Geometrical Characterization of p-Hyperelliptic Planar ... - CiteSeerX

Computational Methods and Function Theory Volume 2 (2002), No. 1, 267–279

Geometrical Characterization of p-Hyperelliptic Planar Klein Surfaces Beatriz Estrada (Communicated by Mario Bonk) Abstract. A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface can be seen as the quotient of the hyperbolic plane H2 under the action of a non-Euclidean crystallographic group (NEC group) Γ. The q-hyperelliptic Klein surfaces are characterized by the existence of an order two automorphism φ, called q-hyperelliptic involution, such that the quotient X/hφi has algebraic genus q. In this work, the geometry of the q-hyperelliptic involution is studied for planar surfaces. It is made by constructing fundamental regions that are right-angled hyperbolic polygons. These polygons are also interesting in the study of the Teichm¨ uller space of planar q-hyperelliptic Klein surfaces. Keywords. Non-euclidean crystallograhic groups, Klein surfaces, fundamental regions. 2000 MSC. 20H10, 30F50, 30F60.

1. Introduction Klein surfaces, introduced from a modern point of view by Alling and Greenleaf [AG], are surfaces endowed with a dianalytic structure. A compact Klein surface X is said to be q-hyperelliptic if and only if it admits an involution φ, that is an order two automorphism, such that the quotient space X/hφi is an orbifold with algebraic genus q. In the particular cases q = 0, 1, the surface X is hyperelliptic and elliptic-hyperelliptic respectively. Non-Euclidean crystallographic groups (NEC groups in short) were introduced by Wilkie and Macbeath. They are an important tool in the study of Klein surfaces. As the results of Preston [P] and May [My] show, Klein surfaces can be seen to be quotients of the hyperbolic plane H2 under the action of an NEC group Γ. Received January 16, 2003. Partially supported by BFM2002-04801. c 2002 Heldermann Verlag ISSN 1617-9447/$ 2.50 °

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In this paper we study q-hyperelliptic Klein surfaces that are topologically twospheres with k holes, k ≥ 3, usually called planar surfaces. Such a surface has the form X = H2 /Γ, where Γ is a surface NEC group with signature (1)

σ(Γ) = (0; +; [−]; {(−)k }),

k ≥ 3.

Each NEC group Γ has associated a fundamental region R which is a hyperbolic polygon with identified pairs of sides by transformations of Γ. We can obtain the surface X by making identifications in R. Furthermore, the geometry of appropriate fundamental regions can reflect the automorphisms of the quotient surface X = H2 /Γ. The q-hyperellipticity may be expressed, in terms of NEC groups, as follows: A planar Klein surface X = H2 /Γ is q-hyperelliptic if and only if there exists an NEC group Γ1 with algebraic genus q, such that Γ C2 Γ1 . The q-hyperelliptic involution is obtained as the quotient Γ1 /Γ = hφi. If k > 4q + 2 the group Γ1 is unique (the q-hyperellipticity group), and φ is central and unique in Aut(X). Furthermore, Γ1 is an NEC group with signature σ(Γ1 ) = (0; +; [−]; {(−)q , (2, 2(k−2q) . . . , 2)}) This characterization was obtained in [BE]. The aim of this work is the geometrical study of the q-hyperelliptic involution. To do it we construct fundamental regions for the NEC group Γ that are right-angled hyperbolic polygons. This generalizes results obtained in [CM] for hyperelliptic surfaces. In Section 3 we present a method to obtain canonical regions from Wilkie regions. In Section 4 the q-hyperelliptic character of planar surfaces is characterized by the existence of a symmetric canonical right-angled region. For the role of hyperbolic polygons in the study of Teichm¨ uller spaces see [ZVG] and [Bs]. These spaces parametrize classes of holomorphically equivalent marked surfaces. In Section 5 we study the Teichm¨ uller space of q-hyperelliptic planar Klein surfaces. Its dimension is calculated and it is described geometrically by means of congruence classes of polygons. In Section 2 we give the necessary preliminaries about Klein surfaces, NEC groups and fundamental regions.

2. NEC groups and fundamental regions An NEC group Γ is a discrete subgroup of isometries of the hyperbolic plane H2 , including reversing-orientation elements, with compact quotient X = H2 /Γ. Each NEC group Γ is given a signature [Ma] (2)

σ(Γ) = (g, ±, [m1 , . . . , mr ], {(ni1 , . . . , nisi ), i = 1, ..., k}),

where g, mi , nij are integers satisfying g ≥ 0, mi ≥ 2 and nij ≥ 2. The number g is the topological genus of X. The sign determines the orientability of X.

