UBC Theses and Dissertations

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UBC Theses and Dissertations

The genus of a group Medalen, David Norman 1989

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THE GENUS OF A GROUP By DAVID NORMAN MEDALEN B . A . , S t . O l a f C o L l e g e , 1966 A T H E S I S SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES ( D e p a r t m e n t o f M a t h e m a t i c s ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1989 David Norman Medalen, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) TABLE OF CONTENTS I n t r o d u c t i o n jf1* ' C h a p t e r One: The C a y l e y Genus o f a Group p. 2 B a s i c G r a p h T h e o r y p . 2 The C a y l e y C o l o r Graph o f a Group p . 9 The Genus of a Graph p . 15 The C a y l e y Genus o f a Group p . 21 1) Groups o f C a y l e y Genus 0, 1, 2: M a s c h k e , P r o u l x , and T u c k e r p . 22 2) The Q u a t e r n i o n Group Q p . 27 3) The Groups G = ( Z 2 m ) n : p . 28 4) H a m i l t o n i a n Groups p . 29 5) T h e o r e t i c a l R e s u l t s - p . 30 6) A p p l i c a t i o n s to S n and A n p . 34 7) The Q u a s i - H u r w i t z Bound p . 37 C h a p t e r Two: The B u r n s i d e Genus o f a Group p . 40 The F u n d a m e n t a l Group p . 40 F u n d a m e n t a l P o l y g o n s p . 42 C o v e r i n g Spaces p . 44 F u n d a m e n t a l Groups and C o v e r i n g Spaces p . 47 fl B r a n c h e d C o v e r i n g Spaces p . 49 The Riemann - H u r w i t z F o r m u l a p . 50 Riemann S u r f a c e s p . 51 A b s t r a c t Riemann S u r f a c e s p . 53 Automorphisms o f a Riemann S u r f a c e p . 55 The H u r w i t z Bound p . 55 The B u r n s i d e Genus o f a Group p . 58 The H y p e r b o l i c P l a n e p . 58 F u c h s i a n G r o u p s p . 61 F u c h s i a n Groups and Riemann S u r f a c e s p . 68 The A c t i o n on a Riemann S u r f a c e p . 70 E x i s t e n c e o f the B u r n s i d e Genus p . 72 Improved H u r w i t z Bounds p . 73 Genus F o r m u l a s f o r Groups p . 75 1) B u r n s i d e ' s work p . 75 2) M a c h l a c h l a n 1 s work p . 77 3) G l o v e r and S j e r v e ' s work p . 83 C o n c l u s i o n p . 90 R e f e r e n c e s p . 93 Hi I n t r o d u c t i o n T h i s a r t i c l e i s a s u r v e y o f the two d e f i n i t i o n s o f the genus o f a f i n i t e g r o u p t h a t a re p r o m i n e n t i n the m a t h e m a t i c a l l i t e r a t u r e . We c a l l them the C a y l e y genus and the B u r n s i d e genus o f a g r o u p , but we do not i n t e n d to s u g g e s t t h a t e i t h e r m a t h e m a t i c i a n was r e s p o n -s i b l e f o r the o r i g i n a l d e f i n i t i o n s . Each d e f i n i t i o n g i v e s a scheme f o r c l a s s i f y i n g f i n i t e g r o u p s by t o p o l o g i c a l c r i t e r i a . The C a y l e y genus i s d e t e r m i n e d by e m b e d d i n g a c e r t a i n g r a p h f o r the group i n a s u r f a c e o f l e a s t p o s s i b l e genus . The genus o f t h i s s u r f a c e i s t h e n d e f i n e d as the C a y l e y genus o f the g r o u p . The B u r n s i d e genus o f a group i s d e t e r m i n e d by s h o w i n g t h a t the group i s i s o m o r p h i c to a g r o u p o f c o n f o r m a l homeomorphisms o f a s u r f a c e o f f i n i t e genus , and i s n o t i s o m o r p h i c to s u c h a group f o r any s u r f a c e o f l e s s e r genus . The genus o f the s u r f a c e i s d e f i n e d to be the B u r n s i d e genus o f the g r o u p . T h i s p a p e r t r i e s to p r e s e n t a b r o a d m a t h e m a t i c a l b a c k g r o u n d f o r e a c h d e f i n i t i o n o f group g e n u s . We a t t e m p t t o show most o f the i d e a s and methods e m p l o y e d i n the r e s e a r c h , and most o f the s i g n i f i c a n t known t h e o r y and r e s u l t s . P r o o f s a r e s p o r a d i c a l l y i n c l u d e d , but o n l y to c o n v e y a s e n s e o f the n a t u r e o f the r e s e a r c h . 