On the connectedness of the branch loci of moduli spaces of orientable Klein surfaces

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On the connectedness of the branch loci of

moduli spaces of orientable Klein surfaces

Antonio F. Costa, Milagros Izquierdo and Ana M. Porto

Linköping University Post Print

N.B.: When citing this work, cite the original article.

The original publication is available at www.springerlink.com:

Antonio F. Costa, Milagros Izquierdo and Ana M. Porto, On the connectedness of the branch

loci of moduli spaces of orientable Klein surfaces, 2014, Geometriae Dedicata, 1-18.

http://dx.doi.org/10.1007/s10711-014-9983-1

Copyright: Springer Verlag (Germany)

http://www.springerlink.com/?MUD=MP

Postprint available at: Linköping University Electronic Press

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On the connectedness of the branch loci of moduli

spaces of orientable Klein surfaces

Antonio F. Costa

Departamento de Matematicas UNED 28040 Madrid, Spain acosta@mat.uned.es

Milagros Izquierdo

y Matematiska institutionen Linköpings universitet 581 83 Linköping, Sweden miizq@mai.liu.se

Ana M. Porto

z Departamento de Matematicas UNED 28040 Madrid, Spain asilva@mat.uned.es

April 12, 2012

2010 Mathematics Subject Classi…cation: 30F50, 30F10, 14H37.

Abstract. _{Let M}K

(g;+;k) be the moduli space of orientable Klein surfaces of

genus g with k boundary components (see [AG], [N2]). The space MK

(g;+;k) has

a natural orbifold structure with singular locus BtK consisting of symmetric Klein

surfaces. We prove that BK

(g;+;k) is connected for every k, and in particular we give

an alternative proof to the one in [BCIP], of the connectedness of BK (g;+;0).

1

Introduction

The concept of Klein surface was introduced by Felix Klein in the last chapter of his book “On Riemann’s Theory of Algebraic Functions and their Integrals”(1882). A Klein surface is a surface with a dianalytic structure and here we are concerned with compact surfaces. For dianalytic structure and Klein surfaces see [AG], [BEGG] or [N2]. Topologically (compact) Klein surfaces are surfaces which might be non-orientable and might have boundary. Klein surfaces are important in the study of

Partially supported by MTM2011-23092.

y_{Work done during a visit to the Institut Mittag-Le- er (Djursholm, Sweden).} z_{Partially supported by MTM2011-23092.}

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real algebraic curves (see [G], [CG], [N2]) and in the development of topological …eld theory ([AN]).

The deformations of Riemann surfaces is the cause of the introduction of the moduli space Mg consisting of the complex structures on (orientable) compact

surfaces of genus g. This space has called constant attention of mathematicians and theoretical physicist due to the connections between such space and string theory ([Na]). As for Riemann surfaces there is as well the moduli space MKt

for the dianalytic structures on surfaces with …xed topological type t = (h; ; k), where h is the genus, the sign determines the orientability and k is the number

of connected components of the boundary. The spaces MK

t have similarities and

di¤erences with Mg. For instance Mgis a complex orbifold of (complex) dimension

3g 3 but for the topological type t = (h; _{; k), M}K

t is an orbifold with universal

covering of (real) dimension 3("h + k 1) 3, where " is either 2 if the sign in the topological type t is + or 1 if the sign is .

The singular sets Bg and BtK of the moduli space of Riemann and respectively

Klein surfaces are called the branch loci of such orbifolds and consist of the surfaces having no trivial automorphisms, i.e. having symmetries. The fact that Bg or BKt

is connected means that a symmetric surface can be continuously deformed to any other symmetric surface preserving the symmetry property along the whole path of deformation. Recently, the connectedness of Bghas been studied and, for instance,

it is shown that B4is connected [CI2] and Bg for g 60 is always disconnected see

[CI3] (see also [BI]).

