q hyperelliptic compact nonorientable Klein surfaces without boundary

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q

-HYPERELLIPTIC COMPACT NONORIENTABLE KLEIN

SURFACES WITHOUT BOUNDARY

J. A. BUJALANCE and B. ESTRADA

Received 21 September 2001

LetXbe a nonorientable Klein surface (KS in short), that is a compact nonorientable sur-face with a dianalytic structure defined on it. A Klein sursur-faceXis said to beq-hyperelliptic

if and only if there exists an involutionΦ onX(a dianalytic homeomorphism of order two) such that the quotientX/Φhas algebraic genusq.q-hyperelliptic nonorientable KSs without boundary (nonorientable Riemann surfaces) were characterized by means of non-Euclidean crystallographic groups. In this paper, using that characterization, we deter-mine bounds for the order of the automorphism group of a nonorientableq-hyperelliptic Klein surface X such thatX/Φ has no boundary and prove that the bounds are at-tained. Besides, we obtain the dimension of the Teichmüller space associated to this type of surfaces.

2000 Mathematics Subject Classification: 30F50, 20H10.

1. Introduction. AKlein surfaceX (KS in short) is a compact surface with a di-analytic structure defined on it [1]. Nonorientable KSs without boundary are also called nonorientable Riemann surfaces. A dianalytic homeomorphism ofX onto it-self is called an automorphism ofX. We say that a Klein surfaceX isq-hyperelliptic if and only if there exists an involution Φ onX such that the quotientX/Φ has algebraic genus q. (If q=0, then X is called hyperelliptic, and if q =1, then X is elliptic-hyperelliptic.)

LetXbe a nonorientableq-hyperelliptic KS without boundary and Aut(x)the full group of its automorphisms. It is known [11] that|Aut(x)| ≤84(p−1), wherepis the algebraic genus ofX. This type of surfaces was characterized in [7] by means of non-Euclidean Crystallographic groups, and there, the bound 84(p−1)was reduced to 12(p−1)if the quotientX/Φhas boundary.

In this paper, we improve the bound 84(p−1) when X/Φ has no boundary. The new bound will depend on the parity of|Aut(X)|. That is made in Section 3. InSection 4, we prove that the bounds are attained by showing different examples. In

Section 5, the dimension of the Teichmüller space associated to this type of surfaces is obtained.

Similar characterizations for compact orientable KS without boundary (i.e., Riemann surfaces) and compact KS with boundary have been obtained in [2,3,4,10].

Wilkie [14] and Macbeath [9] associated to each NEC group a symbol, calledsignature, that determines its algebraic structure and has the following form:

σ=g;±;m1, . . . , mr;ni1, . . . , nis_i, i=1, . . . , k, (2.1) where g, mi, andnij are integers satisfying g≥0, mi≥2, and nij ≥2; g is the topological genus ofX. The sign determines the orientability ofX. The numbersmiare theproper periodscorresponding to cone points inX. The brackets(ni1, . . . , nis_i)are theperiod-cycles. The numberkof period-cycles is equal to the number of boundary components ofX. The numbersnij are the periods of the period-cycle(ni1, . . . , nisi) also calledlink-periods, corresponding to corner points in the boundary of X. The numberp=αg+k−1 is called thealgebraic genusofX, whereα=1 or 2, if the sign ofσ (Γ)is “−” or “+”, respectively. If the sign ofσis “+” andk=0, thenΓis a Fuchsian group.

An NEC groupΓ with signatureσ has the following presentation: Generators:

xi, i=1, . . . , r;

ei, i=1, . . . , k;

cij, i=1, . . . , k;j=0, . . . , si;

ai, bi, i=1, . . . , g, (ifσ has sign+);

di, i=1, . . . , g, (ifσ has sign−).

(2.2)

Relations:

ximi, i=1, . . . , r;

cij−12=cij2=cij−1cijnij, i=1, . . . , k; j=1, . . . , si;

ei−1ci0eicisi=1, i=1, . . . , k;

x1···xre1···eka1b1a−11b1−1···agbga−g1b−g1=1 (ifσ has sign+),

x1···xre1···ekd12···d2g=1 (ifσ has sign−).

