Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 439–456

AUTOMORPHISM GROUPS OF ORIENTABLE ELLIPTIC-HYPERELLIPTIC KLEIN SURFACES

Beatriz Estrada

Universidad Nacional de Ecucacion Distancia, Depto de Matemáticas Fundamentales c/. Senda del Rey 9, E-28040 Madrid, Spain; bestra@mat.uned.es

Abstract. A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface is said to be elliptic-hyperelliptic if it admits an involution ϕ, that is an order two automorphism, such that X/⟨ϕ⟩ has algebraic genus 1. Klein surfaces can be seen as quotients of the hyperbolic plane by the action of NEC groups, and their automorphism groups as quotients of NEC groups. Using this, we determine the full automorphism groups of orientable elliptichyperelliptic Klein surfaces of algebraic genus p > 5.

1. Introduction

Klein surfaces, introduced from a modern point of view by Alling and Greenleaf [1], are surfaces endowed with a dianalytic structure. A compact Klein surface is said to be elliptic-hyperelliptic if it admits an involution ϕ, that is an order two automorphism, such that X/⟨ϕ⟩ has algebraic genus 1.

Non-Euclidean crystallographic groups (NEC groups in short) where introduced by Wilkie and Macbeath, and they are an important tool in the study of Klein surfaces since the results of Preston and May. Klein surfaces can be seen as quotients of the hyperbolic plane under the action of an NEC group, and the automorphism groups of such surfaces as quotients of NEC groups; hence the relevance of the work about normal subgroups of NEC groups in [2], [3] and [9].

In [6] elliptic-hyperelliptic Klein surfaces (EHKS in short) were characterized by means of NEC groups. In this paper we determine all groups that are the automorphism group of an orientable EHKS of algebraic genus p > 5. Similar studies have been made for hyperelliptic and cyclic-trigonal Klein surfaces (see [5], [8]).

2. Preliminaries on NEC groups

Let 𝒟 denote the hyperbolic plane and 𝒢 its group of isometries. A non-Euclidean crystallographic group Γ, is a discrete subgroup of 𝒢 with compact quotient X = 𝒟/Γ. NEC groups were introduced by Wilkie [14], and Macbeath [10] associated to each NEC group a signature that determines its algebraic structure and has the following form:

1991 Mathematics Subject Classification: Primary 30F50, 20H10; Secondary 14J50.

Partially supported by CICYTPB95-0017.

(2.1)  σ(Γ) = (g, ±, [m_{1}, …, m_{r}], {(n_{i1}, …, n_{is_{i}}), i = 1, …, k}),

where g, m_{i}, n_{ij} are integers verifying g ≥ 0, m_{i} ≥ 2, n_{ij} ≥ 2; g is the topological genus of X. The sign determines the orientability of X. The numbers m_{i} are the proper periods corresponding to cone points in X. The brackets (n_{i1}, …, n_{is_{i}}) are the period-cycles. The number k of period-cycles is equal to the number of boundary components of X. Numbers n_{ij} are the periods of the period-cycle (n_{i1}, …, n_{is_{i}}) also called link-periods, corresponding to corner points in the boundary of X. The number p = αg + k − 1, where α = 1 or 2 if the sign of σ(Γ) is − or +, respectively, is called the algebraic genus of X.

An NEC group Γ with signature (2.1) has the following presentation [10]: Generators:

x_{i}, i = 1, …, r;
e_{i}, i = 1, …, k;
c_{ij}, i = 1, …, k, j = 0, …, s_{i};
a_{i}, b_{i}, i = 1, …, g, (if σ has the sign +);
d_{i}, i = 1, …, g (if σ has the sign −).

Relations:

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 = 1, …, s_{i};
e_{i}^{−1}c_{i0}e_{i}c_{is_{i}} = 1, i = 1, …, k;
x_{1} ⋯ x_{r}e_{1} ⋯ e_{k}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{g}b_{g}a_{g}^{−1}b_{g}^{−1} = 1 (if σ has the sign +);
x_{1} ⋯ x_{r}e_{1} ⋯ e_{k}d_{1}^{2} ⋯ d_{g}^{2} = 1 (if σ has the sign −).

By [10] cyclic permutations of periods in period-cycles or arbitrary permutations of proper periods in the signature of an NEC group Γ, lead to a signature σ′ corresponding to an NEC group isomorphic to Γ. Every NEC group Γ with signature (2.1) has associated to it a fundamental region whose area μ(Γ), called the area of the group (see [13]), is

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

Automorphism groups of orientable elliptic-hyperelliptic Klein surfaces 441

An NEC group with signature (2.1) actually exists if and only if the right-hand side of (2.2) is greater than 0 (see [15]). If Γ is a subgroup of an NEC group Γ′ of finite index N, then also Γ is an NEC group and the following Riemann–Hurwitz formula holds

(2.3)  μ(Γ) = Nμ(Γ′).

Let X be a Klein surface of topological genus g having k boundary components. Then, by [12], there exists an NEC group Γ with signature

(2.4)  (g; ±; [−], {(−), …^{k}, (−)}),

such that X = 𝒟/Γ, where the sign is “+” if X is orientable and “−” if not. An NEC group with signature (2.4) is called a surface group.

May [11] proved that if G is a group of automorphisms of a surface X = 𝒟/Γ of algebraic genus p ≥ 2, then G can be presented as a quotient Γ′/Γ for some NEC group Γ′. The full group of automorphisms of X is Aut(X) = N_{𝒢} (Γ)/Γ, where N_{𝒢} (Γ) is the normalizer of Γ in the group 𝒢 of isometries of 𝒟.

In order to decide if a group G of automorphisms of a Klein surface X equals Aut(X), the concept of maximality we are going to expose now is very useful.

An NEC group is said to be maximal if there does not exist another NEC group containing it properly. In particular if Γ′ is maximal (following the notation above) then G = Aut(X).

An NEC signature τ is said to be maximal if for any NEC group Γ with signature τ and for every NEC group Γ′ containing Γ, the equality d(Γ) = d(Γ′) (dimensions of the associated Teichmüller spaces [8]) implies Γ = Γ′.

Let σ and σ′ be NEC signatures of two NEC groups Γ and Γ′, respectively. We will say that (σ, σ′) is a normal pair, and we write σ ◁ σ′, if Γ ◁ Γ′ and d(Γ) = d(Γ′). The pair is said to be proper if σ′ has period-cycles. The list of normal proper pairs can be seen in [4].

The following results from [8] will be very useful.

Theorem 2.1. Given a maximal NEC signature τ there exists a maximal NEC group Γ with signature τ.

Theorem 2.2. Let Λ be an NEC group containing a surface group Γ as a normal subgroup. If σ(Λ^{+}) is maximal, the topological surface 𝒟/Γ can be endowed with a structure of a Klein surface such that Aut(𝒟/Γ) = Λ/Γ, where Λ^{+} is the normal subgroup of Λ of orientation preserving transformations.