1 C h a p t e r O n e : The C a y l e y Genus o f a Group B a s i c Graph T h e o r y A g r a p h K i s a t r i p l e ( V ( K ) ,E (K) , f ) , w h e r e V ( K ) i s a f i n i t e s e t c a l l e d the v e r t i c e s o f K, E(K) i s a f i n i t e s e t c a l l e d the edges o f K, and f i s a f u n c t i o n d e f i n e d on E(K) t h a t maps an edge e to e i t h e r a s e t {u,v> o f two v e r t i c e s or to an o r d e r e d p a i r (u ,v) o f two v e r t i c e s . I f e i t h e r f ( e ) = { u , v } o r f ( e ) = ( u , v ) , we s a y t h a t e d g e e i s i n c i d e n t w i t h v e r t i c e s u and v , and we say t h a t u and v are a d j a c e n t v e r t i c e s . The v e r t i c e s u and v a r e a l s o c a l l e d the e n d p o i n t s o f e. I f two edges have an e n d p o i n t i n common, the edges are a d j a c e n t . I f f ( e ) = ( u , v ) , we s a y t h a t e i s a d i r e c t e d e d g e , g o i n g f r o m u t o v . A d i r e c t e d g r a p h o r d i g r a p h has e v e r y edge d i r e c t e d . I f e i t h e r f ( e ) = { v , v } o r f ( e ) = ( v , v ) , e d g e e i s c a l l e d a l o o p . I f two edges have the same e n d p o i n t s , they a r e c a l l e d m u l t i p l e e d g e s . A m u l t i g r a p h a l l o w s m u l t i p l e e d g e s , and a p s e u d o g r a p h a l l o w s l o o p s and m u l t i p l e e d g e s . The d e g r e e d(v) o f a v e r t e x i s the number o f edges i n c i d e n t w i t h v , a l o o p c o u n t i n g t w i c e . I f d(v) = r f o r a l l v e r t i c e s v , then the g r a p h i s s a i d to be r e g u l a r o f d e g r e e r . We w i l l make the f o l l o w i n g b a s i c d e f i n i t i o n s f o r graphs w i t h no d i r e c t e d e d g e s . The c o r r e s p o n d i n g d e f i n i t i o n s f o r g r a p h s w i t h d i r e c t e d edges s h o u l d be o b v i o u s . S u p p o s e K 2 = ( V ( K 1 ) , E ( K 1 ) , f 2 ) a n d K 2 = ( V ( K 2 > , E ( K 2 > , f 2 ) a r e t w o g r a p h s . They a r e s a i d to be i s o m o r p h i c i f t h e r e e x i s t b i j e c t i v e 2 f u n c t i o n s p : ViK^ -> V ( K 2 ) a n d q : E ( K X ) -> E ( K 2 ) s u c h t h a t , i f f j C e ) = { u , v } , t h e n f 2 ( q ( e ) ) = { p ( u ) , p ( v ) > . I f K j and K 2 a r e i s o m o r p h i c g r a p h s , we w r i t e 2_ K<>. T h e o r e m 1 .1 : L e t K ] [ ^ T h e n d ^ = d ( P ^ v ^ f o r a 1 1 v e r t i c e s v i n K ^ . C o r o l l a r y 1.2: I f i s r e g u l a r o f d e g r e e r and 2. ^ 2 ' t n e n K 2 i s r e g u l a r o f d e g r e e r . We a l s o have the f o l l o w i n g n e c e s s a r y c o n d i t i o n f o r g r a p h i s o -m o r p h i s m , though we w i l l soon o b s e r v e t h a t i t i s not s u f f i c i e n t . C o r o l l a r y 1 . 3 : L e t the v e r t i c e s o f have d e g r e e s d^ < d 2 <^  . . . <_ d n and the v e r t i c e s o f K 2 have d e g r e e s c^ c 2 £ . . . £ c n . I f d^ i s not e q u a l to c^ f o r some 1 < i £ n , t h e n i s not i s o -m o r p h i c to K 2 « L e t t h e g r a p h K = ( V ( K ) , E ( K ) , f ) . T h e c o m p l e m e n t o f K i s t h e g r a p h K c = ( V ( K c ) , E ( K c ) , g ) , w h i c h i s d e f i n e d to have no l o o p s o r m u l t i p l e e d g e s , and has V ( K C ) = V ( K ) , a n d E ( K C ) = { e : g ( e ) = { u , v } i f and o n l y i f f ( e ) £ <u ,v } f o r any e i n E ( K ) }. The u n i o n U K 2 o f g r a p h s and K 2 i s the g r a p h K w h i c h has V ( K ) = V ( K j ) U V ( K 2 ) and E ( K ) = E C K ^ U E ( K 2 > . T h e j o i n K 1 - « 2 o f t w o g r a p h s i s t h e g r a p h K = ( V ( K ) , E ( K ) , f ) w h i c h h a s V ( K ) = V C K j ) U V ( K 2 > a n d E ( K ) = E C K j ) U E ( K 2 ) U E , w h e r e the s e t o f edges E , d e f i n e d to have no m u l t i p l e e d g e s , s a t i s f i e s E = { e : f ( e ) = <u ,v> ; u i n V ( K X ) , v i n V ( K 2 > }. C o n s i d e r g r a p h s K J L = (vCKj) , E ( K 1 ) , f ^ a n d K 2 = ( V ( K 2 ) , E ( K 2 ) , f 2 ) . The C a r t e s i a n p r o d u c t X K 2 o f the two g r a p h s i s the g r a p h K = ( V ( K ) , E ( K ) , g ) w h i c h h a s V ( K ) = V d C j ) X V ( K 2 ) a n d E ( K ) = 3 { e = ( e j , e 2 ) ) s a t i s f y i n g the f o l l o w i n g c o n d i t i o n s . 1) ej . a n d e2 a r e e d g e s i n t h e j o i n K ^ + I ^ = ( V ( K 1 + K 2 ) , E ( K 1 + K 2 ) , f ) 2) i f f ( e ^ ) = {upUj} a n d f ( e 2 ) = ( v p V 2 ) , t h e n a n d a r e or u 2 = v 2 a n c * ^ l ^ e ^ = ^ u l > v l ^ ^ o r s o m e edge e i n E ( K ^ ) . We w i l l now g i v e the d e f i n i t i o n s o f some i m p o r t a n t w e l l - k n o w n g r a p h s and show e x a m p l e s . 