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The numbers mi are the proper periods corresponding to cone points in X. The brackets (ni1 , . . . , nisi ) are the period-cycles. The number k of period-cycles is equal to the number of boundary components of X. The numbers nij are the periods of the period-cycle (ni1 , . . . , nisi ). They are also called link-periods corresponding to corner points in the boundary of X. The number p = ηg + k − 1, where η = 2 or 1 if the sign of σ(Γ) is “+” or “−” respectively, is called the algebraic genus of X. The signature determines a canonical presentation [Ma] of Γ. The generators are xi , ei , ci,j , ai , b i , di ,

i = 1, . . . , r; i = 1, . . . , k; i = 1, . . . , k; j = 0, . . . , si ; i = 1, . . . , g; if σ has sign “+”; i = 1, . . . , g; if σ has sign “−”.

The relations are ci,j−1 2 = ci,j 2 r Y i=1

xi

k Y i=1

ei

i xm i = 1, i = 1, . . . , r; = (ci,j−1 ci,j )ni,j = 1, i = 1, . . . , k; j = 1, . . . , si ; ei −1 ci,0 ei ci,si = 1, i = 1, . . . , k;

g Y

(ai bi ai −1 bi −1 ) = 1, i = 1, . . . , g; if σ has sign “+”;

i=1

r Y i=1

xi

k Y i=1

ei

g Y

di 2 = 1, i = 1, . . . , g; if σ has sign “−”.

i=1

The isometries xi are elliptic, ei , ai , bi are hyperbolic, ci,j are reflections and di are glide reflections. Wilkie [W] found a fundamental region RW from which he obtained the algebraic structure of NEC groups. The region RW is called a Wilkie region. For an NEC group Γ with signature (2), RW is a hyperbolic polygon with sides labeled in anticlockwise order as follows . . . ξi , ξi0 , . . ., . . . εi , γi,0 , . . . , γi,si , ε0i , . . ., . . . , αi , βi0 , αi0 , βi , . . ., | {z } | {z } {z } | i=1,...,r

i=1,...,k

i=1,...,g

if the sign is “+”, or

. . . ξ , ξ 0 , . . ., . . . ε , γ , . . . , γ , ε0 , . . ., . . . , δi , δi∗ , . . ., {z } | i{z i } | i i,0 {z i,si i } | i=1,...,r

i=1,...,k

i=1,...,g

if the sign is “−”, where the side pairings are ei (ε0i ) = εi ,

ci,j (γi,j ) = γi,j ,

ai (αi0 ) = αi ,

bi (βi0 ) = βi ,

di (δi∗ ) = δi .

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Let us denote by ∠(s1 , s2 ) the angle between two consecutive sides. In RW we have ∠(εi , γi,0 ) + ∠(γi,si , ε0i ) = π, 2π , ∠(ξr , ξr0 ) = mi π ∠(γi,j−1 , γi,j ) = , ni,j and the sum of the remaining angles (accidental cycle) is 2π. Without any loss of generality (cf. [Mr]) we may suppose that RW is a convex polygon. Let Γ be an NEC group. Then the hyperbolic area of all fundamental regions of Γ are equal. This number is called the area of the group. If Γ has signature (2) it satisfies (see [S]) " ¶ ¶# si µ r µ k X 1 1 XX 1 1− (3) µ(Γ) = 2π ηg + k − 2 + . + 1− mi 2 i=1 j=1 ni,j i=1 An NEC group with signature (2) actually exists if and only if the right hand side of (3) is positive (see [ZVG]). If Γ is a subgroup of an NEC group Γ0 of finite index N then Γ is an NEC group too, and the following Riemann-Hurwitz formula holds. (4)

µ(Γ) = N µ(Γ0 ).