3. Characterization of EHKS in terms of NEC groups

Definition 3.1. Let X be a Klein surface of algebraic genus p ≥ 2. We say that X is an elliptic-hyperelliptic Klein surface (EHKS in short) if it admits an involution ϕ (an automorphism of order 2), such that X/⟨ϕ⟩ has algebraic genus 1.

In the sequel Klein surface will mean compact bordered Klein surface. Now we give some results about EHKS and NEC groups obtained in [6].

Proposition 3.2. Let X = 𝒟/Γ be a bordered Klein surface of algebraic genus p ≥ 2. X is an EHKS if and only if there exists an NEC group Γ_{1} of algebraic genus 1 such that [Γ_{1} : Γ] = 2.

The group Γ_{1} is called the group of the EH character of X, and Γ_{1}/Γ = ⟨ϕ⟩ the group generated by ϕ.

Proposition 3.3. If X is an EHKS of algebraic genus p > 5, ϕ is unique and central in the full group of automorphisms of X. We will call it the EH-involution.

Theorem 3.4. Let X = 𝒟/Γ be an orientable bordered EHKS of topological genus g and k boundary components (p ≥ 2), and let Γ_{1} be the group of the EH character of X. Then Γ_{1} has one of the following signatures:

(i) If g = 0, (0; +; [−], {(2, …^{2k−4}, 2), (−)}).

(ii) If g = 1, (0; +; [−], {(2, …^{k−2}, 2), (−)}) or (0; +; [−], {(2, …^{s}, 2), (2, …^{2k−s}, 2)}),

s > 0, s even, or (0; +; [−], {(2, …^{2k}, 2)}).

(iii) If g ≥ 0, 2 ≤ k ≤ 4, (0; +; [2, …^{p−1}, 2], {(−), (−)}).

Corollary 3.5. An orientable EHKS must have one of the following topological types: topological genus 0 and more than 2 boundary components, topological genus 1 and at least 1 boundary component or topological genus greater than 2 and 2, 3 or 4 boundary components.

Remark 3.6. In [6] it was also proved that if X is an orientable EHKS of algebraic genus p > 5, then | Aut(X)| ≤ 4(p − 1), except for two special cases:

σ(Γ) = (\frac{1}{2}(p − 3), +, [−], (−)^{4}),  Aut(𝒟/Γ) = (D_{p−1} × Z_{2}) ∝ Z_{2};
σ(Γ) = (\frac{1}{2}(p − 1), +, [−], (−)^{2}),  Aut(𝒟/Γ) = D_{2(p−1)} ∝ Z_{2}.

4. Signatures associated to automorphism groups of EHKS

Let X = 𝒟/Γ be an orientable EHKS of topological genus g, k boundary components and algebraic genus p > 5; this condition gives unicity and centrality properties for the EH-involution. Let Γ_{1} be the group of the EH character. If G is an automorphism group of X containing ϕ (the EH-involution), then G = Γ′/Γ for a certain NEC group Γ′ such that Γ◁Γ_{1}◁Γ′. Let us suppose that N = [Γ′ : Γ_{1}], then the following three propositions determine a finite set of possible signatures of Γ′, for each topological type:

Proposition 4.1. If X has topological genus 0 then the signature of Γ′ is one of the following:

τ_{1} = (0, +, [−], {(2, …^{2(p−1)/N}, 2), (−)}),  N some divisor of 2(p − 1);
τ_{2} = (0, +, [−], {(2, …^{(2(p−1)/N)+4}, 2)}),  N some even divisor of 2(p − 1).

Proof. Let Γ′ be an NEC group of signature of the general form

(4.1.1)  σ(Γ′) = (g′, ±, [m_{1}, …, m_{r}], {(n_{i1}, …, n_{is_{i}}), i = 1, …, k′})

and let θ_{1}: Γ′ → Γ′/Γ_{1} be the canonical epimorphism such that Ker(θ_{1}) = Γ_{1} . Let G_{1} = Γ′/Γ_{1} and

l_{i} the order of θ_{1}(e_{i}) in G_{1} for i = 1, …, k′; p_{i} the order of θ_{1}(x_{i}) in G_{1} for i = 1, …, r; q_{ij} the order of θ_{1}(c_{ij−1}c_{ij}) in G_{1} for i = 1, …, k′, j = 1, …, s_{i};

n^{l}(i, j) the order of θ_{1}(c_{li−1}c_{lj+1}) in G_{1} for i = 1, …, s_{l}, j = 1, …, s_{l} − 1, i ≤ j.

Firstly we are looking for signatures for Γ′ such that μ(Γ′) = Nμ(Γ_{1}), where σ(Γ_{1}) = (0; +; [−], {(2^{2k−4}), (−)}) (see 3.4), and then we will try to obtain θ_{1} .

By the Riemann–Hurwitz formula

(4.1.2) N(αg′ + k′ − 2 + ∑_{i=1}^{r}(1 − \frac{1}{m_{i}} ) + \frac{1}{2} ∑_{i=1}^{k′} ∑_{j=1}^{s_{i}}(1 − \frac{1}{n_{ij}} )) = \frac{k − 2}{2}.

Period-cycles of Γ_{1} come from those of Γ′ having some reflection in Γ_{1} . Two possibilities appear ([2], [9]):

(i) If the two period-cycles of Γ_{1} come from different period-cycles of Γ′, namely C_{1} and C_{2} , then C_{i} = (2, …^{s_{i}}, 2), i = 1, 2. Let us suppose that all reflections of C_{1} and C_{2} are in Γ_{1} , then Ns_{1} = 2k − 4, s_{2} = 0, l_{1} = l_{2} = N and from (4.1.2)

N(αg′ + k′ − 2 + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{4}(\frac{2k − 4}{N}) + \frac{1}{2} ∑_{i=3}^{k′} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} )) = \frac{k − 2}{2};

hence g′ = 0, k′ = 2, r = 0 and σ(Γ′) = (0, +, [−], {(2^{(2(p−1))/N}), (−)}). The epimorphism θ_{1} is defined by

θ_{1}(e_{1}) = x, θ_{1}(e_{2}) = x^{−1}, θ_{1}(c_{ij}) = 1, for all i, j, where G_{1} = ⟨x : x^{N} = 1⟩.

When not all reflections of C_{1} or C_{2} are in Γ_{1} , the area of Γ′ increases and does not satisfy (4.1.2).