(2.3)

It is known [9] that cyclic permutations of periods in period-cycles or arbitrary permutations of proper periods in the signatureτof an NEC groupΓlead to a signature

τof an NEC groupΓisomorphic toΓ. EveryΓ with signature (2.1) has associated a fundamental region whose area,|Γ|, is also calledthe area of the group[12]:

|Γ| =2π 

αg+k−2+ r

i=1

1−_m1 i

+1₂

k

i=1 si

j=1

1−_n1 ij



. (2.4)

An NEC groupΓ with signatureσ actually exists if and only if the right-hand side of (2.4) is greater than zero [15]. IfΓis a subgroup of an NEC groupΓ of finite index

N, then it is also an NEC group and the following Hurwitz-Riemann formula holds:

LetXbe a nonorientable Klein surface without boundary of topological genusg≥3. Then by [11], there exists an NEC groupΓ with signature

g;−;[−];{−} (2.6)

such thatX=Ᏸ/Γ.

An NEC group with signature (2.6) is said to be anonorientable surface group. In [11] it was proved that ifGis a group of automorphisms of a nonorientable Klein surface without boundaryX=Ᏸ/Γ, thenGcan be presented as a quotientΓ_{/Γ, for some NEC} groupΓsuch thatΓΓ. The full group of automorphisms ofXis Aut(X)=NᏳ(Γ)/Γ, whereN_Ᏻ(Γ)is the normalizer ofΓ inᏳ.

In general, it is very difficult to decide whether a group of automorphismsG=Γ_/Γ of a Klein surfaceX=Ᏸ/Γ equals Aut(X)or not. This problem is equivalent to the problem of deciding whetherΓ_{equals the normalizer ofΓ inᏳor not.}

From now on, KS is assumed to have algebraic genus not smaller than 2. The following results, that appear in [7], will be used through this paper.

Theorem_2.1. _LetX=_D/Γ _{be a nonorientable KS without boundary of algebraic}

genusp. Then,Xisq-hyperelliptic if and only if there exists an NEC groupΓ1, containing

Γ as a subgroup of index2and having the signature of one of the following types: (a) (h;+;[2,(p+1)−2q. . . ,2];{(−)q−2h+1}),0≤h≤q/2;

(b) (h;−;[2,(p+_{. . .}1)−2q_,2];{(−)q−h+1_{}),0≤h≤qifpis even,1≤h≤q+1ifpis odd.} Moreover,

(1) for each0≤h≤q/2, there exists a nonorientable q-hyperelliptic KS without boundaryD/Γ whoseq-hyperellipticity group has the signature (a);

(2) for each 0 ≤h ≤q (if p is even) and for each 1≤ h≤ q+1 (if p is odd), there exists a nonorientableq-hyperelliptic KS without boundaryD/Γ whoseq -hyperellipticity group has the signature (b).

Theorem2.2. LetX=D/Γ be a nonorientableq-hyperelliptic KS without boundary

of algebraic genusp >4q+1, then theq-hyperelliptic involutionΦis central and unique inAut(X).

3. Bounds for the order of the automorphism group. LetX=Ᏸ/Γ be a nonori-entableq-hyperelliptic KS without boundary, and letΦbe theq-hyperellipticity invo-lution. ThenΦ Γ1/Γ for a certain NEC groupΓ1and, byTheorem 2.1,X/Φ Ᏸ/Γ1 is an orientable KS with boundary, or a nonorientable KS (with or without boundary). A group of automorphisms ofXcan be represented as a quotientΓ_{/Γ whereΓΓ.} SinceΦis central and unique, we haveΓΓ1Γ.

In the next theorem we prove that the bound 84(g−1), for the order of an auto-morphism group of a nonorientable KS without boundaryX, can be improved when

Xisq-hyperelliptic andX/Φhas empty boundary.

Theorem_3.1. _LetX=_D/Γ _{be a nonorientableq-hyperelliptic KS without boundary}

of algebraic genusp >4q+1. LetΦ be theq-hyperelliptic involution and X/Φ a surface without boundary. Then,

(2) if|Aut(X)| =2N,Nis even, and

(a) signature ofΓ_{has sign “−”, then|Aut(X)| ≤12(q−1),} (b) signature ofΓ_{has sign “+”, then|Aut(X)| ≤24(q−1).}

