On the order of a group of automorphisms of a compact bordered Klein surface

11 (1989), 3 - 8

On the order of a group of

automorphisms of a compact bordered

Klein surface

Grzegorz G rom adzki

We will prove the announced results by means of NEC groups. An

NEC-group is a discrete subgroup T of the group of isometries Q of

the hyperbolic piane C+ (including those which reverse orientation— reflections and glide reflections) with compact ąuotient space C+/T . Let Q+ denote the subgroup of index 2 in Q consisting of orientation preserving isometries. An NEC group T contained in Q+ is called a

Fuchsian group, and a proper NEC-group in the other case. In what

follows r + = T fi Q+ is the canonical Fuchsian subgroup of an NEC group T.

M acbeath [7] and Wilkie [13] associated to every NEC group a sig- nature th a t has the form

(1)

(g\

and determines the algebraic structure of the group. The numbers m,- are called proper periods, the brackets ( n a , • • . , n,Sl) period cycles and g > 0 is called orbit genus. The group with signature 1 has the presentation with the following generators

(2) X i , i = l , . . . , r ,

C i j, t 1, • • • , k , J 0 ,. . . , S i,

e,-, t 1 , . . . , h )

cii, b , i = 1 , . . . ,g (if the sign is + ) d{, i = 1 , . . . , g (if the sign is —)

4 G. G rom adzki

subject to the relations

1. x™' = 1, i = l , . . . , r ,

2* ^isi ^ 1? . . . , A-,

3. c?j _1 = cC = (c ,j_ ic tj )nij = 1, i = 1 , . . . , k; j = 1 , . . . , a,-, 4. X\ . . . x re\ . . . e ^ a i^ a j" 1^ 1 . . . cigbga^b^1 = 1 if the sign is + ,

X\ . . . x re\ .. . &kd\ ■.. d2g — 1 if the sign is —. In what follows these generators are saicl to be the canonical gener- ators of T. It is known th a t the only elements of finite order in T are those th a t are conjugate to powers of cp-, l5 X{. Every NEC group has a fundamental region associated, whose area depends only on the group. It is given by

(3) , . ( 0 = 2* (a g + k -2 + g ( l - i - ) + i Y .(l - ^ - ) ) .

where a = 1 if the sign is — and a = 2 in the other case.

It is known th a t the necessary and sufficient condition for a group T with presentation 2 to be realized as an NEC group with signature 1 is th a t the right hand side of 3 is greater than 0.

If T is a subgroup of finite index in an NEC group A, then it is an NEC group itself and the following Hurwitz-Riemann formuła holds

(4)

[a : r] = /i(r)//<(A).

An NEC group with signature

(5) [ - ] ; { ( - ) > •*•>(-)})

(k > 1) is said to be a bordered surface group of genus g with k boundary

components orientable or non-orientable according as the sign is + or —. The number p = ag + k — 1 is called the algebraic genus of T and it ecjuals the algebraic genus of the corresponding Klein surface

x

= c+/r.

It is known [11] th a t a compact bordered Klein surface of algebraic genus p > 2 can be represented as C+ /T, where T is a bordered surface

group of algebraic genus p. Moreover given a surface so represented, a fmite group G is a group of its automorphisms if and only if there exists a proper NEC group A containing F as a normal subgroup such th a t G = A / r [8].

L e m m a 1 The only proper N E C groups with area sm aller than 7r/6

are those which have a signature (0 ;+ ; [—]; { ( n i , n2,U3)}), where 5/6 <

l/7?.i+ 1/??.2 + 1/h3 < 1 o / '(0;+;[???]; {(77)}), where 5/6 < 2 / 7 7 7 + 1 / 7 7 < 1.

P r o o f . Straiglitforward verification.

L e m m a 2 Nonę o f the groups listed in the preuious lemma adm it a

bordered surface gro up as a normal subgroup o f finite index.

P r o o f . Notice th a t a canonical Fuchsian subgronp F + of a bordered surface group F is torsion lree.

It is easy to check th a t an NEC group A with a signature (°; +; [-]; {(»i,»2,»3)})

is generated by three reflections Co, ci and c2 obeying the relations

(c0c ,)ni = (c,c2p = (c0c2 ) n 3 = 1.