(ii) If the period-cycles of Γ_{1} come from the same period-cycle of Γ′, namely C_{1} , then by [9], N is even and C_{1} = (2, …^{s_{1}}, 2), where

\begin{aligned}s_{1} & ≥ (\frac{2k − 4}{N}) + 4, \\ n^{1}(1, (\frac{2k − 4}{N}) + 1) & = n^{1}(\frac{2k − 4}{N} + 2, \frac{2k − 4}{N} + 4) = \frac{N}{2} , \\ \frac{2(p − 1)}{N} & ≥ 2. \end{aligned}

Thus, from (4.1.2) we have

\begin{aligned}N(αg′ + k′ − 2 & + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{4}(\frac{2k − 4}{N} + 4) \\ & + \frac{1}{2} ∑_{j=(2k−4)/N+5}^{s_{1}}(1 − \frac{1}{n_{1j}} ) + \frac{1}{2} ∑_{i=2}^{k′} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} )) = \frac{k − 2}{2}, \end{aligned}

and therefore g′ = 0, k′ = 1, r = 0 and σ(Γ′) = (0, +, [−], {(2, …^{(2(p−1)/N)+4}, 2)}). The epimorphism θ_{1} is defined by θ_{1}(e_{1}) = 1, θ_{1}(c_{10}) = θ_{1}(c_{1,((2k−4)/N)+4}) = x, θ_{1}(c_{1,((2k−4)/N)+2}) = y, θ_{1}(c_{1j}) = 1 for the remaining reflections, with G_{1} = ⟨x, y : x^{2} = y^{2} = (xy)^{N/2} = 1⟩ ≃ D_{N/2} .

Proposition 4.2. If X has topological genus 1 then the signature of Γ′ is one of the following, with N some divisor of 2k:

If σ(Γ_{1}) = (0, +, [−], {(2, …^{2k}, 2), (−)});
τ_{3} = (0, +, [−], {(2, …^{2k/N}, 2), (−)});
τ_{4} = (0, +, [−], {(2, …^{(2k/N)+4}2)}), N even.
If σ(Γ_{1}) = (0, +, [−], {(2, …^{s}, 2), (2, …^{2k−s}, 2)}),  0 < s < 2k, s even,
τ_{5} = (0, +, [−], {(2, …^{(2k−s)/N}, 2), (2, …^{s/N}, 2)}),  N|2k − s and N|s;
τ_{6} = (0, +, [−], {(2, …^{2k/N}, 2), (−)}),  s = k;
τ_{7} = (0, +, [2, 2], {(2, …^{2k/N}, 2)}),  s = k;
τ_{8} = (1, −, [−], {(2, …^{2k/N}, 2)}),  s = k;
τ_{9} = (0, +, [2], {(2, …^{(2k/N)+2}, 2)}),  s = k, N ≡ 0 (mod 4);
τ_{10} = (0, +, [−], {(2, …^{(2k/N)+4}, 2)}),  N even.
If σ(Γ_{1}) = (1, −, [−], {(2, …^{2k}, 2)})
τ_{11} = (0, +, [−], {(2, …^{2k/N}, 2), (−)}),  N even,
τ_{12} = (1, −, [−], {(2, …^{2k/N}, 2)}),
τ_{13} = (0, +, [2], {(2, …^{2k/N}, 2)}),  N ≡ 2 (mod 4),
τ_{14} = (0, +, [−], {(2, …^{(2k/N)+4}, 2)}),  N ≡ 0 (mod 4).

Proof. We will use the notation of the proof of Proposition 4.1.

Case 1. If σ(Γ_{1}) = (0, +, [−], {(2, …^{2k}, 2), (−)}). Then σ(Γ′) can have signature τ_{3} or τ_{4} . These cases are completely analogous to τ_{1} and τ_{2} (in Proposition 4.1), respectively. The epimorphisms are:

– for τ_{3} , θ_{1}(e_{1}) = x_{1} , θ_{1}(e_{2}) = x_{1}^{−1} , θ_{1}(c_{2,0}) = 1, θ_{1}(c_{1,j}) = 1, for j = 1, …, s_{1} ; with G_{1} = ⟨x : x^{N} = 1⟩;

– for τ_{4} , we have θ_{1}(e_{1}) = 1, θ_{1}(c_{2,0}) = 1, θ_{1}(c_{1,0}) = θ_{1}(c_{1,(2k/N)+4}) = x, θ_{1}(c_{1,(2k/N)+1}) = y, θ_{1}(c_{1,j}) = 1 for the rest of reflections in C_{1} ; with G_{1} = ⟨x : x^{N} = 1⟩.

Case 2. If σ(Γ_{1}) = (0, +, [−], {(2, …^{s}, 2), (2, …^{2k−s}, 2)}), 0 < s < k, s even, by the Riemann–Hurwitz formula we have

(4.2.1)  αg′ + k′ − 2 + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{2} ∑_{i=1}^{k′} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} ) = \frac{k}{2N} ,

and we must consider the following cases:

(i) If the two period-cycles of Γ_{1} come from different period-cycles C_{1} , C_{2} in Γ′, as in 4.1(i), all reflections of those period-cycles must be in Γ_{1} , and then C_{1} = (2, …^{s/N}, 2) and C_{2} = (2, …^{(2k−s)/N}, 2), l_{1} = l_{2} = N. This, together with (4.2.1)

forces g′ = 0, k′ = 2 having σ(Γ′) = τ_{5} = (0, +, [−], {(2, …^{s/N}, 2), (2, …^{(2k−s)/N}, 2)}).

The epimorphism θ_{1} is defined by θ_{1}(e_{1}) = x_{1} , θ_{1}(e_{2}) = x_{1}^{−1} , θ_{1}(c_{i,j}) = 1, for i = 1, 2, j = 1, …, s_{1} , where G_{1} = ⟨x : x^{N} = 1⟩.

(ii) If the period-cycles of Γ_{1} come from the same period-cycle of Γ′, namely C_{1} , then N even (see [9]) and we have to consider s = k or s ≠ k.

When s = k and all reflections in C_{1} are in Γ_{1} then C_{i} = (2, …^{s_{i}}, 2), l_{1} = \frac{1}{2}N, \frac{1}{2}Ns_{1} = k, and (4.1.2) becomes

(4.2.2) αg′ + k′ − 2 + ∑_{i=1}^{r}(1 − \frac{1}{m_{i}} ) + \frac{1}{4}(\frac{2k}{N} ) + \frac{1}{2} ∑_{i=2}^{k′} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} ) = \frac{k}{2N} .

Then Γ′ can have one or two boundary components:

– If k′ = 2, then r′ = 0, g′ = 0, s_{2} = 0, and we have σ(Γ′) = τ_{6} = (0, +, [−], {(2, …^{2k/N}, 2), (−)}). Here θ_{1} is defined as

θ_{1}(e_{1}) = x_{1}, θ_{1}(e_{2}) = x_{1}^{−1}, θ_{1}(c_{1,j}) = 1, for j = 1, …, s_{1}, θ_{1}(c_{2,0}) = y;

with G_{1} = ⟨x_{1}, y_{1} : x_{1}^{N/2} = y_{1}^{2} = [x_{1}, y_{1}] = 1⟩. (The relation [x_{1}, y_{1}] = 1 comes from e_{2}c_{2,0}e_{2}^{−1} = c_{2,0} in Γ′.)