Proof. Sinceσ (Γ)=(p+1;−;[−]{−}),p >4q+1, andX/Φhas no boundary, by

Theorem 2.1there exists a unique NEC groupΓ1with the signature

σΓ1

=q+1;−;2,(p+_{. . .}1)−2q_{,2, pis odd. (3.1)}

(I) Suppose[Γ_:Γ

1]=Nis odd. We know by [5, Chapter 2] that the signature ofΓ is

σΓ_=g;−;m

1, . . . , mt. (3.2)

Let xi ∈Γ, i=1, . . . , t, be the generators of proper-periods such that pi is the smallest integer satisfyingx_ip∈Γ1(pi≠0). As the only elements of finite order inΓ1 have order two, then (see [5, Section 2.2])

pi=

mi

2 or pi=mi. (3.3)

We can suppose, without loss of generality, that for 1≤i≤n,pi=mi/2 and for

n+1≤i≤t,pi=mi. Thus, by [5, Section 2.2]

p−2q+1= n

i=1 N pi = n

i=1

2N mi

. (3.4)

From the Riemann-Hurwitz formula|Γ1| =N|Γ|, we have

(q−1)+(p−2q+1)

2 =N



g−2+ n

i=1

1− 1 mi + t i=n+1

1− 1 mi



. (3.5)

By (3.4) and (3.5),

(q−1)=N 

g−2+ n

i=1

1−_m1 i

−

n

i=1 1 mi + t

i=1

1−_m1 i

 

=N



g−2+ n

i=1

1− 2 mi + t i=n+1

1− 1 mi

 ,

(3.6)

therefore,

N=Γ1 |Γ_| =

(q−1)

g₋₂₊n

i=11−2/mi+t_i=n+11−1/mi

Sincep >4q+1, then by (3.4) we have n i=1 N mi

> (q+1). (3.8)

We fixqin (3.7) and denote

A=g−2+ n

i=1

1− 2 mi + t i=n+1

1− 1 mi

, 2≤mi≤ ∞. (3.9)

We will consider several cases to determine the minimum positive value ofA(that makesNmaximal). Fornandtfix

(i) ifg≥2, thenA(1)_≥n

i=1(1−2/mi)+ t

i=n+1(1−1/mi) >0; (ii) ifg=1, then

A(2)= n

i=1

1−_m2 i + t i=n+1

1−_m1 i

−1>0,

A(1)_{> A}(2)_>0,

A(2)≥k2

1−2 2

+k4

1−2 4

+k6

1−2 6

+k2

1−1 2

+k3

1−1 3

−1

=1₂k4+k2

+2₃k6+k3

−1;

(3.10)

wherek4,k6,k2, andk3are nonnegative integers withk4+k6+k2+k3≤t. The values ofkiandki, for which the above expression is minimum occurs only fork4+k2=1,

k6+k3=1, but these imply thatNis even, a contradiction. Then, the suitable ones arek4+k2=0 andk6+k3=2, that is,

k6=2, k4=k2=k3=0;

k3=2, k4=k6=k2=0;

k₃=k6=1, k2=k4=0;

(3.11)

andN(max)=3(q−1). Consequently,|Aut(X)| ≤6(q−1)and the signature ofΓhas one of the following signatures:

τ1=

1,−,2,_{. . .,}k2 _{2,6,6, q >}4 3

1

k2+

1, p=3k2+4

(q−1)+1;

τ2=

1,−,2,_{. . .,}k2 _{2,3,3, q >}

3k2+2 3k2−2

, p=(q−1)3k2+2

+1;

τ3=

1,−,2,_{. . .,}k2 _{2,6,3, q >}

3k2+3 3k2−1

, p=(q−1)3k2+3

+1.

(3.12)

(II) Now, suppose that[Γ_{:Γ1]=Nis even. We know (see [5,6]) that the signature} ofΓ_is

σΓ_=g;±;m

1, . . . , mt];

ni1, . . . , nisi

The proper-periods ofΓ1may proceed from elliptic elementsxior period-cycles ofΓ. Ifpiis the order ofΓ1xi∈(Γ/Γ1), thenxigenerates inΓ1,

n

1(N/pi)proper periods, all of them being equal to 2.