For the case of BKt some results are known, for instance B(h; ;0)K is connected

for h _{5 [BEM] and in [BCIP] using [Se] (see also [BSS]) is proved that B}K (h;+;0)

is connected. In this paper we obtain that BK

(h;+;k) is connected for every k, and

in particular we get a proof of the connectedness of B(h;+;0)K without the use of

[BSS]. The proof of the main result is based in the existence of Klein surfaces having two automorphisms with some given topological types and these are the lemmae in Section 5. In the Section 6 we give the proof of the main result which is a consequence of results in Section 5 together with the fact that the set of points

in MK

t corresponding to surfaces with an automorphism with a …xed topological

type is connected (this fact is a consequence of [MS], see Section 4).

Similar techniques can be used on the branch loci of the moduli space of non-orientable Klein surfaces, yielding identical main result; they will appear elsewhere.

2

Klein surfaces and non-euclidean crystallographic groups

A Klein surface X is a compact surface (may be non-orientable and with boundary) endowed with a dianalytic structure, where a dianalytic structure is a class of atlases where the transition maps are analytic or anti-analytic maps of C. The topological

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type of X is given by t = (h; ; k) where h is the genus, + if X is orientable and if X is non-orientable and k is the number of connected components of the boundary.

A non-Euclidean crystallographic group or NEC group is a discrete subgroup of the group Aut (D) of conformal and anticonformal automorphisms of the unit disc D of C and in this paper we shall assume that the orbit space D= is compact. If the NEC group _{does not contain any orientation-reversing automorphism of D,}

then we say that is a Fuchsian group.

The so called canonical presentation for NEC groups …rst appeared in [W] and their structure was clari…ed by the introduction of signatures in [M] (see also [Si]).

Given an NEC group , the subgroup of consisting of the orientation-preserving elements is called the canonical Fuchsian subgroup of . The algebraic structure of _{and the geometric and topological structure of the quotient orbifold D= are} given by the signature:

s( ) = (h; ; [m1; :::; mr]; f(n1;1; :::; n1;r1); :::; (nk;1; :::; nk;rk)g): (1) The orbit space D= is an orbifold with underlying surface of genus h, having r 0 cone points and k boundary components of the underlying surface, each with ri 0

corner points, i = 1; :::; k. The signs 00+00 and 00 00 correspond to orientable and non-orientable orbifolds respectively. The integers mi are called the proper periods

of _{and they are the orders of the cone points of D= . The brackets (n}i;1; :::; ni;ri) are the period cycles of : The integers ni;j are the link periods of and the orders

of the corner points of D= . The group is isomorphic to the fundamental group of the orbifold D= .

A group with signature (1) has a canonical presentation with four types of generators (called canonical generators):

1. Hyperbolic generators: a1; b1; :::; ah; bhif D= is orientable; or glide re‡ection

generators: d1; :::; dh if D= is non-orientable,

2. Elliptic generators: x1; :::; xr,

3. Connecting generators (hyperbolic or elliptic transformations): e1; :::; ; ek;

4. Re‡ection generators: ci;j, 1 i k; 1 j ri+ 1.

And relators: 1. xmi

i ; i = 1; :::; r,

2. c2 i;j;

3. (ci;j 1ci;j)ni;j; j = 1; :::; ri,

4. e_i 1ci;rie

1

i ci;0; i = 1; :::; k;

5. The long relation:

x1:::xre1:::eka1b1a11b11:::ahbhah1b 1

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according to whether D= is orientable or not.

The hyperbolic area of the orbifold D= coincides with the hyperbolic area of an arbitrary fundamental region of and equals:

( ) = 2 ("h 2 + k + r X i=1 (1 1 mi ) +1 2 k X i=1 ri X j=1 (1 1 ni;j )); (2)

where " = 2 if there is a00₊00 _{sign and " = 1 otherwise. If} 0 _{is a subgroup of} _of

…nite index then 0 _{is an NEC group and the following Riemann-Hurwitz formula}

holds:

[ : 0] = ( 0)= ( ): (3)

An NEC or Fuchsian group without elliptic elements is called an NEC or

Fuchsian surface group and it has signature (h; _{; [ ]; f( );}_{: : :; ( )g). Given a}k

Klein surface X then X can be represented as the orbit space X = D= , with an NEC surface group. If a …nite group G is isomorphic to a group of automorphisms

of X then there exists an NEC group and an epimorphism : _{! G with}

ker( ) = . The NEC group is the lifting of G to the universal covering _{: D !}

D= .