– If k′ = 1, then g′ = 0 or 1.

If g′ = 0, then by (4.2.2) r′ = 2, m_{1} = 2, m_{2} = 2, p_{1} = 2, p_{2} = 2,

and σ(Γ′) = τ_{7} = (0, +, [2, 2], {(2, …^{2k/N}, 2)}), with θ_{1}(x_{1}) = x_{1} , θ_{1}(x_{2}) = y_{1} ,

θ_{1}(e_{1}) = y_{1}x_{1} , θ_{1}(c_{1,j}) = 1, for j = 1, …, s_{1} ; and G_{1} = ⟨x_{1}, y_{1} : x_{1}^{2} = y_{1}^{2} = (y_{1}x_{1})^{N/2} = 1⟩. (The choice of θ_{1}(e_{1}) comes from the relation x_{1}x_{2}e_{1} = 1 in Γ′).

If g′ = 1, by (4.2.2), r′ = 0 and σ(Γ′) = τ_{8} = (1, −, [−], {(2, …^{2k/N}, 2)}). Here

θ_{1}(d_{1}) = x_{1}θ_{1}(e_{1}) = x_{1}^{N−2},  θ_{1}(c_{1,j}) = 1,  for j = 1, …, s_{1} ;

and G_{1} = ⟨x_{1} : x_{1}^{N} = 1⟩. (The choice of θ_{1}(e_{1}) comes from the relation d_{1}^{2}e_{1} = 1 in Γ′.)

Now, when k = s and not all reflections in C_{1} are in Γ_{1} , we can obtain the two equal period-cycles of Γ_{1} in two different ways:

(a) From the same “piece” of C_{1} = (2, …, 2^{i} …^{(2k/N)+2}, 2^{j+1}, …, 2), such that c_{i}, …, c_{j} ∈ Γ_{1} , c_{i−1} , c_{j+1} ∉ Γ_{1} and n^{1}(i, j) = \frac{1}{4}N. Hence (4.2.1) becomes

(4.2.3) \begin{aligned}αg′ + k′ − 2 & + ∑_{i=1}^{r}(1 − \frac{1}{m_{i}} ) + \frac{1}{4}(\frac{2k}{N + 2}) \\ & + \frac{1}{2} ∑_{l∉\{i,...,j+1\}}(1 − \frac{1}{n_{1,l}} ) + \frac{1}{2} ∑_{i=2}^{k′} ∑_{j=1}^{s_{i}}(1 − \frac{1}{n_{ij}} ) = \frac{k}{2N} . \end{aligned}

Then g′ = 0, k′ = 1 and

(4.2.4)  ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{2} ∑_{l∉i,...,j+1} (1 − \frac{1}{n_{1,l}} ) = \frac{1}{2}.

– If r′ = 1, we have by (4.2.4), m_{1} = 2, p_{1} = 2 and σ(Γ′) = τ_{9} =

(0, +, [2], {(2, …^{(2k/N)+2}, 2)}); θ_{1} and G_{1} are defined as

\begin{aligned}θ_{1}(x_{1}) & = x_{1}^{−1}, θ_{1}(e_{1}) = x_{1}, θ_{1}(c_{1,0}) = y_{1}, θ_{1}(c_{1,(2k/N)+2}) = z_{1} \\ θ_{1}(c_{1,j}) & = 1, for j = 1, …, \frac{2k}{N} + 1; \\ G_{1} & = ⟨x_{1}, y_{1} : y_{1}^{2} = z_{1}^{2} = (y_{1}z_{1})^{N/4} = 1, x_{1}y_{1} = z_{1}x_{1}⟩. \end{aligned}

Firstly, the order of x_{1} is not fixed by the epimorphism, but since G_{1} has cardinality N, the order of x_{1} equals 2. This group is isomorphic to a semidirect product of type D_{N/4} ∝ Z_{2} .

– If r′ = 0, also by (4.2.4), σ(Γ′) = τ_{10} = (0, +, [−], {(2, …^{(2k/N)+4}, 2)}). Here θ_{1} and G_{1} are defined by

\begin{aligned}θ_{1}(e_{1}) & = 1, θ_{1}(c_{1,0}) = θ_{1}(c_{1,(2k/N)+4}) = x_{1}, \\ θ_{1}(c_{1,(2k/N)+2}) & = y_{1}, θ_{1}(c_{1,(2k/N)+3}) = z_{1}, \\ θ_{1}(c_{1,j}) & = 1, for j = 1, …, (2k/N) + 1; \\ G_{1} & = ⟨x_{1}, y_{1}, z_{1} : x_{1}^{2} = y_{1}^{2} = z_{1}^{2} = (y_{1}x_{1})^{N/4} = (y_{1}z_{1})^{2} = (z_{1}y_{1})^{2} = 1⟩. \end{aligned}

Note that ord(y_{1}x_{1}) = ord(y_{1}z_{1}) = 2 is necessary for Ker(θ_{1}) to have no proper periods.

(b) From different “pieces” of C_{1} (also works for k ≠ s). Then C_{1} =

(2, …^{(2k/N)+4}, 2), c_{1,0} , c_{1,(2k/N)+2} , c_{1,(2k/N)+4} ∉ Γ_{1} , c_{1j} ∈ Γ_{1} for the remaining

reflections, and n^{1}(1, (2k/N) + 1) = n^{1}((2k/N) + 3, (2k/N) + 3) = \frac{1}{2}N. Hence (4.2.1) becomes

(4.2.5) αg′ + k′ − 2 + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{4}(\frac{2k}{N} + 4)+\frac{1}{2} ∑_{i=2}^{k′} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} ) = \frac{k}{2N} .

Then g′ = 0, k′ = 1, r′ = 0 and σ(Γ′) = τ_{10} = (0, +, [−], {(2, …^{(2k/N)+4}, 2)}). We

have obtained the same signature as before but a different epimorphism:

θ_{1}(e_{1}) = 1, θ_{1}(c_{1,0}) = θ_{1}(c_{1,(2k/N)+4}) = x_{1}, θ_{1}(c_{1,(2k/N)+2}) = y_{1},
θ_{1}(c_{1,j}) = 1 for the remaining reflections;
G_{1} = ⟨x_{1}, y_{1} : x_{1}^{2} = y_{1}^{2} = (y_{1}x_{1})^{N/2}⟩.

Case 3. If σ(Γ_{1}) = (1, −, [−], {(2, …^{2k}, 2)}), we proceed in a similar way to obtain the signatures of Γ′.

