to G × ℤ_{4}.

Proof. Let Δ be the (2, 3, n) triangle group, with signature (0; [2, 3, n]) and canonical presentation

⟨ x_{1}, x_{2}, x_{3} : x_{1}x_{2}x_{3} = 1, x_{1}^{2} = x_{2}^{3} = x_{3}^{n} = 1 ⟩.

Also let θ : Δ → G be the epimorphism that corresponds to a chiral regular map of type {3, n}, taking (say) x_{1}, x_{2} and x_{3} to the automorphisms RS, R and S of M, so that ker θ is a surface group, and let b be an element of G such that b^{2} = θ(x_{3}), which is known to exist because n is odd. Now let Λ be an NEC group with signature (1; −; [2, 3]) and canonical presentation

⟨ d, y_{1}, y_{2} : y_{1}y_{2}d^{2} = 1, y_{1}^{2} = y_{2}^{3} = 1 ⟩,

and define an epimorphism ω : Λ → G × ℤ_{4} = G × ⟨a : a^{4} = 1⟩ by setting

ω(d) = (b, a), ω(y_{1}) = (θ(x_{1}), a^{2}) and ω(y_{2}) = (θ(x_{2}), 1).

Then Γ = ker ω is a surface group, and since the image of the subgroup ⟨x_{1}, x_{2}, dx_{1}d, dx_{2}d⟩ of index 2 in Λ is the subgroup G × ⟨a^{2}⟩ of index 2 in G × ⟨a⟩, the surface X = 𝔻/Γ (with automorphism group Λ/Γ ≅ G × ⟨a⟩) is orientable; see [BEGG, Theorem 2.1.3 (2)]. Moreover, every element of G × ⟨a⟩ lying outside the orientation-preserving subgroup G × ⟨a^{2}⟩ is of the form (u, a^{±1}) for some u ∈ G, and it follow that every anticonformal automorphism of X = 𝔻/Γ has order divisible by 4. Hence the surface X is pseudo-real. □

In the following section, we prove that for every integer k ≥ 7, there exist chiral regular maps of type {3, k} on orientable surfaces of infinitely many genera. Using Theorem 12, we therefore obtain the following theorem:

Theorem 13 There exist pseudo-real surfaces with automorphism group of maximal order, for infinitely many genera. In particular there are infinitely many pseudo-real Riemann surfaces with maximal automorphism group.

7 Chiral 3-valent regular maps

In the previous Section we proved that from every chiral regular map of type {3, n} for n odd, we can construct a pseudo-real Riemann surface with maximal symmetry. In this Section we shall find explicit families of chiral 3-valent regular maps that produce such pseudo-real Riemann surfaces.

Theorem 14 For every prime p congruent to 1, 2 or 4 mod 7, there exists a normal subgroup K_{p} of index 168p^{3} in the ordinary triangle group Δ = Δ(2, 3, 7) such that Δ/K_{p} is isomorphic to an extension by PSL(2, 7) of the 3-generator abelian group ℤ_{p}×ℤ_{p}×ℤ_{p} of order p^{3} and exponent p. Moreover, the subgroup K_{p} is not normal in the extended triangle group Δ^{∗}(2, 3, 7), so Δ/K_{p} has no automorphism that inverts the images of the two generators x and y of Δ = Δ(2, 3, 7) .

Note: here Δ has signature (0; +; [2, 3, 7]; {−}), while the extended triangle group Δ^{∗} has signature (1; +; [−]; {(2, 3, 7)}), with (Δ^{∗})^{+} ≅ Δ.

Corollary 15 There exist chiral regular maps of type {3, 7} on orientable surfaces of infinitely many genera.

Proof of Theorem. Most of this follows from observations made by Leech in [L] and pursued by Cohen in [Ch], and explained also in [Cn2, Cn3]. First, the extended triangle group Δ^{∗} = Δ^{∗}(2, 3, 7) has a normal subgroup N of index 336, generated by a_{0} = [y, x]^{4} and its conjugates, such that Δ^{∗}/N is isomorphic to PGL(2, 7). By observations made by Leech [L], this normal subgroup N has a nice presentation in terms of six generators and a single relation (in which each of the generators appears twice, with exponents ±1). Now for each prime p as given in the statement of the theorem, let N_{p} denote the normal subgroup of Δ^{∗} generated by the derived subgroup N′ = [N, N] of N and the set N^{p} of all pth powers of elements of N. Then N_{p} = N′N^{p} has index p^{6} in N, and is normal in Δ^{∗}, with quotient N/N_{p} elementary abelian of order p^{6}. Moreover, by observations made by Cohen [Ch] about the action of PSL(2, 7) on N/N_{p} induced by conjugation of N by elements of Δ = Δ(2, 3, 7), there exist intermediate subgroups L_{1} and L_{2} of N containing N_{p}, such that each L_{i} is normal in Δ, and N = L_{1}L_{2} with L_{1} ∩ L_{2} = N_{p}, and with N/L_{i} elementary abelian of order p^{3} for i ∈ {1, 2}. On the other hand, L_{1} and L_{2} are not normal in the extended triangle group Δ^{∗} = Δ^{∗}(2, 3, 7); indeed every element of Δ^{∗}\Δ conjugates L_{1} to L_{2} and vice versa. Hence we can take K_{p} = L_{1} or L_{2}, to give the required result. □

Theorem 16 For every integer k ≥ 7, all but finitely many of the alternating groups A_{n} can be generated by two elements x and y such that x, y and xy have orders 2, 3 and k respectively, and that there exists no automorphism of ⟨x, y⟩ = A_{n} taking x and y to x^{−1} and y^{−1} respectively.