1) A b i p a r t i t e g r a p h K h a s V ( K ) = V j U V2, w h e r e V± a n d V 2 a r e n o n - e m p t y , and f o r e v e r y edge e i n E ( K ) , w i t h e n d p o i n t s u and v , we h a v e u i n V , a n d v i n V 0 . 2) P i s the graph w h i c h i s a p a t h o f l e n g t h n - 1 . P 3) C n , n>3, i s t h e g r a p h w h i c h i s a e y e l e o f l e n g t h n , a 4 c l o s e d p a t h . 4) K n i s the c o m p l e t e g r a p h on n v e r t i c e s , the graph w i t h a l l p o s s i b l e e d g e s . K 4 : 5) K n i s the t o t a l l y d i s c o n n e c t e d g r a p h on n v e r t i c e s : E(K. n ) i s empty . K. : 6) K m n i s the c o m p l e t e b i p a r t i t e g r a p h : K. _ = K + K_ m, n —m —n ' 2 ,3 -7) Q n i s the n-cube d e f i n e d r e c u r s i v e l y as = K 2 and Q = K , X Q . f o r n > 1. x n 1 ^ n - i . 5 Here we w i l l o b s e r v e t h a t two g r a p h s may have the same number o f v e r t i c e s w i t h m a t c h i n g d e g r e e s and s t i l l be n o n - i s o m o r p h i c . Shown b e l o w are the graphs 2 a n c * X Ky B o t h a r e r e g u l a r o f d e g r e e 3, b u t the f i r s t i s b i p a r t i t e and the s e c o n d i s n o t . An a u t o m o r p h i s m of a g r a p h K i s an i s o r m o r p h i s m f : K -> K. The a u t o m o r p h i s m group o f K, d e n o t e d A ( K ) , c o n s i s t s o f a l l a u t o -m o r p h i s m s o f K under c o m p o s i t i o n . e x a m p l e : As e x a m p l e s o f the a u t o m o r p h i s m group o f a g r a p h , we 6 g i v e h e r e t h o s e f o r K n , the c o m p l e t e graph on n v e r t i c e s , and f o r C n , t h e c y c l e o f l e n g t h n . 1) A ( K n ) = S n , t n e s y m m e t r i c group on n l e t t e r s . 2) A ( C n ) = D n , t h e d i h e d r a l g r o u p o f o r d e r 2 n . In o r d e r to a n t i c i p a t e l a t e r d e v e l o p m e n t s , we h e r e i n t r o d u c e a few i d e a s and terms t h a t w i l l p l a y an i m p o r t a n t and r e c u r r i n g r o l e i n t h i s d i s c u s s i o n . 3 O b s e r v e t h a t an a b s t r a c t g r a p h may be r e a l i z e d i n R as w e l l as i n R as a s e t o f p o i n t s and l i n e s r e p r e s e n t i n g v e r t i c e s and edges - • 3 r e s p e c t i v e l y . In a d d i t i o n , the g r a p h can be c o n s t r u c t e d i n R so t h a t no edges i n t e r s e c t , e x c e p t at the v e r t i c e s . We say t h a t the a u t o m o r p h i s m group A(K) o f a g r a p h K, o r any o f i t s s u b g r o u p s , a c t s on the g r a p h . We n o t i c e t h a t i f K i s t h o u g h t o f as a t o p o l o g i c a l space w i t h the t o p o l o g y i n h e r i t e d f r o m E u c l i d e a n 3 -s p a c e , t h e n the a u t o m o r p h i s m s o f K a r e a c t u a l l y homeomorphisms o f K. I f G i s a subgroup o f A ( K ) , not n e c e s s a r i l y p r o p e r , we a r e i n t e r e s t e d i n the o r b i t o f a v e r t e x v under the a c t i o n o f G, d e n o t e d as Gv a n d d e f i n e d as t h e s e t o f v e r t i c e s Gv = { g ( v ) : g i n G }. The f i x e d p o i n t s o f a s u b g r o u p G o f A(K) a r e the p o i n t s on K g i v e n by F i x ( G ) = {p i n K : g ( p ) = p f o r some n o n - i d e n t i t y g i n G} . The f i x e d p o i n t s c a n o c c u r at v e r t i c e s or m i d p o i n t s o f e d g e s . G i s s a i d to a c t f r e e l y i f F i x ( G ) i s e m p t y . G a c t s t r a n s i t i v e l y on K i f f o r e v e r y p a i r o f v e r t i c e s u , v t h e r e i s some g i n G such t h a t g ( u ) = v . 7 F o r G a subgroup o f A ( K ) , the q u o t i e n t g r a p h K / G has v e r t i c e s i n 1-1 c o r r e s p o n d e n c e w i t h the o r b i t s Gv, v i n V ( K ) , and an edge j o i n s Gv a n d Gu i n K / G i f a n d o n l y i f an e d g e j o i n s v a n d u i n K. In o t h e r w o r d s , the v e r t i c e s i n an o r b i t Gv a r e i d e n t i f i e d to a s i n g l e v e r t e x , and the edges j o i n i n g v e r t i c e s i n the o r b i t a r e i d e n t i f i e d to a s i n g l e e d g e . e x a m p l e : The a u t o m o r p h i s m group o f the g r a p h i s S^ , but the s u b g r o u p Z5 a l s o a c t s on by r o t a t i o n s t h r o u g h an a n g l e o f 2fT / 5 r a d i a