Assume th a t a group A contains a bordered surface group F as a normal subgroup. Thcn a reflection c of A belongs to F. Reflection c is conjugate to one of the canonical ones, say to co and sińce F is normal in A co itself belongs to F. Now sińce //(A) > 0. ??i or 7/3 is greater tlian

2. But then (coCi) 2 or (coc-2 ) 2 is a nontrivial element of finite order in

F+ which is torsion-free as we alrcady mentioned, a contradiction. Now assume th a t A has a signature of the second type. Then A is generated by c0, Ci, and e subject to the relations

cg = c? = (cbc1)B = 1,

em = 1,

cc0e~‘ = C\ .

As in the precious case we argne th a t one of c,- belongs to F. But then the other one does. So cqCi, being an element of order 77 belongs to F+ ,

6 G. G rom adzki

C o r o lla ry The order o f a group o f autom orphism s o f a bordered com­ pact Klein surface o f algebraic genus p > 2 is bounded aboue by 12(p—1). P ro o f. If a finite group G is a group of automorphisms of a bordered

Klein surface of algebraic genus p > 2 then G — A/T, where F is a bordered surface group of area 27r(p — 1) and by lemma 1 p(A) > 7r/6. Thus

|G| =

A

r)/,,(A ) <

M l 7 f ] =

12

-

R e m a r k 1 It turns out that an NEC-group T with signature

( 0 , + ; [ - ] , { ( 3 , 2 , 2 , 2 ) } )

and area 7r/6 is the group which admits bordered surface groups as nor­ mal subgroups o f a finite index [3], [9] and it was shown in m any papers that the bound 12(p — 1) is attained fo r infinitely m any groups [3], [f],

[5], [6], [9], [10], [12].

R e m a r k 2 Recently it was shown in [2] that the necessary and suffi- cient condition fo r an N E C group A to admit a bordered surface group

F as a norm al subgroup o f finite index is that A has a signature with, an

em pty period cycle or with a period cycle with two consecutiue periods equal to 2. A n N E C group A with an em pty period cycle has clearly area > 7t/ 3 while it is easy to obserue that a period cycle with two con­ secutiue periods equal to 2 in an N E C group willi area < 7t/3 has length fo u r and then p(A) > 7t/6. This giues one more proof o f the result in

question.

A c k n o w le d g e m e n t. This paper was written in D epartam ento de

M atem aticas Fundamentales of UNED in Madrid, when the author was the fellow of a Spanish Government postdoctoral grant. The author would like to thank a referee for several helpful remarks and sugges- tions.

R e fe r e n c e s

[1] E. Bujalance, Norm al subgroups o f N E C groups, M ath. Z. 178 (1981), 331-341

[2] E. Bujalance, E. Martinez, A remark on N E C groups o f surfaces

with boundary, Buli. London Math. Soc. 21 (1989), 263-266

[3] J. J. Etayo Gordejuela, Klein surfaces with m axim al sym m etry

and their groups o f autom orphism s, Math. Ann. 268 (1984),

533-538

[4] J. J. Etayo Gordejuela, C. Perez-Chirinos, Bordered and unbor-

dered surfaces with m azim al sym m etry, J. Pure App. Algebra 42

(1986), 29-35

[5] N. Greenleaf, C. L. May, Bordered Klein surfaces with m azim al

sym m etry, Trans. Amer. Math. Soc. 274 (1982), 265-283

[6] G. Gromadzki, On supersoluable M*-groups, Preprint UNED Mad- rid 1987

[7] A. M. M acbeath, The classification o f non-euclidean crystallo-

graphic groups, Gan. J. Math. 19 (1967), 1192-1205

[8] C. L. May, A utom orphism s o f compact Klein surfaces with bound­

ary, Pacific J. Math. 59 (1975), 199-210

[9] C. L. May, Large autom orphism groups o f compact Klein surfaces

with boundary, I, Glasgow Math. J. 18 (1977), 1-10

[10] C. L. May, A fa m ily o f M*-groups, Gan. J. Math. 38 (1986), 1094-1109

[11] R. Preston, Projectwe structures and fundam ental domains on

compact Klein surfaces, Ph. D. Thesis, The University of Texas

(1975)

[12] D. Singerman, PSL(2,q) as an image o f the eztended modular

group with application to group actions on surfaces, Proc. Edinb.

8 G. G rom adzki

[13] H. C. Wilkie, On non-Euclidean crystallographic groups, M ath. Z. 91 (1966), 87-102 W Y Ż S Z A S Z K O Ł A P E D A G O G I C Z N A I N S T Y T U T M A T E M A T Y K I Chodkiewicza 30 85-064 Bydgoszcz, Poland Received before 23.12.1988