Now consider the periods inΓ1provided by reflections ofΓnot inΓ1. Letqijbe the smallest integer for which(cij, cij−1)qij ∈Γ1, since the only elements of finite order are those of order 2, we have(nij/qij)=2 or 1. LetE= {(i, j)|(nij/qij)=2}. Clearly, the only proper-periods inΓ1induced by reflections ofΓare those provided by pairs of reflections corresponding to elements ofE, and each such pair produces(N/2qij) periods, all of them being equal to 2. As a result, we obtain(i,j)∈E(N/2qij) proper-periods inΓ1. Then by (3.4)

p−2q+1= n 1 2N mi + (i,j)∈E N

2qij

. (3.14)

(A) If the sign in the signature ofΓ_{is minus, by the Riemann-Hurwitz formula, we} hold the expression

N(−)=Γ1 |Γ_|

= (q−1)

g+k−2+n_i=11−_m2 i

+t

n+1

1−_m1 i

+1

2

(i,j)∈E

1−_n2 ij

+1

2

(i,j)∉E

1−_n1 ij

.

(3.15)

Sincep >4q+1, then by (3.14) we have

n 1 N(−) mi +

(i,j)∈E

N(−)

2nij

> (q+1). (3.16)

As in previous case (Nodd), we are looking for the value of parameters in (3.15), that makesN(−)the maximum possible. Let

A=g−2+B1+B2, (3.17)

where

B1= n

i=1

1−_m2 i

+

t

i=n+1

1−_m1 i

,

B2=k+ 1 2

(i,j)∈E

1− 2 nij

+1

2

(i,j)∉E

1− 1 nij

.

(3.18)

Forn,t,k, andEfix, we have the following cases: (i) ifg≥2:A(1)_≥B

1+B2; (ii) ifg=1:A(2)_=B

1+B2−1. (ObviouslyA(1)≥A(2).) As we have seen, in the case whereNis odd,

B1≥B1∗= 1 2

k4+k2

+2₃k6+k3

The minimum positive valueA∗ ofAoccurs whenk=0,k4+k

2=1, andk6+k

3=1 inB1∗; soB∗1=7/6 andA∗=7/6−1=1/6≤B1(min)−1≤B1(min)+B2(min)−1.

Therefore,N(max)=6(q−1)and

Aut(X)≤12(q−1). (3.20)

The possible values ofk4,k2,k6, andk3, lead us to conclude thatΓhas one of the following signatures:

τ4=

1,−,2,_{. . .,}k2 _{2,4,6, q >}

6k2+9 6k2+5

, p=6k2+7

(q−1)+1;

τ5=1,−,2,. . .,k2 2,2,6, q >

6k2+2 2k2

, p=6k2+4(q−1)+1;

τ6=

1,−,2,_{. . .,}k2 _{2,4,3, q >}

6k2+5 6k2+1

, p=6k2+5

(q−1)+1;

τ7=

1,−,2,_{. . .,}k2 _{2,2,3, q >}

3k2+1 3k2−1

, p=6k2+2

(q−1)+1.

(3.21)

(B) If the sign in the signature ofΓ_{is positive, we have, by the Riemann-Hurwitz} formula, the following expression:

N(+)=Γ1 |Γ| =

(q−1)

2g−2+B1+B2

, (3.22)

whereB1andB2are as in (3.18).

LetA=2g−2+B1+B2. As in the previous case we are going to minimizeA. We have forn,t,k, andEfix the following cases.

Case₁. _Ifg≥_{1, thenA}≥_B1+_B2.

Case₂. _Ifg=_{0, thenA}≥_A∗=_B1+_B2−_2.

Since the sign in the signature ofΓ1is negative, thenΓmust have some period-cycle, that is,k≠0.

Ift=0, thenA∗=B2−2 and

B2=1+ 1 2

(i,j)∈E

1− 2 nij

+1

2

(i,j)∉E

1− 1 nij

≥1+1₂N2

1−2₂

+1₂N4

1−2₄

+1₂N6

1−2₆

+1₂N2

1−1₂

+1₂N3

1−1₃

=1+1₄N4+N2

+1₃N6+N3

,

then

A∗≥1 4

N4+N2

+1 3

N6+N3

+0N2−1>0, (3.24)

where the nonnegative integersNiare the number ofnij=i,(i, j)∈E, andNiis the number ofnij=i,(i, j)∉E. The value of (3.24) will be minimum for

N4+N

2=3, N6+N

3=1, (3.25)

so,N(max)=12(q−1)and

_{Aut(X)≤24(q−1). (3.26)}

From conditions (3.25), we obtain the following signatures forΓ_:

τ8=0,+, [−],2,. . .,N2 2,2,4,4,6,

τ9=

0,+, [−],2,_{. . .,}N2 _2,2,4,4,3,

τ10=0,+, [−],2,. . .,N2 2,2,2,4,6,

τ11=

0,+, [−],2,_{. . .,}N2 _2,2,2,4,3,

τ12=

0,+, [−],2,_{. . .,}N2 _2,2,2,2,6,

τ13=

0,+, [−],2,_{. . .,}N2 _2,2,2,2,3,

τ14=

0,+, [−],2,_{. . .,}N2 _2,4,4,4,6,

τ15=0,+, [−],2,. . .,N2 2,4,4,4,3.