n s . and the q u o t i e n t g r a p h K ^ / Z ^ a r e shown b e l o w . Edges t h a t a re d o t t e d or s o l i d i n i d e n t i f y to a s i n g l e d o t t e d or s o l i d edge r e s p e c t i v e l y i n the q u o t i e n t g r a p h . OO As c a n be s e e n f r o m t h i s e x a m p l e , the q u o t i e n t g r a p h may a c t u a l l y be a p s e u d o g r a p h , because i t may have l o o p s . I f G i s a s u b g r o u p o f A ( K ) a n d F i x ( G ) i s e m p t y , t h e n t h e p a i r ( K , p ) , K t o g e t h e r w i t h the p r o j e c t i o n map p: K -> K / G , i s s a i d t o be a r e g u l a r c o v e r i n g o f K / G . In t h e l a s t e x a m p l e , p : K^ -> K ^ / Z ^ i s a r e g u l a r c o v e r i n g . I f |G| = n , the c o v e r i n g i s a l s o c a l l e d an £ -s h e e t e d c o v e r i n g . We have a 5 - s h e e t e d c o v e r i n g o f K 5 / Z 5 by K^ i n 8 the e x a m p l e . The C a y l e y C o l o r G r a p h o f a_ G r o u p L e t G be a g r o u p a n d l e t { g 2 , g 2 , ••• ) D e e l e m e n t s o f G . A w o r d W i n g i , g 2 , . . . i s a f i n i t e p r o d u c t f ^ f 2 . . . f n w h e r e e a c h f^ b e l o n g s t o t h e s e t {g^ , g 2 , • g i ~ * > g 2 ~ ^ > ••• ^ e v e r y e l e m e n t o f G c a n be e x p r e s s e d as a w o r d i n g i , g 2 > . " t h e n { g p g 2 , . . . ) i s s a i d to be a s e t o f g e n e r a t o r s f o r G. A r e l a t i o n i s an e x p r e s s i o n o f the f o r m W = 1 f o r some word W. I f G i s g e n e r a t e d by { g l s g 2 , ••• ) a n d e v e r y r e l a t i o n c a n be d e d u c e d f r o m t h e r e l a t i o n s = 1, W 2 = 1, . . . t h e n we w r i t e G = < gl>&2' "' ' W l = ^2 = ••• = 1 ^ a n < * c a l l t h i s a p r e s e n t a t i o n o f G. A p r e s e n t a t i o n i s s a i d to be f i n i t e l y g e n e r a t e d , or f i n i t e l y  r e l a t e d , i f the number o f g e n e r a t o r s o r d e f i n i n g r e l a t i o n s r e s p e c t i v e l y i s f i n i t e . A f i n i t e p r e s e n t a t i o n i s b o t h f i n i t e l y g e n e r a t e d and f i n i t e l y r e l a t e d . Theorem 1.4: E v e r y f i n i t e g r o u p G has a f i n i t e p r e s e n t a t i o n . p r o o f : L e t { g 1 , g 2 , . . . } = G be t h e s e t o f g e n e r a t o r s a n d t a k e a l l r e l a t i o n s o f the f o r m g ^ g j g ^ * = 1. # I t w i l l be v e r y i m p o r t a n t to know how a group i s changed when g e n e r a t o r s i n a p r e s e n t a t i o n a r e d e l e t e d o r when r e l a t i o n s a r e added . L e t G = < g l , . . . , g n : W x = . . . = W m = 1 > be a p r e s e n t a t i o n o f G. C o n s i d e r the p r e s e n t a t i o n H = < g , , . . . , g i _ ! » 8 i + i , « » . g n = U l = — = U m = 1 > s 9 where g e n e r a t o r g £ has been d e l e t e d f r o m the g e n e r a t i n g s e t and where e a c h word Uj i s the word Wj w i t h each o c c u r r e n c e o f g^ r e p l a c e d by the i d e n t i t y . We say t h a t g^ has been t r i v i a l i z e d . Then H i s the q u o t i e n t g r o u p G/<g^>, where <g^> i s the n o r m a l s u b g r o u p o f G g e n e r a t e d by the e l e m e n t g^ . S t a r t i n g w i t h the same p r e s e n t a t i o n f o r G, l e t the group F be g i v e n by the f o l l o w i n g p r e s e n t a t i o n w i t h added r e l a t i o n s : F = < g i , . . . , g n •  Wx = . . . = Wm = W m + 1 = . . . = W m + p >. Then F i s a homomorphic image o f G, t h a t i s , a q u o t i e n t o f G by a normal s u b g r o u p . L e t g r o u p s G and H have p r e s e n t a t i o n s G = < g p g 2 > ••• : W 1 = W 2 = . . . = 1 >, H = K. h ^ , h 2 , . . . : = R 2 = . . . ~ 1 y. Then the s t a n d a r d p r e s e n t a t i o n f o r the d i r e c t p r o d u c t G X H i s G X H = < g 1 , g 2 , . . . , h 1 , h 2 , . . : Wj =...= R j =...