3

Topological classi…cation of automorphisms of orientable Klein

sur-faces

Two automorphisms f and g of a Klein surface X are topologically equivalent if f and g are conjugated by a homeomorphism of X. The topological types of automorphisms are the topological equivalence classes. The topological types of automorphisms are described using topological invariants (see [BCNS], [Y] and [C]). First we shall present the topological types of primer order automorphisms on orientable Klein surfaces.

Assume that X is an orientable Klein surface and let ' : X ! X be an orienta-tion preserving, order p automorphism; where p is a prime. The topological type of ' is given by the rotation indices for the …xed points of ' and the rotation angles of setwise invariant boundary components. If ' leaves r …xed points and left s set-wise invariant boundary components, the topological type of ' is described by the following data _{= (p; +; fn}1; :::; nrg; fm1; :::; msg) where 1 ni; mi p 1. The

ni(respectively mi) means that there is a …xed point of ' (respectively a boundary

component of X) where locally ' acts topologically as a rotation with angle 2 ni=p

(resp. 2 mi=p). By using uniformization by NEC groups X can be uniformized by

a group with signature

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and the fact of admitting an automorphism of topological type = (p; +; fn1; :::; nrg; fm1; :::; mbg)

implies that there is an NEC group with signature

(h; +; [p;_{:::; p]f( );}r _{:::; ( ); ( );}b k_{::: ; ( )g)}pb

an epimorphism ! : _{! C}p= h i such that = ker ! and if ai; bi; xj; cl; el are

a set of canonical generators of , must be:

! (xj) = nj; ! (el) = mj; ! (ew) = 1; for w > b

Note that for Riemann-Hurwitz formula p divides k b.

Assume that X is an orientable Klein surface and let _{: X ! X be an} orienta-tion reversing involuorienta-tion. The topological invariants for are mainly related with F ix( ). The set F ix( ) consists of:

(a) a …nite number q of simple closed curves that we shall call ovals.

(b) a …nite number t of chains, which we de…ne now. A chain of length 2l is a set C of l disjoint arcs properly embedded in X (i. e. the ends of each component of C are in the boundary of X) such that for each boundary component B of X, either C \ B = ? or C \ B consists of two distinct points.

The extra information that we shall need to determine up topological equiv-alence is the orientability of X= h i, where h i is the cyclic group of order two generated by .

All the above information can be presented in a symbol = (2; ; _{; q; fs}1; :::; stg)

that we shall call the species of the involution, where q is the number of ovals, t is the number of chains and si is the length of each chain and the second sign +

is used if X= h i is orientable and the sign is used if X= h i is non-orientable. The species is a complete system of invariants for the topological classi…cation of involutions (see [BCNS], [N1]).

By using uniformization by NEC groups X can be uniformized by a group with signature (g; +; [ ]; f( );_{: : :; ( )g) and the fact of admitting an involution of}k

topological type = (2; ; _{; q; fs}1; :::; stg) implies that = ker ! where ! : !

C2= h i, has signature

(h; _{; f( );}_{:::; ( ); ( );}l _{:::; ( ); (2;}q s_{:::; 2);}1 _{:::; (2;}t _{:::; 2)g);}st

and if either ai; bi; ej; ci; ci;j or di; ej; ci; ci;j is a set of canonical generators of ,

must be:

! (ai) = ! (bi) = ! (ej) = 1 or ! (di) = '; ! (ej) = 1

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4

Moduli spaces

Let s be a signature of NEC groups (1) and G be an abstract group isomorphic to the NEC groups with signature s. We denote by R(s) the set of monomorphisms r : G !Aut (D) such that r(G) is an NEC group with signature s. The set R(s) has a natural topology given by the topology of Aut (D). Two elements r1 and

r22 R(s) are said to be equivalent, r1s r2, if there exists g 2 Aut (D) such that

for each _{2 G, r}1( ) = gr2( )g 1. The space of classes T(s) = R(s)= s is called

the Teichmüller space of NEC groups with signature s (see [MS]). The Teichmüller space T(s) is homeomorphic to Rd(s) _where

d(s) = 3("h 1 + k) 3 + (2 r X i=1 mi+ k X i=1 si X j=1 nij)):

The modular group Mod(G) of G is the quotient Mod(G) = Aut(G)=Inn(G), where Inn(G) denotes the inner automorphisms of G. The moduli space for NEC groups with signature s is the quotient Ms= T(s)=Mod(G) endowed with the

quo-tient topology. Hence Msis an orbifold with fundamental orbifold group Mod(G).