(3.27)

Now, ift≠0, thenA∗=B1+B2−2 and

A∗=1 2

k4+k2)+ 2 3

k6+k3

+1 4

N4+N2

+1 3

N6+N3

−1>0. (3.28)

The smallest value ofAappears under the following conditions:

k6+k3=0, N6+N3=1, N4+N2=3, k4+k2=0 (3.29)

or

k6+k3=0, N6+N3=1, N4+N2=1, k4+k2=1. (3.30)

ThenN(max)=12(q−1)and

By elementary combinatorial methods, we obtain 16 solutions for the signature ofΓ

τ16=0,+,2,. . .,k2 2,2,N2. . .,2,2,2,2,6,

τ17=0,+,2,. . .,k2 2,2,N. . .,2 2,2,2,4,6,

τ18=0,+,2,. . .,k2 2,2,N. . .,2 2,2,4,4,6,

τ19=

0,+,2,_{. . .,}k2 _2,2,N_{. . .,}2 _2,4,4,4,6,

τ20=

0,+,2,_{. . .,}k2 _{2,2,(2,. . .,}N2 _2,2,6,

τ21=

0,+,2,_{. . .,}k2 _2,2,2,N_{. . .,}2 _2,4,6,

τ22=

0,+,2,_{. . .,}k2 _2,4,2,N_{. . .,}2 _2,2,6,

τ23=

0,+,2,_{. . .,}k2 _2,4,2,N_{. . .,}2 _2,4,6,

τ24=0,+,2,. . .,k2 2,2,N. . .,2 2,2,2,2,3,

τ25=0,+,2,. . .,k2 2,2,N. . .,2 2,2,2,4,3,

τ26=0,+,2,. . .,k2 2,2,N. . .,2 2,2,4,4,3,

τ27=

0,+,2,_{. . .,}k2 _2,2,N_{. . .,}2 _2,4,4,4,3,

τ28=

0,+,2,_{. . .,}k2 _2,2,2,N_{. . .,}2 _2,2,3,

τ29=

0,+,2,_{. . .,}k2 _{2,2,(2,. . .,}N2 _2,4,3,

τ30=

0,+,2,_{. . .,}k2 _2,4,2,N_{. . .,}2 _2,2,3,

τ31=

0,+,2,_{. . .,}k2 _{2,4,(2,. . .,}N2 _2,4,3.

(3.32)

Corollary_3.2. _LetX=_D/Γ _{be a nonorientableq-hyperelliptic KS without}

bound-ary of algebraic genusp >4q+1and|Aut(X)| =Γ_{/Γ. Let Φbe the q-hyperelliptic} involution such thatX/Φhas no boundary. Then,

(1) if|Aut(X)| =2N,Nis odd, then|Aut(X)| ≤(3/2)(p−6); (2) if|Aut(X)| =2N,Nis even, and

(a) signature ofΓ_{has sign “−”, then|Aut(X)| ≤3(p−6),} (b) signature ofΓ_{has sign “+”, then|Aut(X)| ≤6(p−6).}

4. Examples. It is natural to ask for what values ofqthe bounds ofTheorem 3.1are attained. We have found examples for low values ofq. Here we present some examples of surfaces with maximal automorphism group for the three different bounds.

In the remainder of this section, θ denotes the canonical epimorphism from Γ ontoGΓ_{/Γ, whereΓhas one of the signatures obtained inSection 3τ1, . . . , τ31. We} denote byΦtheq-hyperelliptic involution,π:G→G/Φthe natural epimorphism and ¯θ=π θ.

The procedure we are going to follow in these examples is as follows:

(a) to prove the existence of epimorphismsθand ¯θas above, such that ker ¯θΓ1, ker(θ)Γ, and ¯θ=π θ;

(b) consequently, we obtain the realization (fulfillment) of the automorphism group

Example4.1. IfNis odd,σ (Γ)=(1,−, [2,2,6,6]), andq=2, then|Aut(X)| ≤6.