= g i h j g £ _ 1 h j " 1 = 1 >, where the e x t r a r e l a t i o n h o l d s f o r a l l i and j and g u a r a n t e e s t h a t the g e n e r a t o r s o f G commute w i t h t h o s e o f H. L e t P be a p r e s e n t a t i o n f o r the group G. T h e r e i s a s s o c i a t e d w i t h P a g r a p h c a l l e d the C a y l e y c o l o r g r a p h o f G w i t h p r e s e n t a t i o n P d e n o t e d by Kp(G) . I t has v e r t i c e s i n b i j e c t i v e c o r r e s p o n d e n c e w i t h the e l e m e n t s o f G. I f v^ and v 2 a r e v e r t i c e s c o r r e s p o n d i n g t o e l e m e n t s g^ and g 2 i n G, t h e n an edge runs f r o m v ^ t o v 2 i f , and o n l y i f , g j h = g 2 f o r some g e n e r a t o r h . E a c h g e n e r a t o r h i s i m a g i n e d t o p r o d u c e edges o f a u n i q u e c o l o r . Thus Kp(G) i s a d i r e c t e d g r a p h , and i f P has r g e n e r a t o r s o f o r d e r 2 and n g e n e r a t o r s o f o r d e r g r e a t e r t h a n 2, t h e n Kp(G) i s 10 r e g u l a r o f d e g r e e 2n + r . A word W i n the g e n e r a t o r s c o r r e s p o n d s to a p a t h , i n f a c t many p a t h s , i n K p ( G ) , and a r e l a t i o n o f the f o r m R = 1 i n the p r e s e n t a -t i o n o f G c o r r e s p o n d s to a c y c l e , many c y c l e s , i n the C a y l e y c o l o r g r a p h , or to an edge f o r a r e l a t i o n l i k e g ^ g ^ - * = 1. I n a c c o r d a n c e w i t h o u r p r e v i o u s d e f i n i t i o n , an a u t o m o r p h i s m o f a C a y l e y c o l o r g r a p h K p ( G ) = ( V ( K p ( G ) ) , E ( K p ( G ) ) , f ) i s d e t e r m i n e d by t w o b i j e c t i v e m a p s , p : V ( K p ( G ) ) -> V ( K p ( G ) ) a n d q : E ( K p ( G ) ) -> E ( K p ( G ) ) s u c h t h a t f o r e a c h g ^ , g 2 i n G and e a c h g e n e r a t o r h , f ( e ) = ( g 1 h , g 2 ) i f , and o n l y i f , f ( q ( e ) ) = ( p ( g 1 ) h , p ( g 2 ) ) . Thus an a u t o m o r p h i s m o f a C a y l e y c o l o r g r a p h p r e s e r v e s d i r e c t e d  a d j a c e n c y and the c o l o r c o r r e s p o n d i n g to e a c h a d j a c e n c y . W h i t e ( [ 3 4 ] , p . 2 5 ) g i v e s a p r o o f o f the next t h e o r e m . T h e o r e m 1.5: L e t K p ( G ) be a C a y l e y c o l o r g r a p h f o r a f i n i t e g r o u p G. T h e n A ( K p ( G ) ) = G , the f u l l g roup o f a u t o m o r p h i s m s o f the g r a p h K p ( G ) i s G. D i s p l a y e d b e l o w i s the C a y l e y c o l o r g r a p h f o r A ^ w i t h the p r e s e n t a t i o n A ^ = < r , s : r = s = ( r s ) = 1 >, w h e r e t h e g e n e r a t o r s a r e the p e r m u t a t i o n s r = ( 1 2 ) ( 3 4 ) and s = ( 1 2 3 ) . 1 1 A member of a g e n e r a t i n g s e t f o r a group G i s s a i d to be r e d u n d a n t i f i t can be w r i t t e n as a word i n the r e m a i n i n g g e n e r a t o r s . A g e n e r a t i n g s e t i s m i n i m a l i f i t has no r e d u n d a n t g e n e r a t o r s . 2 3 -e x a m p l e : The s e t {x ,x } i s a m i n i m a l g e n e r a t i n g s e t f o r the g r o u p Zg•= < x : x^ = 1 >, e v e n t h o u g h {x} i s a g e n e r a t i n g s e t w i t h f e w e r e l e m e n t s . A g e n e r a t o r h i s r e d u n d a n t i f , and o n l y i f , the d e l e t i o n o f a l l edges c o l o r e d h i n Kp(G) l e a v e s a c o n n e c t e d d i r e c t e d g r a p h . I f h i s not r e d u n d a n t , d e l e t i o n o f a l l edges c o l o r e d h l e a v e s a c o l l e c t i o n o f i s o m o r p h i c d i s j o i n t s u b g r a p h s , e a c h r e p r e s e n t i n g the q u o t i e n t group o f G g e n e r a t e d by i t s g e n e r a t i n g s e t w i t h h t r i v i a l i z e d . G i v e n a C a y l e y c o l o r g r a p h K p ( G ) f o r a group G w i t h p r e s e n t a t i o n P, and a s u b g r o u p H o f G, the S c h r e i e r ( r i g h t ) c o s e t  g r a p h has v e r t i c e s c o r r e s p o n d i n g to r i g h t . cosets o f H i n G, and an edge runs f r o m c o s e t Hg^ to c o s e t H g 2 i f , and o n l y i f , g^h = g 2 f o r some g e n e r a t o r h o f G i n P. Loops o r m u l t i p l e edges may be i n t r o d u c e d by the i d e n t i f i c a t i o n o f H w i t h t h e i d e n t i t y e l e m e n t , so a S c h r e i e r c o s e t g r a p h may be a p s e u d o g r a p h . In c a s e H i s a n o r m a l s u b g r o u p o f G, the S c h r e i e r c o s e t g r a p h i s e v i d e n t l y the C a y l e y c o l o r g r a p h o f the q u o t i e n t group G / H . S i n c e the subgroup H a c t s on the g r a p h Kp(G) , the S c h r e i e r c o s e t g r a p h i s a l s o j u s t the q u o t i e n t g r a p h K p ( G ) / H . 