If s is the signature of a surface group uniformizing surfaces of topological type

t, then we denote by T(s) = Tt and Ms = Mt the Teichmüller and the moduli

space of Klein surfaces of topological type t.

Let G and G0 _{be groups isomorphic to NEC groups with signatures s and s}0

respectively. The inclusion mapping _{: G ! G}0 induces an embedding T( ) :

T(s0_{) ! T(s) de…ned by [r] 7! [r} _{]. See [MS].}

If a …nite group G is isomorphic to a group of automorphisms of Klein sur-faces with topological type t = (g; +; k), then the action of G is determined by an epimorphism ! : D ! G, where D is an abstract group isomorphic to NEC groups with a given signature s and ker(! ) = G is a abstract group isomorphic to surface groups uniformizing Klein surfaces of topological type t. Then there is an

inclusion _{: G ! D and an embedding T( ) : T(s) ! T}t. The continuous map

T_{( ) induces a continuous map M}s! Mtand as a consequence:

Remark. _{The set B}G;t of points in Mt corresponding to surfaces having a

group of automorphisms isomorphic to G, with a …xed action , is a connected set.

5

Klein surfaces with two automorphisms with some given topological

types

The following series of lemmae will be essential in the proof of the main theorem.

Lemma 1

Let _{= (p; +; fw}1; :::; wrg; fv1; :::; vsg) be the topological type of an

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surface X of topological type t = (g; +; k). Hence there exists a Klein surface Y of topological type t with an automorphism of topological type and an orientation reversing involution with topological type (2; _{; +; b; fs}1; :::; slg), with b + l > 0.

Proof.

_{Let X= h i be the orbit Klein surface by the action of} on X. Using the uniformization theorems, there is a surface NEC group with signature

s( ) = (g; +; [ ]; f( );_{:::; ( )g)}k

such that is a normal subgroup of an NEC group of signature

s( ) = (h; +; [p;_{:::; p]; f( );}r _{:::; ( ); ( );}b m=_:::kpb_{; ( )g)}

and in such a way that there are Klein surfaces isomorphisms _{: X ! D= and}

: X= h i ! D= making commutative the following diagram:

X _{! D=}

# #

X= h i ! D=

The monodromy epimorphism ! : _! = _{' C}p = h i is determined up

topological equivalence by the topological type of the automorphism . We can

construct a canonical presentation of :

hai; bi; xj; e_Ql; cl; i = 1; :::; h; j = 1; :::; r; l = 1; :::; b + m :

[ai; bi] Qxj Qel= 1; c2l = 1

adapted to the epimorphism ! in such a way that: ! (ai) = ! (bi) = 1; i = 1; :::; h;

! (xj) = wj; j = 1; :::; r; wj2 f1; :::; p 1g;

! (el) = vl; l = 1; :::; b; vl2 f1; :::; p 1g; ! (el) = 1; l = b + 1; :::; b + m;

! (cl) = 1; l = 1; :::; b + m:

In order to …nd the surface Y in the statement of the lemma, we consider now

a NEC group with signature:

s( ) = (d; +; [ ]; f(p;_{:::; p; 2;}r _{:::; 2; 2;}2b 2m_{::: ; 2); ( )}"

g) where d =h₂ and " = 0, if h is even or d = h 1₂ and " = 1; if h is odd.

Let

ha_Qi; bi; el; c1;u; c2; i = 1; :::; d; l = 1; 2; u = 0; :::; r + 2b + 2m :

[ai; bi] Qel= 1; c21u= c22= 1; e1c1;r+2b+2me11c10= 1

be a canonical presentation for in case h odd and

hai; bi; e1; c1u; i = 1; :::; d; u = 0; :::; r + 2b + 2m :

Q

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in the case of h even.