Let ¯θ:Γ_→G¯_Γ/Γ

1be the canonical epimorphism defined as follows: ¯

θd1=1, θ¯x1=y,¯ θ¯x2=y¯−1, θ¯x3=θ¯x4=¯1, (4.1)

whered1,x1,x2,x3, andx4are the generators ofΓ, and ¯yis an element of order three inΓ_{/Γ1.As|Γ/Γ1| =3 and it is a group generated by ¯y, with the relation ¯y}3_{=1, we} haveΓ_/Γ

1C3; so thatG(Γ/Γ)would have order 6 and its quotient by a subgroup generated by a central element must beC3. ThenΓ_{/Γ isC6.}

Finally, we need to prove that there exists a nonorientable 2-hyperelliptic KS without boundary of topological genusp=11 whose automorphism group isC6.

Letθbe the epimorphismθ:Γ_{→C6 y:y}6_{=1defined by}

θd1=1, θx1=y, θx2=y−1, θx3=θx4=y3. (4.2)

We chooseΦ=y3_{as the central element, thus kerθΓ. Moreover, it is straightforward} to prove ker ¯θθ−1(y3_)Γ1.

Example_4.2. _{LetNbe even,σ (}Γ₎=_(1,−_{, [2,2,2,3]), andq}=_{3. Letθ and ¯θ be}

the canonical epimorphisms over

Gx, y, z:x2, y3, z2, (xy)3, [xz], [yz]A4×C2, ¯

Gx, y:x2_{, y}3_{, (xy)}3_A 4,

(4.3)

respectively,

θd1

=xy, θx1

=Φ, θx2

=Φ, θx3

=x, θx4

=y; ¯

θd1

=x¯y,¯ θ¯x1

=¯_{1, θ}¯_x2=¯_{1, θ}¯_{x3=x,¯ θ}¯_x4=y.¯ (4.4)

It is easy to see that Ker(θ)¯ (4,−, [2,2,2,2],{−})andX=Ᏸ/Ker(θ)is a nonori-entable surface without boundary of algebraic genusp=29. Now, considerΦ=z. Sincezis central inGwe haveG/Φ =_G¯_{andX/Φhas algebraic genus 3, therefore,}

Xis a 3-hyperelliptic Klein surface such that|Aut(X)| =12(q−1)=24.

Example_4.3. _{LetNbe even,σ (}Γ₎=_(0,+_{, [2],}{_(2,2,2,6)}_{), andq}=_{2. LetG,θ, ¯G,}

and ¯θbe defined as follows:

G=x, y, z:x2_{, y}2_{, z}2_{, (xy)}2_{, (yz)}2_{, (zx)}6_D 6×C2;

θx1

=θe1

=Φ, θc1,0

=θc1,4

=x, θc1,1

=Φxy, θc1,2

=y, θc1,3

=z; ¯

G=x,¯ y,¯ z¯: ¯x2_,y¯2_,z¯2_,x¯y¯2

,y¯z¯2,z¯x¯3D3×C2, ¯

θx1

=_θ¯_e1=1¯_{, θ}¯_c1,0=θ¯_{c1,4=x,¯ θ}¯_c1,1=x¯y,¯ ¯

θc1,2=y,¯ θ¯c1,2=z.¯

We can see that Ker(θ)¯ has signature τ=(3,−, [2,_{. . .,}14 _{2],{−})and X=Ᏸ/Ker(θ)is} a nonorientable surface without boundary and algebraic genusp=17. ConsiderΦ= (zx)3_{an order-two central element inG, we haveG/Φ=G¯andXΦhas signatureτ.} In particular,X/Φhas algebraic genus 2, then byTheorem 2.1Xis a 2-hyperelliptic Klein surface. Furthermore,Gis an automorphism group ofX. Now, byTheorem 3.1, we have Aut(X)≤24(q−1), thenG=Aut(X)attaining the upper bound.

It remains as an open problem to determine topological types of nonorientable

q-hyperelliptic Klein surfaces without boundary admitting maximal automorphism group for arbitraryq.

5. Teichmüller space. In this section, we are going to study the dimension of the Teichmüller space associated to nonorientableq-hyperelliptic Klein surfaces without boundary of algebraic genusp >4q+1 such that X/Φ has empty boundary. We denote this space byTq−p.