12 I t f o l l o w s f r o m a p r e v i o u s d e f i n i t i o n t h a t the C a r t e s i a n  p r o d u c t K p Q ) ( G ) x K p ( 2 ) ^ ° ^ C a y l e y c o l o r g r a p h s f o r g r o u p s G and H h a s v e r t e x s e t V ( K p Q ) ( G ) ) X V ( K p ( 2 ) ( H ) ) , a n d ( g p g 2 ) i s j o i n e d t o ( 8 3 . 8 4 ) i f e i t h e r (1) = g 3 a n d g 2 h = g ^ f o r a g e n e r a t o r h i n ? 2 > o r (2) g 2 = g ^ a n d g ^ h = g ^ f o r a g e n e r a t o r h i n P ^ . e x a m p l e : H e r e a r e K ( Z 2 ) , K ( Z 3 ) , and K ( Z 2 ) X K ( Z 3 > , w h e r e t h e 7 3 p r e s e n t a t i o n s a r e Z 2 = < y : y = 1 >, Z ^ = < x : x = 1 >, w i t h t h e o s t a n d a r d p r e s e n t a t i o n o f t h e c r o s s p r o d u c t , Z 2 X Z-j = < y , x : y = x 3 = y x y _ 1 x _ 1 = 1 >. 13 W h i t e p r o v e s the f o l l o w i n g theorem i n h i s t e x t G r a p h s , G r o u p s , and S u r f a c e s . Theorem 1.6: L e t K p ( i ) ( G ) a n d K p ( 2 ) ^ ^ e Cay^-ey c o l o r g r a p h s o f a g r o u p G w i t h p r e s e n t a t i o n P ( l ) , and a g r o u p H w i t h p r e s e n t a t i o n P ( 2 ) . L e t P be t h e s t a n d a r d p r e s e n t a t i o n o f G X H . T h e n K p ( G X H) = K P ( i ) ( G ) X K p ( 2 ) ( H ) . Here we n o t i c e t h a t d i f f e r e n t p r e s e n t a t i o n s o f a group can g i v e v e r y d i f f e r e n t C a y l e y c o l o r g r a p h s . F o r e x a m p l e , the group Z 2 X w i t h the s t a n d a r d p r e s e n t a t i o n has the g r a p h shown above . But s i n c e t h i s group i s i s o m o r p h i c t o Z g , i t s p r e s e n t a t i o n as a c y c l i c g r o u p w i l l have a C a y l e y c o l o r g r a p h i s o m o r p h i c to C ^ , the c y c l e o f l e n g t h s i x . Theorem 1.7: The F u n d a m e n t a l Theorem o f A b e l i a n Groups L e t G be a f i n i t e a b e l i a n g r o u p . Then G c a n be e x p r e s s e d as a d i r e c t p r o d u c t o f c y c l i c g r o u p s , G = Z m Q ) X Z m ( 2 ) X . . . X Z m ^ k j , w h e r e m ( i ) d i v i d e s m ( i + l ) , a n d | G | = m( 1 ) m ( 2 ) . . . m ( k ) . T h i s e x p r e s s i o n i s u n i q u e . The number k i s c a l l e d the rank o f G. Theorem 1.8: L e t G be a f i n i t e a b e l i a n group w i t h G = Z m Q ) X Z m ( 2 ) X . . . X Z ^ ^ . L e t e a c h g r o u p Z m ^ ^ h a v e a p r e s e n t a t i o n o f t h e f o r m = < g i : g i m ^ 1 ^ = 1 >. T h e n , i f P i s t h e s t a n d a r d p r e s e n t a t i o n f o r the d i r e c t p r o d u c t Z ^ Q J X Z-m(2) X . . . X Z m(k)> w e h a v e K P ( G ) = C m ( l ) X C m ( 2 ) X - X C m(k)> w h e r e ^ra(i) d e n o t e s t n e d i r e c t e d c y c l e o f l e n g t h m ( i ) . 14 The Genus o f a_ G r a p h L e t us now d e v e l o p the e s s e n t i a l t o p o l o g i c a l i d e a s needed t o d i s c u s s the genus o f a g r a p h . The f o l l o w i n g t h e o r e m i s w e l l known. T h e o r e m 1.9: A c l o s e d o r i e n t a b l e 2 - m a n i f o l d S, a s u r f a c e , i s a s p h e r e w i t h g h a n d l e s , g !> 0. T h e g e n u s o f S i s d e f i n e d t o be g , a n d we w r i t e g e n u s ( S ) = g . A p s e u d o g r a p h K i s s a i d to be embedded i n a s u r f a c e S i f i t i s d r a w n on S so t h a t edges i n t e r s e c t o n l y a t a common v e r t e x . F o r a p s e u d o g r a p h K embedded i n a s u r f a c e S, the components o f S \ K are c a l l e d the r e g i o n s o r f a c e s o f the e m b e d d i n g . An e m b e d d i n g o f a p s e u d o g r a p h K i n a s u r f a c e S i s s a i d to be a 2 - c e l l e m b e d d i n g i f e v e r y r e g i o n o f the e m b e d d i n g i s a 2 - c e l l , t h a t i s h o m e o m o r p h i c to the open u n i t d i s c . Theorem 1.10: L e t K be a c o n n e c t e d p s e u d o g r a p h w i t h a 2 - c e l l e m b e d d i n g i n a c l o s e d o r i e n t a b l e s u r f a c e o f genus g. L e t V , E , and F be the number o f v e r t i c e s , e d g e s , and f a c e s r e s p e c t i v e l y . T h e n V - E + F = 2 - 2 g . T h i s i s known as E u l e r ' s t h e o r e m . I t i s u s u a l l y g i v e n f o r a t r i a n g u l a t i o n o f a s u r f a c e o f genus g , w i t h V , E , and F b e i n g v e r t i c e s , e d g e s , and f a c e s o f the t r i a n g u l a t i o n . The genus o f a g r a p h K, g e n u s ( K ) , i s the minimum genus among a l l c l o s e d , o r i e n t a b l e s u r f a c e s i n w h i c h K c a n be embedded. Theorem 1.11: E v e r y f i n i t e g r a p h has a g e n u s . 