Automorphisms of Klein Surfaces of Algebraic Genus One

Transcription

1 Çankaya Üniversitesi Fen-Edebiyat Fakültesi, Journal of Arts and Sciences Say : 5, May s 006 Automorphisms of Klein Surfaces of Algebraic Genus One Adnan MELEKOĞLU * Abstract Klein surfaces of algebraic genus one are the Möbius band, the annulus and the Klein bottle In this paper, the automorphisms of these surfaces are determined Keywords: Klein surface, Möbius band, annulus, Klein bottle, automorphism Özet Cebirsel cinsi bir olan Klein yüzeyleri; Möbius şeridi, silindir ve Klein şişesidir Bu çal şmada bu yüzeylerin otomorfizmalar belirlenmiştir Anahtar Kelimeler: Klein yüzeyi, Möbius şeridi, silindir, Klein şişesi, otomorfizma 1 Introduction Let X be a compact Riemann surface of genusg 1 An automorphism of X is a conformal or anti-conformal homeomorphism f : X X X is called symmetric if it admits an anti-conformal involution s : X X which we call a symmetry of X The quotient surface S = X / s is a Klein surface By a Klein surface we mean a surface with a dianalytic structure (see [1) Here X is called the complex double of S The algebraic genus of S is then defined to be the topological genus of X It is known that the Klein surfaces of algebraic genus one are the Möbius band, the annulus and the Klein bottle In this paper we study the automorphisms of these surfaces We do not claim originality of the work However, it contains something demonstrative of the method, not readily available in the literature, which may be helpful to those who are not experts but wish to understand the subject * Adnan Menderes Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü AYDIN amelekoglu@aduedutr 71

2 Automorphisms of Klein Surfaces of Algebraic Genus One Lattices and tori A lattice is a group generated by two linearly independent translations in the complex plane C Let be a lattice Then, the quotient space C / is a torus Let z C and let [z denote the -orbit of z Then, [z is a single point on the torus C / If is a lattice generated by the translations z z z1 and z z z, and a C {0}, then a is a lattice and is generated by z z az 1 and z z az We shall use the following theorems to determine the automorphisms of tori and automorphisms of their quotients by symmetries Theorem 1 ([) Let be a lattice Then, the conformal automorphisms of the torus C/ are the transformations [ z [ az b such that a,b C and a = Theorem ([1) Let X be a Riemann surface and s : X X a symmetry of X Then, the automorphisms of the Klein surface X / s consist of the conformal automorphisms of X commuting withs 1 Theorem ([1) Let X be a Riemann surface, and s 1, s be symmetries of X Then, the Klein surfaces X / s 1 and X / s are isomorphic (dianalytically equivalent) if and only if s 1 and s are conjugate in the group of automorphisms of X It is known that every complex torus is conformally equivalent to C /, where is a lattice generated by the translations z z 1 and z z γ Here γ lies in the usual fundamental region F for the modular group PSL(,Z ), that is, γ F = { z C 1/ Re( z) 1/, Im( z) > 0, z 1} (see [1) The symmetric tori correspond to the points of F on the imaginary axis and on the boundary of F The following table is given in [1, where in each case, is the lattice generated by the translations z z 1 and z z γ, and T is the torus C / The quotient surface in the last column is the quotient of T by the corresponding symmetry 7

3 Adnan MELEKOĞLU Case 1 Re( γ ) = 0, Im( γ ) > 1 γ = i γ = 1, 0 < Re( γ ) < 1/ 4 γ = ( 1 i ) / 5 Re( γ ) = 1/, Im( γ ) > / Table 1 Symmetry of T [ z 1/ [ z γ / [ iz [ z 1/ Quotient surface Annulus Annulus Klein Bottle Klein Bottle Annulus Klein Bottle [ γ z [ γ z Remark 1 (i) Any two points in the upper half complex plane lying in the same orbit under the action of the modular group correspond to conformally equivalent tori (see [) For this reason, the cases Im( γ ) > 1, Re( γ ) = 1/ and γ = 1, 1/ < Re( γ ) < 0 have not been considered in Table 1 (ii) As shown in [1, the torus corresponding to γ = i (case ) admits three conjugacy classes of symmetries Since the quotients of a torus by conjugate symmetries are isomorphic Klein surfaces by Theorem, in case, we consider only one symmetry from each conjugacy class The same discussion applies to the case 4 Automorphisms of the Möbius band In this section, we shall determine the automorphisms of the Möbius bands given in Table 1 Let us choose, for example, the Möbius band M corresponding 7