Let X be a Klein surface of topological genus g, with k boundary components and algebraic genus p ≥ 2. Then, by [P], there exists an NEC group Γ with signature (5)

σ(Γ) = (g; ±; [−], {(−), . k. ., (−)})

such that X = H2 /Γ, where the sign is “+” if X is orientable and “−” if not. An NEC group with signature (5) is called a surface group. For each automorphism group G of a surface X = H2 /Γ of algebraic genus p ≥ 2 there exists an NEC group Γ0 such that G = Γ0 /Γ where Γ ⊂ Γ0 ⊂ NG and NG denotes the normalizer of Γ in the group G, the full group of isometries of H 2 (cf. [My]). We give two previous results from [BE] in Propositions 1 and 2 for future reference. Proposition 1. a) The Klein surface X = H2 /Γ is q-hyperelliptic if and only if there exists an NEC group Γ1 with algebraic genus q such that Γ C2 Γ1 . b) Let X be a q-hyperelliptic Klein surface of algebraic genus p ≥ 2 such that p > 4q + 1. Then the group Γ1 is unique and the automorphism φ, hφi = Γ1 /Γ, is central in Aut(X).

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Proposition 2. A Klein surface X = H2 /Γ of topological genus 0 and k boundary components, k > 4q+2, is q-hyperelliptic if and only if there exists a (unique) NEC group Γ1 with signature (0; +; [−]; {(−)q , (2, 2(k−2q) . . . , 2)}), such that Γ C2 Γ1 .

3. Canonical regions Let X = H2 /Γ be a Klein surface of topological genus 0 and k ≥ 3 boundary components, where Γ is an NEC group of signature (1). Let RW be a Wilkie region of Γ. Then RW is a hyperbolic polygon with labeled sides ε1 , γ1 , ε01 , . . . , εk , γk , ε0k . Let us denote ∠(ε0i , εi+1 ) = θi , i = 1, . . . , k − 1 and ∠(ε0k , ε1 ) = θk . As we have seen in the previous section k X θi = 2π. i=1

Definition 3. RW is a minimal Wilkie region if ∠(εi , γi ) = ∠(γi , ε0i ) = π/2. That is, it is obtained by cutting from an interior point of X by orthogonal geodesics (εi ) to the boundaries (γi ).

Figure 1. Minimal Wilkie region of Γ.

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Let q be a natural number such that k > 2q +1, and RW a minimal Wilkie region of Γ. We consider in RW the following geodesic segments µi : λj :

orthogonal to γi and γi+1 , i = 1, . . . , k − 2q − 1; orthogonal to γ1 and γj , j = k − 2q + 1, . . . , k

(cf. Figure 1). Because of the convexity of RW the geodesics µi , λj do not intersect each other. To transform RW in a right-angled fundamental region of Γ let us define the following “cut and paste” procedures. Procedure Qµi , i = 2, . . . , k − 2q − 1: Cut in Ri−1 along µi the polygon Pi which contains the side ε0i and paste it via fi = e1 ◦ · · · ◦ ei . Let Ri = Qµi (Ri−1 ) = (Ri−1 − Pi ) ∪ fi (Pi ),

where R0 = RW .

Each step Qµi eliminates the angle θi , i = 1, . . . , k − 2q − 1, see Figures 2 and 3. The boundary components γi , i = 2, . . . , k − 2q − 1, become divided in two segments γ i and γ bi .

Figure 2. R1 . We now separate the 2q geodesics λi in two groups λi , λi ,

i = k − 2q + 1, . . . , k − q, i = k − q + 1, . . . , k.

For the first group we define the cuts as follows.

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Figure 3. Rk−2q−1 . Procedure Qλi , i = k − 2q + 1, . . . , k − q: Cut in Ri−1 along λi , the polygon Pi which contains the side εi and paste it via ei −1 . Let Ri = Qλi (Ri−1 ), i = k − 2q + 1, . . . , k − q, where Rk−2q = Rk−2q−1 . For the second group, we define the cuts in the following manner. Procedure Qλi , i = k, . . . , k − q + 1: Cut in Ri+1 along λi the polygon Pi which contains the side ε0i and paste it via ei . Let Rk+1 = Rk−q , Ri = Qλi (Ri+1 ), i = k, . . . , k − q + 1. Then Rq,k = Qλk−q+1 · · · (Qλk (Qλk−q · · · (Qλk−2q+1 (Qµk−2q−1 · · · (Qµ1 (RW )) . . .))) is a right angled fundamental region of Γ with 4(k − 1) sides (see Figure 4). We relabel the sides in anticlockwise order and obtain the perimeter

(6)

γ1,k−q , . . . , λi , γi , λ0i , γ1,i−1 , . . ., µ1 , . . . γ i , µi . . ., γk−2q , | {z } | {z } i=2,...,k−2q−1

i=k−q,...,k−2q+1

. . . µ0i , γ bi

|

{z

. . ., µ01 , . . . , γ1,j+1 , λ0j , γj , λj , . . ., }

i=k−2q−1,...,2

|

{z

i=k,...,k−q+1

}

where γ1,i , i = k − 2q, . . . , k, denotes the images of the segments in which γ1 is divided by λi , and γ i , γ bi , i = 2, . . . , k − 2q − 1, are the transforms of γ i , γ bi .

