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Vol.30,1978

435

A remark

on nonsymmetric

compact Riemann
By

surfaces

D~AR

G~B~

Let G be a group of automorphisms of a compact Riemann surface of genus g > 1. It is well known that G is a factor group of a Fuchsian group _F of signature (h; ll, 4 .... ,/~), that means a factor group of

Xlz 1

= ~2 -

= ~2 = ~ .x i =!V~[a~,
i=i i=i

bi] = 1

where [a, b] = aba-lb -1. The kernel _N is a group of signature (g; --). We use the notation of Singerman [3]. G is called a _K-automorphism group if F is of signature (0; ll, 12, 2). There is a close connection between K-automorphism groups and r%o~lar maps. A Riemann surface is called nonsymmetric if it admits no anticonformal involution. Singerman [3; Theorem 1] has shown that for every automorphism group G of a compact Riemann surface S for which T is not a triangle group (0; 11, le, 13) there exists an automorphism group G1--~-G of a nonsymmetric Riemarm surface $1 homeomorphic to S. Thus it is of interest to study the nonsymmetric giemann surfaces where/' is a triangle group. They are rather exceptional. Singerman Eves one of genus g ---- 17 and some others of higher genus. this note we classify all K-automorphism groups of compact Riemann surfaces of genus 7. Together with earlier results of Coxeter-Moser, Sherk and the author (see [2]) one then knows all K-automorphism groups of genus 2 ~ g < 7. Apart from isomorphic copies there exist exactly two nonsymmetric Riemarm surfaces of genus 2 g g < 7 whose group is a K-automorphism group. They belong to two surface kernels in (0; 9, 6, 2) and are of genus 7. As P. Bergau told me, one of these has been known to him before. Our enumeration is based on the methods described in [2]. Firstly one determines the arithmetical possibilities of (0; ll, 12, 2) --> G by means of the Riemann-Hurwitz formula
(1) 2g-2= [G] zl 4 "

By grouptheoretical reasoning one then decides wether there exist torsionfree normal subgroups of index I G] in (0; lz, 12, 2). We illustrate this procedure by proving: 28*


436

D. GARB~

ARCH.MATH.

Proposition. There exist exactly three classes of con]ormally equivalent compact Riemanr~ surfaces of genus 7 whose automorphism group is o] type G ~_ (0; 9, 6, 2)/N with torsion[tee N. They are defined by the relation~

(i) (iii)

x~ = x~ -----x~ = xl x2 x3 = 1 ; x~ = x26-----x~ = xi x2 x3 = 1 ;

Ix1, ~] = x~; [xi, x~] = 1.

(i) and (ii) belong to nonsymmetric ~iemann surfaces. So they give irreflexible regular

maps.
Proof. Using (I) we ge~ IGI = 54. The 3-Sylow sub~oup Sa of G is normal. The corresponding normal subgroup in (0; 9, 6, 2) is defined by
U=Xi, V = X2:VlX211 U9 = V9 = (UV)3 = 1.

(Use Reidemeister-Sehreier.) If Ss is abelian we have xi ~-x~. Taking 2V as the normal closure of [xi, x~] in (0; 9, 6, 2) we get the K-automorphism group (iii). Now let $3 be nonabelian. The commutator quotient group S3/S's is of order 9. In fact: S S can't be of order 9.Otherwise S" ~ C9 or S a ~ Ca · Ca, and so *~s would a have a normal subgroup of index 9. Ss/S~ cannot be cyclic of order 9. As the automorphism group of S~ has order 2 the centralizer of S~ in $8 would be Sa. Hence Sa would be abelian. Thus Ss/S'~ is elementary abelian of type (3, 3). The commutator quotient group of (u, v) is isomorphic to C9 · Cs. Thus the normal subgroup H in (u, v) which corresponds to S~ is uniquely determined. Hence u3 ~ H, uv-lu-lv e H. The eosets ~3 and ~-1~-1~ in G necessarily generate the same cyclic group of order 3, so
/ r

(2)

~-~-l~___~a

or

4~-1~-1~=~

-a.

Adding the relations which correspond to (2) to ~hose of (0; 9, 6, 2) one gets cases (i) and (ii). They satisfy the well-known criterion of nonsymmetry resp. irreflexibility (see [2; p. 41] and [3; Theorem 2]) whereas (iii) does not. We want to remark that (i) and (ii) are 'mirror images' of each other. The following list gives all K-automorphism groups of genus 7.


Vol. 30, 1978

Nonsymmetric compact Riemann surfaces Defining relators of N in (0; 11, h, 2)
x~,3x; 1

437

([1, 12, 2) ll ~_~12

]G I

References

1.
2. 3. 4. 5. 6. 7. 8.
9. 10.

(28, 28, 2)
(30, (16, (16, (21, (12, ( 9, ( 9, 15, 16, 16, 6, 6, 6, 6, 2) 2) 2) 2) 2) 2) 2)

28
30 32 32 42 48 54 54
54

[1; p. 105] {28, 28}1.o
[1; [1; [2; [2; [2; p. 139] (30, 15}2 p. 139] {16, 16)2 Satz 6.2] Satz 6.2] Satz 6.4] ---

xll"x; ~

x~ x~
x~~ x~x~ ~ xlx~, [x~, x2] ~-

x; S [xl, x~] x~ [x~, x~]
[x~, x~] x~x~(x~Sx2) ~

( 9, 6, 2)

( 7, 7, 2)

56

[1; p. 103] Edmonds' map

11.
12. 13. 14. 15.

(28, 4, 2)
(16, (16, (12, ( 7, 4, 4, 3, 3, 2) 2) 2) 2)

56
64 64 144 504

~,"x]
(x: ~x~)~
x~x~x;Sx; z

[1; p. 115] {28, 4h.~
[1 ; p. 110] {16, 412) [2; Satz 6.4] [4; p. 70]

~1 [x~,~]x~[~, xD 32
((x[ x;')S x;') 2 Referenees

[1] H. S. M. COXETER and W. 0. J. MOSES, Generators and relations for discrete groups. 2nd edition. Berlin 1965. [2] D. G~RBE, ]~ber die regul~ren Zerlegungen geschlossener orientierbarer Fl~ehen. J. reine angew. Math. 237, 39--55 (1969). [3] D. Sn~GER.~A:~N, Symmetries of Riemann surfaces with large automorphism group. Math. Ann. 210, 17--32 (1974). [4] A. SINKOV, Necessary and sufficient conditions for generating certain simple groups. Amer. J. Math. 59, 67--76 (1937). Eingegangen am22.11.1977 Anschrift des Autors: Dietmar Garbe Fakultat fiir Mathematik Universit~t Bielefeld Universit~tsstr. 4800 Bielefeld 1