Proposition 4.3. If X has topological genus g ≥ 2 and 2 ≤ k ≤ 4, then the signature of Γ′ is one of the following:

\begin{aligned}τ_{15} & = (0, +, [2, …^{l+1}, 2, 4,], \{(−)\}), l ≥ 0, N = \frac{2(p − 1)}{2l + 1}; \\ τ_{16} & = (0, +, [2, …^{l}, 2, 4, 4], \{(−)\}), l ≥ 0, N = \frac{p − 1}{l + 1}; \\ τ_{17} & = (0, +, [2, …^{l_{1}}, 2, 4], \{(2, …^{l_{2}+2}, 2)\}), l_{1}, l_{2} ≥ 0, N = \frac{2(p − 1)}{2l_{1} + l_{2} + 2}; \\ τ_{18} & = (0, +, [−], \{(2, …^{l+4}, 2)\}), l > 0, N = \frac{2(p − 1)}{l}; \\ τ_{19} & = (0, +, [−], \{(2, …^{l+2}, 2, 4, 4)\}), l ≥ 0, N = \frac{2(p − 1)}{l + 1}; \\ τ_{20} & = (0, +, [−], \{(2, …^{l+3}, 2, 4)\}), l ≥ 0, N = \frac{4(p − 1)}{2l + 1}; \end{aligned}

The proof of this proposition is analogous to the previous one.

5. The automorphism group of EHKS

Let X = 𝒟/Γ be an orientable EHKS of topological genus g, k boundary components and algebraic genus p > 5. Remember that p > 5 makes the EHinvolution unique and central in Aut(X). Let Γ_{1} be the group of the EH character. If G is an automorphism group of X containing ϕ (the EH-involution), then G = Γ′/Γ for a certain NEC group Γ′ with signature of one of the types described in Section 4. Let us suppose that |G| = 2N and denote by θ: Γ′ → G and π: G → G/⟨ϕ⟩ the canonical epimorphisms such that θ_{1} = πθ.

The method to obtain θ (and then G) is to use the epimorphism θ_{1} already defined in Section 4 and consider the diagram

θ_{1}: Γ′ ⟶^{θ} G ⟶^{π} G_{1} = G/⟨ϕ⟩.

Elements in G_{1} will be denoted by letters like x_{1} , y_{1} , z_{1} , t_{1} , …; and those in G, like x, y, z, t, …, subject to π(x) = x_{1} , π(y) = y_{1}, ….

The study of automorphism groups of orientable EHKS splits into three parts corresponding to the three propositions in Section 4.

Theorem 5.1. Let X be an orientable EHKS of algebraic genus p > 5, topological genus 0 and k boundary components. Then G ≃ Z_{N} ×Z_{2} or D_{N/2}×Z_{2} for some integer N (even in the last case). Moreover,

(i) There exists such an EHKS having Z_{N}×Z_{2} as the full group of automorphisms if and only if N is a proper divisor of p − 1.

(ii) There exists such an EHKS having D_{N/2} × Z_{2} as the full group of automorphisms if and only if N is an even divisor of 2(p − 1).

Proof. By Theorem 3.4, σ(Γ_{1}) = (0; +; [−], {(2, …^{2(p−1)}, 2), (−)}) and there are two possibilities for the signature of Γ′ namely τ_{1} and τ_{2} described in 4.1.

We begin with τ_{1} = (0, +, [−], {(2, …^{2(p−1)/N}, 2), (−)}). The epimorphism θ_{1} is defined as

θ_{1}(e_{1}) = x_{1}, θ_{1}(e_{2}) = x_{1}^{−1}, θ_{1}(c_{ij}) = 1 for all reflections in Γ′

with G_{1} = ⟨x_{1} : x_{1}^{N} = 1⟩. Let θ(e_{1}) = x, now as ord(x_{1}) = N and θ_{1} = πθ then ord(x′) = N or 2N:

(i) If ord(x) = 2N, then G ≃ Z_{2N} and we will show there exists no epimorphism θ: Γ′→Z_{2N} verifying Ker(θ) = Γ. Empty period-cycles of Γ come from consecutive pairs of period-cycles with associated reflections

c_{ij−1}, c_{ij}, c_{ij+1} : c_{ij−1}, c_{ij+1} ∉ Γ, c_{ij} ∈ Γ;

from each of these pairs we obtain N/ord(θ(c_{ij−1}, c_{ij+1})) period-cycles in Ker(θ). Then we will have as maximum p − 1 = k − 2 from the non empty period-cycle of Γ′. The other two will come from the empty period-cycle of Γ′, but since ord(x) = 2N, we only can obtain 2N/ord(θ(e_{2})) = 1.

(ii) If ord(x) = N, let us consider the EH-involution ϕ, we can see that ⟨x⟩ ∩ ⟨α⟩ = 1, otherwise, since ϕ has order 2 in G, we have ϕ = x^{N/2} and consequently π(ϕ) = π(x^{N/2}) = x_{1}^{N/2} = 1, a contradiction with the order of x_{1} in G_{1} . Furthermore, since ϕ is central in Aut(X), then G ≃ Z_{N} × Z_{2} and θ is defined by

(5.1.1)  \begin{aligned}θ(e_{1}) & = x, θ(e_{2}) = x^{−1}, θ(c_{2,0}) = 1, \\ θ(c_{1,2j}) & = y, j = 0, …, \frac{p − 1}{N}; \\ θ(c_{1,2j+1}) & = 1, j = 0, …, \frac{p − 1}{N} − 1 (π(y) = 1). \end{aligned}

In this case 2(p − 1)/N is necessarily even, implying that N divides p − 1. If 2(p − 1)/N is odd the epimorphism would be impossible.

If the signature of Γ′ is τ_{2} = (0, +, [−], {(2, …^{(2(p−1)/N)+4}, 2)}), θ_{1} is defined as

θ_{1}(e_{1}) = 1, θ_{1}(c_{10}) = θ_{1}(c_{1,((2k−4)/N)+4}) = x_{1}, θ_{1}(c_{1,((2k−4)/N)+2}) = y_{1},
θ_{1}(c_{1j}) = 1 for the remaining reflections;
with G_{1} = ⟨x_{1}, y_{1} : x_{1}^{2} = y_{1}^{2} = (x_{1}y_{1})^{N/2} = 1⟩ ≃ D_{N/2}.