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Figure 4. Rq,k . Let us denote by |l| the length of a geodesic segment l. Then |γ1 | = |γ1,k−2q | + · · · + |γ1,k |, |γi | = |γ i | + |b γi |,

for i = 2, . . . , k − 2q − 1.

The identifications in Rq,k are made as follows. λ0i = gi (λi ), gi = ek−q −1 ◦ · · · ◦ ei −1 , for i = k − q, . . . , k − 2q + 1; λ0i = hi (λi ), hi = ek−q+1 ◦ · · · ◦ ei , for i = k − q + 1, . . . , k; µ0i = li (µi ), li = hk ◦ fi ◦ gk−2q+1 −1 , for i = 1, . . . , k − 2q − 1. We have thus proved the following result. Proposition 4. Let X = H2 /Γ be a planar Klein surface with k ≥ 3 boundary components, where Γ is an NEC group of signature (0, +, [−], {(−) k }), and let q be a natural number such that k > 2q +1. Then there exists a fundamental region of Γ that is a right-angled polygon with 4(k − 1) sides following the identification pattern of Rq,k . Definition 5. A right-angled polygon with 4(k − 1) and side-pairings as in (6) is called a q-canonical right-angled region of Γ. Let us note that Γ has different q-canonical right-angled regions. The method shown above to obtain this type of regions starts with a Wilkie region that is not unique.

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4. The q-hyperelliptic character Let X = H2 /Γ be a Klein surface of topological genus 0 and k ≥ 3 boundary components, where Γ is an NEC group of signature (0, +, [−], {(−)k }).

(7)

In this section we assume k > 4q + 2. Under this condition, by Proposition 1, if X is q-hyperelliptic then the q-hyperelliptic involution is unique and central in the group Aut(X). Let us call µ the common orthogonal to the sides γ1,k−q and γk−2q in a canonical right-angled region (see Figure 4). The aim of this section is to prove the following geometrical characterization of the q-hyperelliptic character. Theorem 6. The surface X = H2 /Γ is q-hyperelliptic if and only if there exists a q-canonical right-angled region symmetric with respect to µ. Proof. Let R be a q-canonical right-angled region symmetric with respect to µ. The reflection in R that fixes the geodesic µ, induces in X an automorphism. which we denote by σµ . The quotient space S1 = S/hσµ i has as fundamental region any of the two polygons in which µ divides R. Let us take R ∗ as in Figure 4. The side-pairings are λ0i = hi (λi ),

i = k − q + 1, . . . , k.

Then S1 = H2 /Γ1 , where Γ1 is an NEC group of signature (0, +, [−], {(−)q , (2, 2(k−2q) . . . , 2)}),

(8)

and, by Proposition 2, X is q-hyperelliptic. Now let us suppose that X is q-hyperelliptic and that φ is the q-hyperelliptic involution. Then, by Proposition 2, there exists an NEC group Γ1 with signature (8) such that Γ C2 Γ1 . Furthermore, Γ1 is unique and hφi = Γ1 /Γ. Let us consider the canonical presentation of Γ1 . The generators are e1 , . . . , eq+1 , hyperbolics; ci,0 , i = 1, . . . , q, reflections generating the empty boundaries; b c0 , c1 , b c 1 , c2 , b c2 , . . . , ck−2q , b ck−2q , reflections generating the non-empty boundary.

The relations are

ei −1 ci,0 ei eq+1b c0 eq+1 −1 cj 2 e1 . . . eq+1

= = = =

ci,0 , b ck−2q ; 1; 1.

i = 1, . . . , q;

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Let R1 be a Wilkie minimal region of Γ1 with labeled sides ε1 , γ1,0 , ε01 , . . . , εq , γq,0 , ε0q , εq+1 , γˆ0 , γ1 , γˆ1 , . . . , γk−2q , γˆk−2q , ε0q+1 ci (ˆ γi ) = γî , ci (γi ) = γi , such that ei (ε0i ) = εi , ci,0 (γi,0 ) = γi,0 , i = 1, . . . , q, b i = 0, . . . , k − 2q.