4 Automorphisms of Klein Surfaces of Algebraic Genus One to the case of Table 1 In this case, the translations z z 1 and z z i generate a lattice such that C / is a torus T T admits a symmetry s : T T defined by ) = [ iz such that M = T / s The Möbius band M can also be considered as C /, where is the group generated by the translations z z 1, z z i and the reflection z iz, that is, =, s So, is a group containing with index Now let us calculate the automorphisms of M By Theorem, the automorphisms of M consist of the conformal automorphisms of T commuting withs Let F = { z 0 Re( z) 1,0 Im( z) 1}, which is a fundamental region for It follows from [1 and Theorem 1 that T admits the following conformal automorphisms: f = ( b F ), 1([ z ) [ z b f ( b F ), ( ) = [ z b f ( b F ), ( ) = [ iz b f ( b F ) 4 ( ) = [ iz b Let us now determine those conformal automorphisms of T commuting withs The automorphism f1 of T commutes with s if and only if for every z C, ( 1 f1o s)([ z ) = ( s o f )([z ) [ iz b = [ iz ib [ b = [ ib Re( b ) = Im( b) b = ( 1 i) x, (0 x < 1) Therefore, by using f 1, we obtain the following automorphisms for the Möbius band M : z [ z (1 i) ( 0 < 1 ) x Similarly, from f we obtain the following automorphisms for M : z (1 i) ( 0 < 1 ) x 74

5 Now, consider the automorphism [ z ) = [ iz if and only if for every z C, ( f f ( ) = [ iz b o s)([ z ) = ( s o f )( ) [ i( iz) b = [ i( iz b) [ z b = [ z ib [ z = [ ib b Adnan MELEKOĞLU of T It commutes with [ z = [ Re( b) Im( b) ( i 1) ( ) ( ) [ z = [(1/)(1 i) Re( b) Im( b) So, the equation ( f o s)([ z ) = ( s o f )( ) is not satisfied for all z, and hence f and s do not commute In the same way, we can show that f 4 and s do not commute and therefore, from f and f 4 we cannot obtain any automorphisms for the Möbius band M As a result, all the automorphisms of M are as follows: [ z [ ± z (1 i) x, ( 0 x < 1) By doing similar calculations to those above, we can determine the automorphisms of the Möbius bands corresponding to the other cases, and we give the results in Table Table Case Symmetry of T Automorphisms of T / s γ = i ) = [ iz [ z [ ± z (1 i) x, ( 0 x < 1) γ = 1, 0 < Re( γ ) < 1/ γ = ( 1 i ) / γ > 1, Re( γ ) = 1/ ) = ) = [ z ) = ) = [ z ) = ) = [ z γ ) x, [ ± z (1 ( 0 x < 1) [ ± z (1 γ ) x, ( 0 x < 1) [ z [ ± z x, ( 0 x < 1) [ ix z [ ± z, ( 0 x < ) z [ ± z, ( 0 x < 1 ) [ ± z [ z ix, ( 0 x < Im( γ)) 75