Considering θ_{1} = πθ, since ord(x_{1}y_{1}) = \frac{1}{2}N then ord(xy) = \frac{1}{2}N or N. If ord(xy) = N, then G = D_{N} and as in the previous case, there is no epimorphism θ: Γ′→D_{N} satisfying Ker(θ) = Γ. Let l = 2(p − 1)/N; the way to obtain a maximal number of empty period-cycles in Ker(θ), having in mind that consecutive reflections cannot be in Ker(θ), is to define

θ(c_{1,l+3}) = 1 having one period-cycle from c_{l+2} , c_{l+3} , c_{l+4} (see [9]), and θ(c_{1,2j+1}) = z for j = 0, …, \frac{1}{2}(l − 2), θ(c_{1,2j}) = 1 for j = 1, …, \frac{1}{2}(l − 2),

where z^{2} = 1 and π(z) = 1. From here we obtain k − 2 − N more period-cycles. There remain N + 1 period-cycles to be obtained by playing with the values of θ(c_{l}), and θ(c_{l+1}). In any case we have

\frac{N}{ord(θ(c_{i,l−1}, c_{i,l+1}))}  or  \frac{N}{ord(θ(c_{i,l}, c_{i,l+2}))}

depending on θ(c_{1,l}) = 1 or θ(c_{1,l+1}) = 1, respectively. In the two cases we have at most k − 1 empty period-cycles in Ker(θ). Then ord(xy) = \frac{1}{2}N.

Let us consider ϕ, the EH-involution. We can see that ⟨x, y⟩∩⟨ϕ⟩ = 1 because otherwise, since ϕ has order 2 in G, then ϕ = (xy)^{N/4} or ϕ = (xy)^{k}x, for some k in \{0, …, \frac{1}{2}N\}. The first possibility contradicts the order of x_{1}y_{1} in G_{1} , because π(ϕ) = π((xy)^{N/4}) = x_{1}y_{1}^{N/4} = 1. The second implies π((xy)^{k}x) = 1, or equivalently (c_{1,l+2}c_{1,l+4})^{k}c_{l+2} ∈ Γ_{1} and since Γ_{1} ◁ Γ′, c_{l+4} ∈ Γ_{1} , a contradiction. Thus, since |G| = 2N, x, y, ϕ generate G, and since ϕ is central in G the epimorphism is defined by

(5.1.2)

If l even:
θ(e_{1}) = 1, θ(c_{1,0}) = θ(c_{1,l+4}) = x θ(c_{1,l+2}) = y, θ(c_{1,l+3}) = 1;
θ(c_{1,2j+1}) = z for j = 0, …, \frac{1}{2}l, θ(c_{1,2j}) = 1 for j = 1, …, \frac{1}{2}l.

(5.1.3)

If l odd:
θ(e_{1}) = 1, θ(c_{1,0}) = θ(c_{1,l+4}) = x, θ(c_{1,l+2}) = y, θ(c_{1,l+3}) = 1,
θ(c_{1,2j+1}) = z for j = 0, …, \frac{1}{2}(l − 1),
θ(c_{1,2j}) = 1 for j = 1, …, \frac{1}{2}(l + 1).

In any case G = ⟨x, y, z : x^{2} = y^{2} = z^{2} = (xy)^{N/2} = (xz)^{2} = (yz)^{2} = 1⟩ ≃ D_{N/2} × Z_{2} , ϕ = z.

This proves the first part of the theorem. For the second, let X be an EHKS in the conditions of the theorem:

(i) If Aut(X) ≃ Z_{N} × Z_{2} , then Γ′ must have signature τ_{1} and N | p − 1, as we saw before. Let N be a proper divisor of p − 1 and consider the signature τ_{1} . If N divides p − 1 properly, τ_{1}^{+} is maximal (see Section 2), and there exists a maximal NEC group Γ′ with signature τ_{1} . Considering the epimorphism θ (5.1.1), the surface X = 𝒟/ Ker(θ) is elliptic-hyperelliptic, ϕ = y being the EHinvolution and Aut(X) ≃ Z_{N} × Z_{2} . Now if N = p − 1 we consider Γ′, θ, X as before. If τ_{1} is not maximal, we will see that whenever Z_{p−1} × Z_{2} is a group of automorphisms of X, so is D_{p−1} × Z_{2} . To do it let us consider Theorem 4.1 in [7] and the following normal proper pair

(τ_{1}, τ_{2}), τ_{2} = (0, +, [−], {(2, 2, 2, 2, 2)}).

We only need to argue that the epimorphism θ: Γ′→Z_{N} × Z_{2} is unique up to automorphisms of Γ′ and Z_{N} × Z_{2} .

(ii) If Aut(X) = D_{N/2} × Z_{2} (N even), then Γ′ must have signature τ_{2} and N | 2(p − 1), this last condition makes τ_{2} maximal. Let N be an even divisor of 2(p − 1) and consider the signature τ_{2} . Since τ_{2}^{+} is maximal (see Section 2) there exists a maximal NEC group Γ′ with signature τ_{2} ; and considering the epimorphism θ (5.1.2), the surface X = 𝒟/ Ker(θ) is elliptic-hyperelliptic. The EH-involution ϕ is (xy)^{N/4} and Aut(X) ≃ D_{N/2} × Z_{2} .

Theorem 5.2. Let X be an orientable EHKS of algebraic genus p > 5, topological genus 1 and k boundary components. Then Aut(X) is isomorphic to one of the following groups: Z_{N} ×Z_{2} , Z_{N/2}×Z_{2}×Z_{2} , D_{N/2}×Z_{2} , (D_{N/4}×Z_{2})×Z_{2} or a semidirect product of type (D_{N/4} ∝ Z_{2})×Z_{2} for some even integer N, N ≡ 0 (mod 4) in the last two cases.

Moreover, in all cases there exists such an EHKS having full group of automorphisms isomorphic to

– Z_{N} × Z_{2} if and only if N is a proper divisor of k;

– Z_{N/2} × Z_{2} × Z_{2} if and only if N is an even proper divisor of k;

– D_{N/2} × Z_{2} if and only if N is an even divisor of 2k;

– D_{N/4} × Z_{2} × Z_{2} if and only if N is a divisor of 2k, N ≡ 0 (mod 4);

– (D_{N/4} ∝ Z_{2}) × Z_{2} if and only if N is a divisor of 2k, N ≡ 0 (mod 4).

Proof. Let G be an automorphism group of X as described at the beginning of this section. Since X has topological genus 1, by Proposition 4.2, the group of the EH character can be of three different types:

1. If σ(Γ_{1}) = (0, +, [−], {(2, …^{2k}, 2), (−)}), we have two possibilities for σ(Γ′), to consider, τ_{3} and τ_{4} . The study is completely analogous that of τ_{1} and τ_{2} , respectively, in the previous theorem. We only show the epimorphisms for each case.