We consider the canonical epimorphism θ : Γ1 → Γ1 /Γ, where Γ1 /Γ ∼ = Z2 = hx : x2 = 1i, defined by θ(ci,0 ) θ(ei ) θ(ˆ ci ) θ(ci )

= = = =

1, 1, x, 1,

i = 1, . . . , q; i = 1, . . . , q + 1; i = 0, . . . , k − 2q; i = 1, . . . , k − 2q.

Then R = R1 ∪ b c0 (R1 ) is a fundamental region of Γ. Now we can obtain a canonical right-angled region of Γ from R. To do it let us consider in R the geodesic segments λi : common orthogonal to γ1 and γi,0 , b c0 (λi ) : common orthogonal to b c0 (γ1 )yb c0 (γi,0 ),

i = 1, . . . , q; i = 1, . . . , q.

We define the following cut and paste procedures.

Procecure Qλi : Cut along the geodesic λi , the polygon Pi which contains εi , and paste it via ei −1 , i = 1, . . . , q. Procedure Q∧ : Cut along the geodesic b c0 (λi ), the polygon Pbi which contains c 0 (λi )

b c0 (εi ), and paste it via b c0 ei −1b c0 , i = 1, . . . , q. Then,

ˆ = (Qbc (λq ) . . . (Qbc (λ ) (Qλq . . . (Qλ1 (R)) . . .))) R 0 0 1 is a q-canonical right-angled region of Γ with the desired symmetry. Similar characterizations (by means of right-angled hyperbolic polygons) of qhyperelliptic bordered tori (surfaces of genus 1 with boundary) can be seen in [EMr].

5. The Teichm¨ uller space In this section we study the Teichm¨ uller space associated to q-hyperelliptic planar Klein surfaces. Let G be the full group of isometries of the hyperbolic plane D. Given an NEC group Λ, considered as an abstract group, let us denote by R(Λ, G) the set of monomorphisms r : Λ → G such that r(Λ) is a discrete group and the quotient

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D/r(Λ) is compact. Two elements r1 , r2 ∈ R(Λ, G) are equivalent, r1 ∼ r2 , if and only if there exists an element g ∈ G satisfying r1 (λ) = gr2 (λ)g −1 , for every λ ∈ Λ. The quotient space T (Λ, G) = R(Λ, G)/ ∼ is called the Teichm¨ uller space of Λ. This space is homeomorphic to a cell with dimension d(Λ). If Λ is an NEC group P with signature (2) it is well known that d(Λ) = 3p − 3 + 2r + s, where s = ki=1 si is the number of periods, r is the number of proper periods and p is the algebraic genus. Let us denote by Tk the Teichm¨ uller space of planar Klein surfaces with k boundary components, that is Tk = T (Γ, G) where Γ is an NEC group with signature (0, +, [−], {(−)k }). This space parametrizes classes of holomorphically equivalent (marked) planar Klein surfaces. Then the set Tkq = {[r] ∈ Tk : H2 /r(Γ) is a q-hyperelliptic planar surface} is the Teichm¨ uller space of q-hyperelliptic planar Klein surfaces. Let us suppose k > 4q + 2. If H2 /Γ is the q-hyperelliptic planar Klein surface in Proposition 2. Then the q-hyperelliptic group Γ1 is unique and has signature (0; +; [−]; {(−)q , (2, 2(k−2q) . . . , 2)}). Moreover, in [BE] it was proven that the canonical epimorphism θ : Γ1 → Γ1 /Γ w Z2 is unique up to automorphisms of Γ1 and Z2 . The uniqueness of Γ1 and φ are sufficient conditions to apply Maclachlan’s method ([Mc, Lemma 3]) and to conclude that Tkq is a submanifold of Tk of dimension d(Γ1 ) = 2k − q − 3. We have proven the following result. Theorem 7. The Teichm¨ uller space of q-hyperelliptic planar Klein surfaces, with k > 4q + 2 boundary components Tkq is a submanifold of Tk of dimension 2k − q − 3. Other approxinations to the Teichm¨ uller space can be done by means of congruence classes of marked hyperbolic polygons. From the study of these polygons different parametrizations of Teichm¨ uller spaces have been obtained (cf. [ZVG] and [Bs] for the case of Riemann surfaces). The idea is as follows: as we have seen, two elements r1 , r2 ∈ R(Γ, G) belong to the same class in T (Γ, G) if and only if there exists g ∈ G such that r1 (γ) = gr2 (γ)g −1

for all γ ∈ Γ.