Corollary 17 For each integer k ≥ 7, there exist chiral regular maps of type {3, k} on orientable surfaces of infinitely many genera.

Proof of Theorem. In all cases our argument relies heavily on a construction used by the second author in [Cn1] to prove that (for every k ≥ 7) all but finitely many A_{n} are homomorphic images of the extended triangle group

Δ^{∗}(2, 3, k) = ⟨ x, y, t | x^{2} = y^{3} = (xy)^{k} = t^{2} = (xt)^{2} = (yt)^{2} = 1 ⟩,

a group with signature (1; +; [−]; {(2, 3, k)}). We refer the reader to [Cn1] for important details. In that construction, permutation representations of Δ^{∗}(2, 3, k) are depicted by Schreier coset diagrams, and specially chosen examples of such diagrams are linked together to form representations of arbitrarily large degree n, in a way that makes the resulting permutations generate A_{n} or S_{n}. We will amend that construction by adding one more small diagram that depicts a permutation representation of the ordinary triangle group Δ = (Δ^{∗})^{+}, but not depict one of the extended triangle group Δ^{∗}(2, 3, k) itself. Note that Δ = Δ(2, 3, k) is the index 2 subgroup of Δ^{∗}(2, 3, k) generated by x and y. We do this first for the case k = 7, and then explain in less detail how the theorem can be proved for larger k using the same method.

b a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .•

Figure 1: Additional coset diagram R(7, 0) with 7 vertices

When k = 7, consider the permutation representation of Δ(2, 3, 7) on 7 points given by the diagram R(7, 0) in Figure 1.Like the diagrams S(7, 0), T (7, 0), U(7, 0) and V (7, 0) in [Cn1], this has a (1)-handle [a, b]_{1}, consisting of two points a and b such that x fixes both a and b, and y takes a to b. Note that the point a is fixed by the commutator xyxy^{−1}, while b lies in a 2-cycle of xyxy^{−1}, and the other four points lie in a 4-cycle. Similarly, if [a′, b′]_{1} is a (1)-handle of the diagram S(7, 0), then a′ is fixed by xyxy^{−1}, while b′ lies in a 13-cycle of xyxy^{−1}, consisting of the 13 points of the cycle of xyt in the representation of Δ(2, 3, 7) that it depicts. Indeed it follows from the relations for the extended triangle group Δ^{∗}(2, 3, 7) that (xyt)^{2} = xytxyt = xyxtyt = xyxy^{−1}, and hence the cycle structure of xyxy^{−1} can be derived easily from that of xyt on the points of the diagram S(7, 0). Next suppose that a single copy of the diagram R(7, 0) is linked together with a single copy of the diagram S(7, 0), by adding the transpositions (a, a′) and (b, b′) to the permutation induced by x (while not altering the permutation induced by y). Then the resulting diagram is easily seen to be a coset diagram for the ordinary triangle group Δ(2, 3, 7), by the same argument as in [Cn1]. Also in the corresponding permutation representation, the two points a and a′ are still both fixed by xyxy^{−1}, while the cycles containing b and b′ and the other four points of the diagram R(7, 0) are joined together to form a new cycle of xyxy^{−1}, of length 19. (This is easily verified, either by writing out the permutations, or by chasing points around the combined diagrams.) The construction in [Cn1] explains how a transitive permutation representation of Δ^{∗}(2, 3, 7) on n = 42f + 71g + 36 points (when f > g ≥ 0) can be formed by linking together f copies of diagram S(7, 0) and then adjoining g copies of T (7, 0) and a single copy of U(7, 0), by composition of (1)-handles. In the resulting representation, the element xyt has cycle structure

1^{f+1−g}2^{f+g}5^{1}6^{g}8^{1}11^{1}13^{f+1−g}15^{g}20^{g}24^{1}26^{f−1}42^{g},

so the commutator xyxy^{−1} has cycle structure

1^{3f+1}3^{2g}4^{2}5^{1}11^{1}13^{3f−1−g}10^{2g}12^{2}15^{g}21^{2g}.

The unique 11-cycle here comes from the single copy of U(7, 0), and this can be used (with the help of Jordan’s theorem from [W]) to prove that the permutations induced by x, y and t generate S_{n}, while those induced by x and y generate A_{n}. Now suppose that a single copy of the diagram R(7, 0) is linked to one of the copies of S(7, 0) still having a free (1)-handle in this representation. Then we have a new transitive permutation representation of Δ(2, 3, 7) on n + 7 points, in which xyxy^{−1} has cycle structure

1^{3f+2}3^{2g}4^{2}5^{1}11^{1}13^{3f−2−g}10^{2g}12^{2}15^{g}19^{1}21^{2g}.