LetΓ be an NEC group. We define theWeil spaceassociated toΓ,R(Γ), as the set of monomorphismsr:Γ→Ᏻsuch thatr (Γ)is discrete andᏰ/r (Γ)is compact.

We say that two elementsr1,r2∈R(Γ)are equivalents if there existsg∈Ᏻsuch thatr1(Γ)=gr2(Γ)g−1. The quotient space by this relation is called the Teichmüller space ofΓ and denoted byT (Γ). WhenΓ is a Fuchsian group (only orientation pre-serving elements) with signature (g,+, [m1, . . . , mr]), it is known thatT (Γ)is a cell of dimensiond(Γ)=6(g−1)+2r. Singerman [13] proved that ifΓ is a proper NEC group, thend(Γ)=(1/2)d(Γ+_)whereΓ+ _{is the canonical Fuchsian group associated} toΓ. The TeichmüllerModular group ofΓ,M(Γ), is the quotient group Aut(Γ)/I(Γ)

where Aut(Γ)is the full group of automorphisms ofΓ andI(Γ)the set of inner au-tomorphisms. The Modular groupM(Γ)acts on T (Γ) as follows: if[τ]∈T (Γ)and

¯

α∈M(Γ), then ¯α[τ]=[τ·α].

LetX=Ᏸ/Γ be a nonorientableq-hyperelliptic Klein surfaces without boundary of algebraic genusp >4q+1 such thatX/Φhas empty boundary. Following Harvey [8],

Tq−p=

¯ α∈M(Γ)

¯

α 



Φ∈Φ(Γ,Γ1,Γ1/Γ) Imˆıφ



, (5.1)

whereΓ1is the group of theq-hyperellipticity ofXwith signature

q+1,−,2,p+_{. . . ,}1−2q_{2,{−}, ΦΓ,Γ}

1,Γ1/Γ (5.2)

is the set of all equivalence classes of surjectionsφ:Γ1→Z2with Ker(φ)=Γ modulo the action of Aut(Γ1)and Aut(Z2), and Imˆıφ is the image set ˆıφ(T (Γ1)) under the isometry ˆıφinduced by the inclusioniφ: Kerφ→Γ1.

class is defined by

φi

dj

=¯_{0 forj=1, . . . , i;}

φi

dj

=¯_{1 forj=i+1, . . . , q+1;}

φixj=¯1 forj=1, . . . , p+1−2q.

(5.3)

Then,

Tq−p=

¯ α∈M(Γ)

¯

αImˆı0∪···∪Imˆıq+1, (5.4)

where ˆıj:T (Γ1)T (Γ),τ∈R(Γ1) [τ]→[τ·ij]is the real analytic homeomorphism (see [8]) induced by the inclusionij: Kerφj→Γ1.

The situation allows us to use Maclachlan’s method (see [10, Lemma 3]), to prove that (5.4) is a disjoint union.

Suppose that there existl, m∈ {0, . . . , q+1}, such that

¯

αˆıl

TΓ1

∩ˆım

TΓ1

≠∅. (5.5)

Therefore, there existτ1, τ2∈R(Γ1)satisfying

τ1·il·α=τ2·im (5.6)

if and only if there existst∈Ᏻsuch that

τ1·il·α(f )=t

τ2·im

(f )t−1, ∀f∈Γ1. (5.7)

Considerτ3:τ3(f )=tτ2(f )t−1, for allf∈Γ1; thenτ1·il·α=τ3·im. Now, asΓ1is unique we can consider the inverse image ofτ3to obtain

τ3−1·τ1

·il·α=im (5.8)

beingτ3−1·τ1 an automorphism ofΓ1. This leads us to conclude thatl=m. Thus,

Tq−p is a disjoint union of copies ˆıl(T (Γ1)), that is,Tq−p is a submanifold ofT (Γ)of dimension

dΓ1

=2p−q−1. (5.9)

Acknowledgment. The authors wish to thank Professor E. Bujalance for his

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J. A. Bujalance: Departamento de Matemáticas Fundamentales, Facultad de Cien-cias, UNED, Paseo Senda del Rey9,28040Madrid, Spain

E-mail address:[email protected]

B. Estrada: Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, Paseo Senda del Rey9,28040Madrid, Spain