Now we consider the epimorphism

$ : _{! D}p= ; : p= 2= ( )2= 1

de…ned by:

$(ai) = $(bi) = 1; i = 1; :::; d,

$(c1;0) = ; $(c1;u) = gu ; where gu 1 gu= wuand g0= 0

$(c1;r+2l+1) = 1; $(c1;r+2l) = fl , where fl 1 fl = vl; l = 1; :::; b; and

f0= gr

$(c1;r+2l+2b+1) = 1; $(c1;r+2b+2l) = fr+2b .

Note that if h is even fr+2b= 0 and we de…ne $(e1) = 1.

For the case h odd, $(c2) = , $(e1) is determined by the relation

e1c1;r+2b+2me11c1;0= 1

and $(e2) is determined by the long relation

Y [ai; bi]

Y el= 1:

The surface Y uniformized by ker $, Y = D= ker $, admits a conformal auto-morphism of topological type (see Section 3) that is given by deck transforma-tion of the cyclic covering:

D= ker $ = Y ! D=$ 1h i .

As well Y admits an orientation reversing involution

D= ker $ = Y ! D=$ 1h i

having non-empty …xed point set (note that the NEC group $ 1_{h i contains}

re-‡ections).

Lemma 2

Let _{= (p; +; f g; f g) be the topological type of an orientation} pre-serving automorphism of order p, without any …xed point and no invariant bound-ary component, on a Klein surface X of topological type t = (g; +; k). Hence there exists a Klein surface Y of topological type t with an automorphism of topological type _{and an orientation preserving involution with topological type (2; +; f1;}_{:::; 1g; f1}x "_g),

with x > 0 and " = 0 if g_p1 is even and " = 0 if g_p1 _{is odd (where f0g means} f g).

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Proof.

_{Let X= h i be the orbit Klein surface. Using the uniformization theorems,}

there is a surface NEC group with signature

s( ) = (g; +; [ ]; f( );_{:::; ( )g)}k

s( ) = (h = g 1

p + 1; +; [ ]; f( );

m=k=p

::: ; ( )g)

and in such a way that there are Klein surfaces isomorphisms _{: X ! D= and}

: X= h i ! D= making commutative the following diagram:

X _{! D=}

# #

X= h i ! D=

The monodromy epimorphism ! : _! = _{' C}p = h i is determined up

topological equivalence by the topological type of the automorphism . We can

construct a canonical presentation of :

hai; bi_Q; el; cl; i = 1; :::; h; l = 1; :::; m :

[ai; bi] Qel= 1; c2l = 1

such that the epimorphism ! has the following form:

! (ai) = ! (bi) = 1; i = 1; :::; h 1; ! (ah) = ; ! (bh) = 1

! (el) = 1; l = 1; :::; m;

! (cl) = 1; l = 1; :::; m:

s( ) = (0; +; [2;2h+2 "_::: _{; 2]; f( );}m_{::: ; ( ); ( )}2" "_g) where " = 0, if h is even and " = 1; if m is odd.

Let

xj; el; cl; j = 1; :::; 2h + 2_Q "; l = 1; :::;m "₂ + " :

xj Qel= 1; c2l = 1

be a canonical presentation for . Now we consider the epimorphism

$ : _{! D}p= ; : p= 2= ( )2= 1 de…ned by: $(xj) = ; j = 1; :::; 2h; $(xj) = ; j = 2h + 1; 2h + 2 " $(cl) = 1; l = 1; :::;m "₂ + "; $(el) = 1; l = 1; :::;m "₂ ; if " = 1, $(em " 2 +1) = .

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The surface Y uniformized by ker $, Y = D= ker $, admits a conformal auto-morphism of order p of topological type that is given by deck transformation of the cyclic covering:

D= ker $ = Y ! D=$ 1h i .

As well S admits an orientation preserving involution

D= ker $ = Y ! D=$ 1h i

having non-empty …xed point set.

Lemma 3

Let = (2; ; _{; q; fs}1; :::; stg) be the topological type of an orientation

reversing involution on a Klein surface X of topological type t = (g; +; k). Hence there exists a Klein surface Y of topological type t with an orientation reversing involution of topological type and an orientation preserving involution.