Again the unique 11-cycle here comes from the single copy of U(7, 0), and can be used to prove that the permutations induced by x and y generate A_{n+7}. An important difference this time, however, is that because the point fixed by y in the single copy of R(7, 0) is the only point fixed by y that lies close to a fixed point of xyxy^{−1} or xy^{−1}xy in the resulting coset diagram (on n + 7 points), this diagram has no axis of reflectional symmetry. Thus we have a homomorphism from Δ(2, 3, 7) to A_{n+7} that does not extend to a representation of the extended triangle group Δ^{∗}(2, 3, 7), as claimed.

b a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} •

Figure 2: Additional coset diagram R(7, d) with 7 + 6d vertices

When k = 7 + 6d for some positive integer d, we can apply the same construction using S(7, d), T (7, d) and U(7, d) from [Cn1], and add a single copy of the new coset diagram R(7, d) for Δ(2, 3, 7 + 6d) on 7 + 6d points given in Figure 2. In the permutation representation of Δ(2, 3, 7 + 6d) depicted by R(7, d), the commutator xyxy^{−1} fixes the point a, and has two 2-cycles, two 4-cycles, and 2(d − 1) 3-cycles. Linking a single copy of R(7, d) to a copy of S(7, d) by their free (1)-handles gives rise to a new permutation representation of Δ(2, 3, 7+6d) in which one of the 2-cycles and one of the 4-cycles from R(7, d) are combined together with two of the cycles from S(7, d), to form a 7-cycle and a 10-cycle when d = 1, or a 6-cycle and an 8-cycle when d ≥ 2. Thus again we can form transitive permutation representations of Δ(2, 3, 7+ 6d) of arbitrarily large degree, and use the unique 11-cycle from the single copy of U(7, d) to prove that the resulting permutations generate an alternating group, and the single copy of R(7, d) to eliminate the possibility of a reflectional symmetry. The proof for other cases (with k in different congruence classes mod 6) is analogous to the above, using the additional diagrams given in Figures 3 to 7 below.

b a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}..............................................................................................

Figure 3: Additional coset diagram R(8, d) with 8 + 6d vertices

For the case k = 8 + 6d, joining a single copy of diagram R(8, d) replaces cycles of xyxy^{−1} by one cycle of length 15 if d = 0, or cycles of length 7 and 9 if d = 1, or cycles of length 6 and 8 if d ≥ 2, leaving a unique 11-cycle from the single copy of diagram U(8, d) for application of Jordan’s theorem.

β  α^{......................................}^{......................................} ^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}..........................................

Figure 4: Additional coset diagram R(9, d) with 9 + 6d vertices

For the case k = 9 + 6d, diagrams are composed using (2)-handles [α, β]_{2}, consisting of fixed points α and β such that y^{2} takes α to β. Joining a single copy of diagram R(9, d) replaces cycles of xyxy^{−1} by cycles of length 12 and 14 if d = 0, or cycles of length 3, 5, 9 and 10 if d = 1, or cycles of length 3, 5, 6 and 7 if d ≥ 2, leaving a unique 13-cycle from the single copy of diagram U(9, d) for application of Jordan’s theorem.

β  α^{......................................}^{......................................} ^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} ^{......................................}^{......................................}^{.....................................................}•

Figure 5: Additional coset diagram R(10, d) with 10 + 6d vertices

For the case k = 10 + 6d, joining a single copy of diagram R(10, d) replaces cycles of xyxy^{−1} by cycles of length 6, 7, 7 and 10 if d = 0, or cycles of length 5, 5, 6 and 7 if d = 1, or cycles of length 4, 5, 6 and 6 if d ≥ 2, leaving a unique 13-cycle from the single copy of diagram U(10, d) for application of Jordan’s theorem.

B  A^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} •

Figure 6: Additional coset diagram R(11, d) with 11 + 6d vertices

For the case k = 11 + 6d, joining a single copy of diagram R(11, d) replaces cycles of xyxy^{−1} by cycles of length 9 and 19 if d = 0, or cycles of length 6, 8 and 9 if d ≥ 1, leaving a unique 11-cycle from adjoining the single copy of diagram U(11, d) for application of Jordan’s theorem.

B  A^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ^{......................................}^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{......................................}^{......................................} ^{......................................}^{......................................} ^{......................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{..........................................}

Figure 7: Additional coset diagram R(12, d) with 12 + 6d vertices

For the case k = 12 + 6d, joining a single copy of diagram R(12, d) replaces cycles of xyxy^{−1} by cycles of length 9 and 10 if d = 0, or cycles of length 5, 7 and 9 if d = 1, or cycles of length 4, 6 and 9 if d ≥ 2, leaving a unique 13-cycle from adjoining the single copy of diagram U(12, d) for application of Jordan’s theorem. □

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