2 p r o o f : Suppose K has E e d g e s . Embed e a c h v e r t e x i n S , the 2 - s p h e r e . A t t a c h E h a n d l e s , and embed one edge i n each h a n d l e . Then 15 g e n u s ( K ) < E . # Theorem 1.12: ( K u r a t o w s k i ) A g r a p h K i s p l a n a r , genus(K) = 0, i f , and o n l y i f , i t c o n t a i n s no s u b g r a p h i s o m o r p h i c w i t h e i t h e r K 5 o r K 3 , 3 -e x a m p l e : The d i a g r a m b e l o w i s an e m b e d d i n g o f 3 i n the t o r u s . T o g e t h e r w i t h K u r a t o w s k i ' s t h e o r e m , t h i s shows t h a t g e n u s ( K o 3) = 1. An e m b e d d i n g o f a g r a p h K i n a s u r f a c e o f genus g i s c a l l e d m i n i m a l i f genus(K) = g. Theorem 1.13: I f a c o n n e c t e d g r a p h K i s m i n i m a l l y embedded i n a s u r f a c e S, t h e n the e m b e d d i n g i s a 2 - c e l l e m b e d d i n g . p r o o f : W h i t e ( [ 3 4 ] , p .54) g i v e s the f o l l o w i n g i n f o r m a l a r g u m e n t . We may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t each v e r t e x o f K l i e s on the s p h e r e p a r t o f S, not on a h a n d l e , and t h a t e a c h h a n d l e has a t l e a s t one edge o f K m e e t i n g i t . Suppose F i s a f a c e t h a t i s not a 2 - c e l l . I t must t h e n c o n t a i n a s i m p l e 16 c l o s e d c u r v e C t h a t cannot be c o n t i n u o u s l y d e f o r m e d , e n t i r e l y w i t h i n F, t o a p o i n t . Then the s u r f a c e S i s not a s p h e r e , so i t has a t l e a s t one h a n d l e . Case I : I f the c l o s e d c u r v e C l i e s e n t i r e l y on a h a n d l e , the s u r f a c e may be c u t a l o n g C , and when the two h o l e s t h a t r e s u l t a r e c a p p e d , an e m b e d d i n g o f K i n a s u r f a c e of l e s s e r genus i s o b t a i n e d . T h i s c o n t r a d i c t s the a s s u m p t i o n o f a m i n i m a l e m b e d d i n g . Case I I : I f the c u r v e C l i e s p a r t l y on a h a n d l e H and p a r t l y on S \ H , the edges o f K t h a t l i e on H may be r e d r a w n t o f o l l o w the c u r v e C on S \ H , g i v i n g an e m b e d d i n g o f K i n a s u r f a c e o f l e s s e r genus a g a i n . The f o l l o w i n g s k e t c h i l l u s t r a t e s t h i s p o s s i b i l i t y . # Since these two cases exhaust the p o s s i b i l i t i e s , the theorem has been proved. This theorem allows us to use the Euler formula V - E + F = 2 - 2 g when a graph i s minimally embedded in a surface. 17 The f o l l o w i n g l o w e r bounds on the genus o f a g r a p h K now f o l l o w f r o m what we know. They a r e v e r y c r u d e , but a r e o f some i n t e r e s t . C o r o l l a r y 1.14: I f K i s a c o n n e c t e d g r a p h , w i t h V >^ 3, t h e n g e n u s ( K ) > E / 6 - V / 2 + 1. E q u a l i t y h o l d s i f , and o n l y i f , K has a t r i a n g u l a r e m b e d d i n g . p r o o f : Assume K i s m i n i m a l l y embedded i n a s u r f a c e o f genus g so t h a t V - E + F = 2 - 2 g . L e t the s i z e o f a f a c e be the number o f edges t r a v e r s e d i n a c i r c u i t o f i t s b o u n d a r y . L e t F n be the number o f f a c e s o f s i z e n , n>3, i n t h e e m b e d d i n g . T h e n 2E = 3 F 3 + 4 F 4 + . . . , and so 2E >_ 3 F , w i t h e q u a l i t y h o l d i n g o n l y i f a l l f a c e s a r e t r i a n g u l a r . The r e s u l t f o l l o w s i m m e d i a t e l y . # C o r o l l a r y 1.15: I f K i s a c o n n e c t e d g r a p h , w i t h V ^ 3, and the e m b e d d i n g o f K has no t r i a n g u l a r r e g i o n s , t h e n g e n u s ( K ) > E / 4 - V / 2 + 1. E q u a l i t y h o l d s i f , and o n l y i f , K has a q u a d r i l a t e r a l e m b e d d i n g . p r o o f : Same as a b o v e . C o r o l l a r y 1.16: I f K i s a c o n n e c t e d b i p a r t i t e g r a p h h a v i n g a q u a d r i l a t e r a l e m b e d d i n g , t h e n g e n u s ( K ) = E / 4 - V / 2 + 1. p r o o f : Same as a b o v e . e x a m p l e : From c o r o l l a r y 1.14 a b o v e , we f i n d t h a t the g r a p h K^Q ^Q, a c o m p l e t