For τ_{3} , G = ⟨x, y : x^{N} = y^{2} = [x, y] = 1⟩ ≃ Z_{N} × Z_{2} , and θ is defined by

\begin{aligned}θ(e_{1}) & = x, θ(e_{2}) = x^{−1}, θ(c_{2,0}) = θ(c_{1,2j}) = 1 for j = 0, …, \frac{k}{N}; \\ θ(c_{1,2j+1}) & = y for j = 0, …, \frac{k}{N} − 1 (π(y) = 1). \end{aligned}

For τ_{4} , G = ⟨x, y, z : x^{2} = y^{2} = z^{2} = (xy)^{N/2} = (xz)^{2} = (yz)^{2} = 1⟩ ≃ D_{N/2} × Z_{2} and θ is defined by

θ(e_{1}) = 1, θ(c_{1,0}) = θ(c_{1,l+4}) = x, l = \frac{2k}{N} , θ(c_{1,l+2}) = y, θ(c_{1,l+3}) = 1,
θ(c_{1,2j+1}) = z for j = 0, …, \frac{1}{2}l, θ(c_{1,2j}) = 1 for j = 1, …, \frac{1}{2}l, l even.

2. If σ(Γ_{1}) = (0, +, [−], {(2, …^{s}, 2), (2, …^{2k−2}, 2)}), 0 < s < 2k, s even. By

Proposition 4.2 we have six possibilities for the signature of σ(Γ′): τ_{5}, …, τ_{10} . The following table shows the results obtained for each one.

σ(Γ′)  Γ′/Γ
τ_{5} = (0, +, [−], {(2, …^{(2k−s)/N}, 2), (2, …^{s/N}, 2)}), N | k  Z_{N} × Z_{2}
τ_{6} = (0, +, [−], {(2, …^{2k/N}, 2), (−)})  Z_{N/2} × Z_{2} × Z_{2}
N | k  Z_{N} × Z_{2} , s = k
τ_{7} = (0, +, [2, 2], {(2, …^{2k/N}, 2)}), N | k  D_{N/2} × Z_{2} , s = k
τ_{8} = (1, −, [−], {(2, …^{2k/N}, 2)}), N | k  Z_{N} × Z_{2} , s = k
τ_{9} = (0, +, [2], {(2, …^{2k/N}, 2)}), 4|N|2k  (D_{N/4} ∝ Z_{2}) × Z_{2} , s = k
τ_{10} = (0, +, [−], {(2, …^{(2k/N)+4}, 2)})  D_{N/2} × Z_{2}
N | k  (D_{N/4} × Z_{2}) × Z_{2} , s = k, 4 | N

For the sake of brevity we only show the complete proof for τ_{9} . The remaining cases are obtained in a similar way.

Let us consider τ_{9} = (0, +, [2], {(2, …^{2k/N}, 2)}). Here θ_{1} was defined as

\begin{aligned}θ_{1}(x_{1}) & = x_{1}^{−1},  θ_{1}(e_{1}) = x_{1},  θ_{1}(c_{1,0}) = y_{1},  θ_{1}(c_{1,2k/N+2}) = z_{1}, \\ θ_{1}(c_{1,j}) & = 1,  for j = 1, …, \frac{2k}{N} + 1, \end{aligned}

where G_{1} = ⟨x_{1}, y_{1}, z_{1} : x_{1}^{2} = y_{1}^{2} = z_{1}^{2} = (y_{1}z_{1})^{N/4} = 1, x_{1}y_{1} = z_{1}x_{1}⟩ ≃ D_{N/4} ∝ Z_{2} .

To obtain θ let us define θ(x_{1}) = x^{−1} , θ(e_{1}) = x, θ(c_{1,0}) = y, θ(c_{1,2k/N+2}) = z, θ(c_{1,2l}) = 1, θ(c_{1,2l+1}) = t, 2l, 2l + 1 ∈ {1, …, (2k/N) + 1}. Here x, y, z, t are order two elements of G, that must verify, for Ker(θ) to be a surface group, and θ to be an epimorphism, (yt)^{2} = (tz)^{2} = 1, xyx^{−1} = z.

Moreover, since we want θ_{1} = πθ and the order of π(xy) = x_{1}y_{1} is \frac{1}{4}N, then the order of xy has to be \frac{1}{4}N or \frac{1}{2}N. If the order is \frac{1}{2}N, we will have Ker(θ) non orientable because (c_{1,0}c_{1,(2k/N)+2})^{N/4}c_{1,1} would belong to it. Then the order of xy is \frac{1}{4}N. Now, since π(t) = 1 (in other words t⟨ϕ⟩ = ⟨ϕ⟩), we have t = ϕ, and since ϕ is central in G the following presentation holds:

G = ⟨x, y, z, t : x^{2} = y^{2} = z^{2} = t^{2} = (yz)^{N/4} = (yt)^{2} = (zt)^{2} = (xt)^{2} = 1, xyx = z⟩.

We can see that this group is isomorphic to a semidirect product (D_{N/4}∝_{φ}Z_{2})×Z_{2} , φ being the automorphism

\begin{matrix}φ: Z_{2} = ⟨a : a^{2}⟩ & ⟶ Aut(D_{N/4}),  D_{N/4} = ⟨b, c : b^{2} = c^{2} = (bc)^{N/4} = (bd)^{2}⟩ \\ 1 & ⟶ Id_{D_{N/4}} \\ a & ⟶ ψ: D_{N/4} ⟶ D_{N/4},  b ⟶ c,  c ⟶ b. \end{matrix}

3. If σ(Γ_{1}) = (1, −, [−], {(2, …^{2k}, 2)}), the following table shows, for each signature of Γ′, the obtained quotient G = Γ′/Γ

σ(Γ′) Γ′/Γ
τ_{11} Z_{N} × Z_{2}, N even divisor of k
τ_{12} There is no epimorphism with Ker θ = Γ
τ_{13} D_{N/2} × Z_{2}, N divides 2k, N ≡ 2 (mod 4)
τ_{14} D_{N/2} × Z_{2}, N divides 2k, N ≡ 0 (mod 4).

This proves the first part of the theorem.

For the second part, let us suppose that Aut(X) = Z_{N} × Z_{2} , N even, then Aut(X) = Γ′/Γ were Γ′ has signature τ_{3} , τ_{5} , τ_{8} or τ_{11} . In any case N divides k, as we saw in the first part of the theorem. Now, if N is a proper even divisor k we can consider the signature τ_{5} . Since τ_{5}^{+} is maximal (see Section 2) there exist a maximal NEC group Γ′ with signature τ_{5} . Define θ as follows:

θ(e_{1}) = x, θ(e_{2}) = x^{−1}, θ(c_{i,j}) = y for i = 1, 2, j even,
θ(c_{i,j}) = 1 for i = 1, 2, j odd,

where the EH-involution ϕ = y and Γ′/Γ = ⟨x, y : x^{N} = y^{2} = (xy)^{2} = 1⟩. Then, the surface X = 𝒟/ Ker(θ) is elliptic-hyperelliptic and Aut(X) ≃ Z_{N} × Z_{2} .

If N = k, we have no maximality conditions for the signatures of Γ′, and we can prove that Z_{k} × Z_{2} can be extended to D_{k} × Z_{2} as an automorphism group.