Proof.

_{Let X= h i be the orbit Klein surface. Using the uniformization theorems,}

there is a surface NEC group with signature

s( ) = (g; +; [ ]; f( );_{:::; ( )g)}k

s( ) = (h; _{; [ ]; f( );}_{:::; ( ); ( );}q _{:::; ( ); (2;}m s_{:::; 2);}1 _{:::; (2;}t _{:::; 2)g)}st

in such a way that there are Klein surfaces isomorphisms _{: X ! D= and} :

X= h i ! D= making commutative the following diagram:

X _{! D=}

# #

X= h i ! D=

The monodromy epimorphism ! : _! = _{' C}2 = h i is determined up

topological equivalence by the topological type of the involution . We can

construct a presentation of : ai; bi; el; cl; cjw; i = 1; :::; h; l = 1; :::; q + m + t; j = q + m + 1; ::::; q + m + t; w = 0; :::; tj : Q [ai; bi] Qel= 1; c2l = cjw2 = 1; (cjwcjw+1)2= 1; ejcj;tje 1 j cj;0= 1 or di; el; cl; cjw; i = 1; :::; h; l = 1; :::; q + m + t; j = q + m + 1; ::::; q + m + t; w = 0; :::; tj : Q d2 i Q el= 1; c2l = c2jw = 1; (cjwcjw+1)2= 1; ejcj;tje 1 j cj;0= 1

depending on the sign in s( ), and such that the epimorphism ! has the following form:

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! (ai) = ! (bi) = 1; i = 1; :::; h if the sign is + and ! (di) = ; i = 1; :::; h if the sign is ; ! (el) = 1; l = 1; :::; q + m; ! (cl) = ; l = 1; :::; q; ! (cl) = 1; l = q + 1; :::; q + m; ! (cj;2i) = , 0 i sj=2; j = q + m + 1; :::; q + m + t; ! (cj;2i+1) = 1, 0 i sj=2 1; j = q + m + 1; :::; q + m + t: By the construction of !, 2m +Psj 2 = k.

an NEC group with signature:

s( ) = (d; ; [2"_{]; f(2;}_{:::; 2; 2;}2b 2m_{::: ; 2; 2;}s1=2+2_{::: ; 2; :::; 2;}st=2+2_{::: ; 2)g)}

where d = h; " = 0 if the sign in s( ) is +; d = h=2; " = 0 if h is even and the sign

in s( ) is and d =h 1

2 ; " = 1 if h is odd and the sign in s( ) is .

Let hdi; x; e; cj; i = 1; :::; d; j = 0; :::; 2m + 2b +P(sj=2 + 2) : Q d2_i x" e = 1; x"+1= 1; c2_j = 1; (cjcj+1)2= 1; ec2m+2b+Psj=2+2e 1_c 0= 1

be a canonical presentation for . Now we consider the epimorphism

$ : _{! D}2= ; : 2= 2= ( )2= 1 de…ned by: $(di) = ; i = 1; :::; d, if " = 1; $(x) = ; $(e) = , if " = 0; $(e) = 1; $(c2j) = ; $(c2j+1) = ; where j = 0; :::; q 1; $(c2m) = ; $(c2m+2j) = ; $(c2m+2j+1) = 1 where j = 0; ::::; m 1 $(c2m+2b) = ; $(c2m+2j+Psi=2+2j+1) = ; $(c2m+2j+Psi=2+2j) = 1; $(c2m+2j+Psi=2) = :

The surface Y uniformized by ker $ admits an orientation reversing involution of topological type that is given by deck transformation of the cyclic covering:

D= ker $ = Y ! D=$ 1h i .

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Lemma 4

Let _{= (2; +; f1;}_{:::; 1g; f1;}r _{:::; 1g) be the topological type of an orienta-}b

tion preserving involution , with r +b > 0, on a Klein surface X of topological type t = (g; +; k) with k even. Hence there exists a Klein surface Y of topological type t with an orientation preserving involution of topological type and an orientation reversing involution with empty …xed point set, i.e. topological type (2; ; _{; 0; f g).}

Proof.