6 Automorphisms of Klein Surfaces of Algebraic Genus One 4 Automorphisms of the Annulus It follows from section two that we obtain annuli in the cases where Re( γ ) = 0, Im( γ ) > 1 and γ = i Let Re( γ ) = 0, Im( γ ) > 1 and be the lattice generated by z z 1 and z z γ Let T be the torus C / From [1 and Theorem 1, it follows that T admits the following conformal automorphisms: [ z b ( b F ), [ ± z where F = { z C 0 Re( z) 1,0 Im( z) γ } is a fundamental region for As shown in [1, T admits two non-conjugate symmetries s1 ( ) = and s ( ) = [ z such that the quotient spaces T / s 1 and T / s are annuli Let A 1 = T/ s1 By Theorem, the automorphisms of A 1 are the conformal automorphisms of T commuting withs 1 Let b F Then the automorphism [ z [ z b of T commutes with s 1 if and only if for every z C, [ z b z b [ b = [ b = [ 0 b < 1 or b = x γ /, (0 x < 1) Thus, the automorphisms of A 1 obtained from [ z b are as follows: [ z z [ z ( 0 < 1) [ γ x, z [ z x / ( 0 < 1) x, where =, s 1 In the same way, from the automorphism [ z b of T we obtain the following automorphisms for A 1 : z ( 0 < 1 ) [ γ x, z x / ( 0 < 1 ) x 76

7 Adnan MELEKOĞLU Now let A = T/ s and =, s As before, we can observe that the automorphisms [ z [ z b and [ z b of T commute with s if and only if b = ix ( 0 x < γ ) or b = 1 / ix ( 0 x < γ ) Therefore, we find that the authomorphisms of A are as follows: [ z ix ( 0 x < γ ), [ ± z [ z ix ( 0 x < γ ) [ ± z 1/ The automorphisms of the annulus corresponding to the case γ = i can be found in the same way, and we give the results of this section in Table Table Case Symmetry of T Automorphisms of T / s Re( γ ) = 0, Im( γ ) > 1 ) = [ z [ ± z x, ( 0 x < 1) [ z [ ± z x γ /, ( 0 x < 1) [ z [ ± z ix, ( 0 x < γ ) ) = [ z [ z [ ± z 1/ ix, ( 0 x < γ ) γ = i ) = z [ ± z, ( 0 x < 1) z [ ± z x /, ( 0 x < 1) [ i 5 Automorphisms of the Klein bottle We know that Klein bottles can be obtained as the quotients of tori by symmetries in the cases where Re( γ ) = 0, Im( γ ) > 1 and γ = i Let Re( γ ) = 0, Im( γ ) > 1 and be the lattice generated by z z 1 and z z γ Let T be the torus C / As shown in [1, T admits two nonconjugate symmetries s1 ( ) = [ z 1/ and s ( ) = [ z γ / such that the quotient spaces T / s 1 and T / s are Klein bottles The automorphisms of these Klein Bottles can be found as before For example, The Klein bottle T / s 1 admits the following automorphisms: [ z x ( 0 x < 1/, =, s1 ), [ ± z 77

8 Automorphisms of Klein Surfaces of Algebraic Genus One [ z γ / ( 0 x < 1/, =, s1 ) [ ± z x These automorphisms have also been determined by Bârz_ [ The automorphisms of the other Klein bottles are given in Table 4 Table 4 Case Symmetry of T Automorphisms of T / s Re( γ ) = 0, Im( γ ) > 1 γ = i = z 1/ ) [ ) [ z = γ / = z 1/ ) [ [ z x, ( 0 x < 1/ ) [ z γ /, ( 0 x < 1/ ) [ ± z [ ± z x [ ± z [ z ix, ( 0 x < γ ) [ z ix, ( 0 x < γ ) [ ± z 1/ [ z x, ( 0 x < 1/ ) [ ± z [ ± z x [ z i /, ( 0 x < 1/ ) References 1 NL Alling and N Greenleaf, Foundation of the Theory of Klein Surfaces (Lecture Notes in Math Vol 19, Springer-verlag, 1971) I Bârz_, The Dianalytic Morphisms of the Klein Bottles, (In: Complex Analysis, Lecture Notes in Math Vol 151 (1988), pp8-51) GA Jones and D Singerman, Complex Functions: An Algebraic and Geometric Viewpoint (Cambridge University Press, 1987) 78