Equivalently, canonical fundamental polygons of the NEC groups r1 (Γ) and r2 (Γ) are isometric. Furthermore, the number of parameters (side-lengths) involved in the construction of such polygons equals the dimension of the Teichm¨ uller space T (Γ, G).

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In the particular case of Tkq we can associate to each class [r] a congruence class of right-angled polygons [Pr(Γ) ], where Pr(Γ) is the q-canonical marked polygon of the marked NEC group r(Γ) defined in the previous section. To compute the number of side-lengths which determine completely a q-canonical polygon Pr(Γ) we consider the following facts: Pr(Γ) is right-angled and has 4(k − 1) sides as in (6). Furthermore, Pr(Γ) is symmetric with respect to µ. So we restrict the necessary side-lengths to the 2k sides of the subpolygon R∗ in Figure 4. Finally, if we consider the q side-pairings in this polygon, we have 2k − q − 3 (the dimension of Tkq ) side-lengths: |µi |, |γi | , |ˆ γi | = 2 |λi |, |γi |, |γ1,i+1 |,

i = 1, . . . , k − 2q − 1, distances between boundary components; i = 2, . . . , k − 2q − 1, lengths of half boundary components; i = k − q + −1, . . . , k, distances between boundary components; i = k − q + −1, . . . , k, lengths of full boundary components; i = k − q + −1, . . . , k, segments of the divided boundary γ.

In special cases of side-lengths of the polygons it is possible to obtain the matrices of the generators of the groups. Acknowledgement. This paper is a part of the author’s PhD thesis. I wish to express my gratitude to my supervisors José A. Bujalance and Ernesto Mart´ınez. Also I thank professor Antonio Costa for his corrections.

References AG. N. L. Alling and N. Greanleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Mathematics, Vol. 219, Springer-Verlag, 1971. BE. E. Bujalance and J. J. Etayo, A characterization of q-hyperelliptic compact planar Klein surfaces, Abh. Math. Sem. Univ. Hamburg 58 (1988), 95–104. Bs. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Vol. 106. Birkhäuser, 1992. CM. A. F. Costa and E. Mart´ınez, Planar hyperelliptic Klein surfaces and fundamental regions of NEC groups, in: Discrete Groups and Geometry, London Math. Soc. Lect. Not. Series 173 (1992), 57–65. EMr. B. Estrada and E. Mart´ınez, On q-hyperelliptic k-bordered tori, Glasgow Math. J. 43 (2001), 343–357. Ma. A. M. Macbeath, The classification of non-Euclidean crystallographic groups, Canad. J. Math. 6 (1967), 1192–1205. Mc. C. Maclachlan, Smooth coverings of hyperelliptic surfaces, Quart. J. Math. Oxford Ser (2) 22 (1971), 117–123. Mr. E. Mart´ınez, Convex fundamental regions for NEC groups, Arch. Math. 47 (1986), 457– 464. My. C. L. May, Large automorphism groups of compact Klein surfaces with boundary, Glasgow Math. J. 18 (1977), 1–10. P. R. Preston, Projective Structures and fundamental domains on compact Klein surfaces, Thesis Univ. of Texas, 1975.

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D. Singerman, On the structure of non-euclidean crystallographic groups, Proc. Cambridge Phil. Soc. 76 (1974), 233–240. W. H. C. Wilkie, On non-Euclidean Crystallographic groups, Math. Z. 91 (1966), 87–102. ZVG. H. Zieschang, E. Vogt, and H. D. Coldewey, Surfaces and Planar Discontinuous Groups, Lecture Notes in Mathematics, Vol. 835, Springer-Verlag, 1980. Beatriz Estrada E-mail: [email protected] Address: Departamento de Matem´ aticas Fundamentales, Facultad de Ciencias, U.N.E.D., Paseo Senda del Rey, 9, 28040 Madrid, Spain