The remaining cases are proved in a similar way.

For the sequel, we set the following group notation:

\begin{aligned}U_{N} & = ⟨x, y, z : x^{4} = y^{2} = z^{N} = xyz = 1, x^{2} = z^{N/2}⟩, \\ V_{N} & = ⟨x, y, z : x^{4} = y^{4} = z^{N} = xyz = 1, y^{2} = x^{2} = z^{N/2}⟩, \\ W_{N} & = ⟨x, y, z : x^{4} = y^{4} = z^{N/2} = xyz = 1, y^{2} = x^{2}⟩, \\ Q_{N} & = ⟨x, y, z : x^{2} = y^{2} = z^{4} = (xy)^{N/2} = 1, zxz^{−1} = y, (xy)^{N/4} = z^{2}⟩. \end{aligned}

Theorem 5.3. Let X be an orientable EHKS of algebraic genus p > 5, topological genus g ≥ 2 and k boundary components, 2 ≤ k ≤ 4. Then, Aut(X) is isomorphic in one of the following groups:

(i) U_{N}, Q_{N}, N = (2(p − 1))/(2l + 1) and l > 0, or V_{N}, N = (p − 1)/(l + 1) and l > 0, or D_{N/2} ∝ Z_{2} , N = (2(p − 1))/(2l + 1) and l ≥ 0 if k = 2.

(ii) W_{N}, N = (p − 1)/(l + 1) and l > 0, 4,

or D_{N/4} ∝ Z_{4} , N = (2(p − 1))/(2l + 1) and l > 0,
or (D_{N/4} × Z_{2}) ∝ Z_{2} , N = (4(p − 1))/(2l + 1) and l ≥ 0 if k = 4.
(iii) Z_{2} if k = 3.

Proof. Let G be an automorphism group of X containing the EH involution ϕ, such that |G| = 2N. Then G = Γ′/Γ for a certain NEC group Γ′ with signature of one of the types described in Proposition 4.3: τ_{15}, …, τ_{20} . The following table summarizes the groups obtained in each case:

σ(Γ′)  k  G = Γ′/Γ
(0, +, [2, …^{l+1}, 2, 4], {(−)}) 2  U_{N}
N = \frac{2(p − 1)}{2l + 1} even  4 D_{4} , l = \frac{1}{4}(p − 3)
(0, +, [2, …^{l}, 2, 4, 4], {(−)}) 2  V_{N}
N = \frac{p − 1}{l + 1} even  4  W_{N}
(0, +, [2, …^{l}, 2, 4], {(2, 2)}) 2  Q_{N}
N = \frac{2(p − 1)}{2l + 1} ≡ 0 (mod 4) 4  D_{N/4} ∝ Z_{4}
(0, +, [−], {(2, 2, 4, 4)})  2  D_{N/2} ∝ Z_{2}
N = \frac{2(p − 1)}{l + 1} ≡ 0 (mod 4) 4 (D_{N/4} × Z_{2}) ∝ Z_{4}
(0, +, [−], {(2, 2, 2, 4)})  2  D_{N/2} ∝ Z_{2}
N = \frac{4(p − 1)}{2l + 1} ≡ 0 (mod 4) 4 (D_{N/4} × Z_{2}) ∝ Z_{2}

Remark. It is easy to see that there exists no epimorphism from Γ′, with signature τ_{18} , onto G, such that Ker(θ) is a surface group. The same occurs for τ_{19} and τ_{20} if l > 0.

When k = 2 and l = 0, we can see that whenever U_{N}, V_{N}, Q_{N} or D_{p−1} ∝ Z_{2} is a group of automorphisms of an EHKS in the conditions of the theorem, it can be extended to D_{2(p−1)} ∝ Z_{2} . When k = 4 and l = 0, the automorphism groups W_{N}, D_{p−1} ∝ Z_{4} or (D_{p−1}×Z_{2}) ∝ Z_{2} can be extended to (D_{p−1}×Z_{2}) ∝ Z_{2} . To do it, we consider Theorem 4.1 in [7] and the following normal proper pairs (see Section 1):

(τ_{15}, τ_{20}), (τ_{16}, τ_{20}), (τ_{17}, τ_{20}), where l = 0, and (τ_{19}, τ_{20}).

In each case we only need to prove that the epimorphism θ: Γ′ → G, having kernel Γ, is unique up to automorphisms of Γ′ and G. This situation occurs when Γ′ has signature τ_{15} , τ_{16} , τ_{17} (with l = 0) or τ_{16} . We show the epimorphisms for each one:

– if σ(Γ′) = τ_{12} , G = U_{N}, then θ(x_{1}) = x, θ(x_{2}) = y, θ(e_{1}) = z, θ(c_{1,0}) = 1;

– if σ(Γ′) = τ_{13} , G = V_{N} or W_{N}, then θ(x_{1}) = x, θ(x_{2}) = y, θ(e_{1}) = z, θ(c_{1,0}) = 1;

– if σ(Γ′) = τ_{14} , G = Q_{N} or D_{p−1} ∝ Z_{4} = ⟨x, y, z : x^{2} = y^{2} = z^{4} = (xy)^{N/2} = 1, zxz^{−1} = y⟩, then θ(x_{1}) = z, θ(e_{1}) = z^{−1} , θ(c_{1,0}) = x, θ(c_{1,1}) = 1, θ(c_{1,2}) = y;

– if σ(Γ′) = τ_{16} , G has one of the following presentations

\begin{aligned}D_{p−1} ∝ Z_{2} & = ⟨x, y, z : x^{2} = y^{2} = z^{2} = (xy)^{p−1} \\ & = (yz)^{4} = (zx)^{4} = 1, (xy)^{(p−1)/2} = (yz)^{2} = (zx)^{2}⟩, k = 2, \end{aligned}

or

\begin{aligned}(D_{p−1}×Z_{2}) ∝ Z_{2} & = ⟨x, y, z : x^{2} = y^{2} = z^{2} \\ & = (xy)^{(p−1)/2} = (yz)^{4} = (zx)^{4} = 1, (yz)^{2} = (zx)^{2}⟩, k = 4. \end{aligned}

The epimorphism is

θ(e_{1}) = 1; θ(c_{1,0}) = x; θ(c_{1,1}) = 1; θ(c_{1,2}) = y; θ(c_{1,3}) = z; θ(c_{1,4}) = x.

Corollary. Let X be an orientable EHKS with algebraic genus p > 5 and topological genus g ≥ 2. If X has a non trivial automophism different from the EH-involution, then it has 2 or 4 boundary components.

Acknowledgments. I would like to express my acknowledgment to Professors José A. Bujalance and Ernesto Martínez who proposed me this work, and Emilio Bujalance for his helpful suggestions. Also, I would thank the referee for his comments.

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456 Beatriz Estrada

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Received 12 February 1999