Using the uniformization theorems, there is a surface NEC group with signature

s( ) = (g; +; [ ]; f( );_{:::; ( )g)}k

s( ) = (h; +; [2;_{:::; 2]; f( );}r _{:::; ( ); ( );}b _{:::; ( )g)}m

X _{! D=}

# #

X= h i ! D=

topo-logical equivalence by the topotopo-logical type of the involution . We can construct a presentation of :

hai; bi; xj_Q; el; cl; i = 1; :::; h; j = 1; :::; r; l = 1; :::; b + m :

[ai; bi] Qxj Qel= 1; x2j = c2l = 1

such that the epimorphism ! has the following form: ! (ai) = ! (bi) = 1; i = 1; :::; h;

! (xj) = ; j = 1; :::; r;

! (el) = ; l = 1; :::; b;

! (el) = 1; l = b + 1; :::; b + m;

! (cl) = 1; l = 1; :::; b + m:

Note that k = 2m + b and r + b is even. Since k is even, then b and r are even. In order to …nd the surface Y in the statement of the lemma, we consider now

s( ) = (h; ; [2;r=2_{::: ; 2]; f( );}b=2+m=2+1::: _{; ( )gg)} if m is even and

s( ) = (h; ; [2;r=2_{::: ; 2]; f(2; 2); ( );}b=2+(m 1)=2::: _{; ( )g)} if m is odd.

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If m is even, let hdi; xj; el; cl; i = 1; :::; h; j = 1; :::; r=2; l = 1; :::; b=2 + m=2 + 1 :_Q d2 i Q xj Qej= 1; x2j = c2l = 1

be a canonical presentation for . For the case m odd we consider the presentation: di; xj; el; c1w; cv; i = 1; :::; h; j = 1; :::; r=2; l = 1; :::; b=2 + (m 1)=2 + 1; w = 0; 1; 2; v = 2; :::; b=2 + (m 1)=2 + 1 : Q d2 i Q xj Qej = 1; x2j = c2l = c1w= 1; e1c13e11c10= 1

Now we consider the epimorphism

$ : _{! D}2= ; : 2= 2= ( )2= 1 de…ned by: $(di) = ; i = 1; :::; h; $(xj) = ; j = 1; :::; r=2; $(el) = ; l = 2; :::; b=2 + 1; $(el) = 1; l = b=2 + 2; :::; b=2 + m=2 + 1 or b=2 + (m 1)=2 + 1;

$(e1) = 1 if r=2 + b=2 is even and $(e1) = if r=2 + b=2 is odd,

$(cl) = 1; l = 2; :::; b=2 + (m 1)=2 + 1;

$(c1) = , if m is even and $(c10) = ; $(c11) = 1; $(c12) = if m is odd.

The surface Y = D= ker $ admits an orientation preserving involution of topo-logical type that is given by deck transformation of the cyclic covering:

D= ker $ = Y ! D=$ 1h i .

D= ker $ = Y ! D=$ 1h i

having empty …xed point set.

Lemma 5

Let _{= (2; +; f1;}_{:::; 1g; f1;}r _{:::; 1g) be the topological type of an orienta-}b

tion preserving involution , with r +s > 0, on a Klein surface X of topological type t = (g; +; k) with k odd. Hence there exists a Klein surface Y of topological type t with an orientation preserving involution of topological type and an orientation reversing involution with topological type (2; ; _{; 0; f1g).}

Proof.

Using the uniformization theorems, there is a surface NEC group with signature

s( ) = (g; +; [ ]; f( );_{:::; ( )g)}k

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X _{! D=}

# #

X= h i ! D=

topo-logical equivalence by the topotopo-logical type of the involution . We can construct a presentation of :

hai; bi; xj_Q; el; cl; i = 1; :::; h; j = 1; :::; r; l = 1; :::; b + m :

[ai; bi] Qxj Qel= 1; x2j = c2l = 1

such that the epimorphism ! has the following form: ! (ai) = ! (bi) = 1; i = 1; :::; h;

! (xj) = ; j = 1; :::; r;

! (el) = ; l = 1; :::; b;

! (el) = 1; l = b + 1; :::; b + m;

! (cl) = 1; l = 1; :::; b + m:

Note that k = 2m + b and r + b is even. Since k is odd, then b and r are odd. In order to …nd the surface Y in the statement of the lemma, we consider now

s( ) = (h; ; [2;(r_:::1)=2_{; 2]; f(2; 2; 2); ( );}(b 1)=2_::: _{; ( ); ( );}m=2_{::: ; ( )g)} if m is even and s( ) = (h; ; [2;(r:::1)=2_{; 2]; f(2; 2; 2; 2; 2); ( );}(b 1)=2::: ; ( ); ( );(m 1)=2::: _{; ( )g)} if m is odd. Let * di; xj; el; ; c1w; cu; i = 1; :::; h; j = 0; :::; (r 1)=2; l = 1; :::; (b 1)=2 + m=2 + 1; u = 2; :::; (b 1)=2 + m=2 + 1; w = 0; ::; 3 : Q d2 i Q xj Qel= 1; x2j = cu2 = c21w= 1; (cl;wcl;w+1)2= 1; e1cl3e11cl0= 1

be a canonical presentation for when m is even. For the case m odd we consider: * di; xj; el; ; c1w; cu; i = 1; :::; h; j = 0; :::; (r 1)=2; l = 1; :::; (b 1)=2 + (m 1)=2 + 1; u = 2; :::; (b 1)=2 + (m 1)=2 + 1; w = 0; ::; 5 : Q d2 i Q xj Qel= 1; x2j = cu2 = c21w= 1; (cl;wcl;w+1)2= 1; e1cl3e11cl0= 1

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Now we consider the epimorphism $ : _{! D}2= ; : 2= 2= ( )2= 1 de…ned by: $(di) = ; i = 1; :::; h; $(xj) = ; j = 1; :::; (r 1)=2; $(el) = ; l = 2; :::; (b 1)=2 + 1; $(el) = 1; l = b=2 + 2; :::; (b 1)=2 + (m 1)=2 + 1;

$(e1) = 1 if (r 1)=2 + (b 1)=2 is even and $(eb=2+m=2+1) = if (r 1)=2 +

(b 1)=2 is odd,

$(c10) = ; $(c11) = ; $(c12) = 1; $(c13) = , if m is even, and

$(c10) = ; $(c11) = ; $(c12) = 1; $(c13) = , $(c12) = 1; $(c13) = , if m

is odd,

$(cu) = 1; l = 2; :::; b=2 + m=2 + 1:

The surface Y = D= ker $ admits an orientation preserving involution of topo-logical type that is given by deck transformation of the cyclic covering:

D= ker $ = Y ! D=$ 1h i .

D= ker $ = Y ! D=$ 1h i

with a chain of length 1 as …xed point set and non-orientable quotient space (see [H] and [HS]).

6

Connectedness of the branch locus of moduli spaces of orientable

Klein surfaces

Theorem 6

If t = (g; +; k) with g _{2, then the branch locus B}K

t of the moduli

space MKt of Klein surfaces with topological type t is connected.

Proof.

Let +

p be set of topological types of actions of orientation preserving

automorphisms of prime order p on Klein surfaces of topological type t and ₂

be the set of topological types of actions of orientation reversing involutions on Klein surfaces of topological type t. If _{2 ([}p +p) [ 2 we shall denote Bt the

set of points in BtK corresponding to Klein surfaces admitting an automorphism of

topological type .

Since each group of automorphism of a Klein surface must contain either an automorphism of prime order or an anticonformal involution we have:

BKt = [ 2[p +p Bt [ [ 2 2 Bt

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As we remark at the end of Section 4, the Bt are connected sets.

By lemmae 4 and 2, if k is even, the set:

C = [

2[ +2 Bt [ B

(2; ; ;0;f g) t

is connected, analogously by lemmae 5 and 2, if k is odd, the set:

C = [ 2[ + 2 Bt[ B (2; ; ;0;f1g) t is connected. Now by lemma 3: C [ [ 2 2 Bt = [ 2[ + 2 Bt [ [ 2 2 Bt

is a connected set. And …nally by lemmae 1 and 2: [ 2[p>2 +p Bt [ [ 2[2 +2 Bt[ [ 2 2 Bt = BKt is connected.

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