e b i p a r t i t e g r a p h , has genus >_ 8. The same c o r o l l a r y y i e l d s t h a t the g r a p h K^Q , the c o m p l e t e g r a p h on t e n 18 v e r t i c e s , has genus > 4. The e x a c t v a l u e f o r the genus o f t h e s e g r a p h s w i l l be o b t a i n e d i n the s e q u e l . In a 1980 p a p e r , "The Number o f G r o u p s o f a G i v e n G e n u s " ( [ 2 8 ] ) , T . W. T u c k e r r e p o r t s the f o l l o w i n g r e s u l t s c o n c e r n i n g g r a p h e m b e d d i n g s . The f i r s t t h e o r e m i s an i m p o r t a n t t e c h n i c a l t o o l t h a t T u c k e r uses r e p e a t e d l y i n much o f h i s work. Theorem 1.17: L e t K be a r e g u l a r graph o f d e g r e e d w i t h V v e r t i c e s , E e d g e s , and F f a c e s w i t h a 2 - c e l l e m b e d d i n g i n an o r i e n t a b l e s u r f a c e of genus g. L e t F n be the number o f f a c e s o f s i z e n , w i t h t h e u n d e r s t a n d i n g t h a t ¥ ^ = ¥ 2 = 0. I f n i s a n y i n t e g e r g r e a t e r t h a n 1, t h e n ( n d / 2 - n - d)V + n(2 - 2g) < ( n - l ) F 1 + ( n - 2 ) F 2 + . . . + ( D F ^ . p r o o f : C l e a r l y 2E = 3 F 3 + 4 F ^ + . . . . T h e r e f o r e , 3 F 3 + 4 F 4 + . . . + ( n - l ) F n _ 1 + n( ¥ - (¥l + ¥2 + . . . + F n _ 1 > ) < 2 E . Now F and E c a n be e l i m i n a t e d f r o m t h i s i n e q u a l i t y u s i n g the r e l a t i o n s dV = 2E and F = E - V + ( 2 - 2 g ) . The r e s u l t f o l l o w s . # One a p p l i c a t i o n of t h i s i n e q u a l i t y i s shown i n the f o l l o w i n g r e s u l t by l e t t i n g d > 6 a n d n = 3. Theorem 1.18: L e t K be a r e g u l a r g r a p h o f d e g r e e d > 6 embedded i n a s u r f a c e o f genus g. Then V < 6(2g - 2). T h e r e f o r e , the number o f c o n n e c t e d r e g u l a r g r a p h s o f d e g r e e d > 6 e m b e d d a b l e i n a g i v e n s u r f a c e i s f i n i t e . On the o t h e r hand, T u c k e r g i v e s an e x p l i c i t c o n s t r u c t i o n , w h i c h we o m i t , to show t h a t : Theorem 1.19: The number o f c o n n e c t e d r e g u l a r g r a p h s o f d e g r e e 3 o r 4 o f a g i v e n g e n u s i s i n f i n i t e . 19 T u c k e r a l s o r e p o r t s t h a t G r o s s and P r o u l x have found a method o f c o n s t r u c t i o n to show t h a t : Theorem 1.20: The number o f c o n n e c t e d r e g u l a r graphs o f d e g r e e 5 o r 6 o f a g i v e n g e n u s i s i n f i n i t e . A r t h u r T . W h i t e has done c o n s i d e r a b l e work on the genus o f g r a p h s and r e p o r t s much o f h i s own work i n t h i s f i e l d i n a 1970 p a p e r e n t i t l e d "The Genus o f R e p e a t e d C a r t e s i a n P r o d u c t s o f B i p a r t i t e G r a p h s " ( [32 ] ) . T h i s paper o f W h i t e ' s i s a l s o a s e c o n d a r y s o u r c e f o r s e v e r a l w e l l - k n o w n r e s u l t s due to the m a t h e m a t i c i a n s G. R i n g e l and J . W. T . Y o u n g s , s o m e t i m e s w o r k i n g t o g e t h e r ( [ 2 5 ] , [ 2 6 ] , [ 2 7 ] ) . F o r e x a m p l e , W h i t e c r e d i t s R i n g e l ( [25] ) w i t h one o f the e a r l i e s t genus f o r m u l a s f o r a g r a p h , when he e s t a b l i s h e d i n 1955 the genus o f the n - c u b e Q n : T h e o r e m 1 . 2 1 : g e n u s ( Q n ) = 1 + 2 n - 3 ( n - 4), n > 2. L e t { x } = n i f n - 1 < x £ n . A c c o r d i n g t o W h i t e , R i n g e l p r o v e d the f i r s t o f the f o l l o w i n g r e s u l t s i n 1965 and R i n g e l and Youngs showed the next one i n 1968, i n b o t h c a s e s by a c t u a l c o n s t r u c t i o n o f an embedding shown to be m i n i m a l . T h e o r e m 1 .22 : g e n u s ( K m n ) = < (m - 2 ) ( n - 2)/4 }, m , n > 2. T h e o r e m 1 .23 : g e n u s ( K n ) = < (n - 3 ) ( n - 4 ) / 1 2 >, n > 3. e x a m p l e : The g r a p h K^Q t h e n has a genus o f 16, and K^Q has a genus o f 4. Our e a r l i e r c o a r s e e s t i m a t e s had g i v e n l o w e r bounds o f 8 and 4 r e s p e c t i v e l y f o r t h e s e two v a l u e s . W h i t e ' s own work i n the same p a p e r g e n e r a l i z e s the r e s u l t f o r 20 c o m p l e t e b i p a r t i t e g r a p h s , a g a i n t h r o u g h e x p l i c i t c o n s t r u c t i o n o f an e m b e d d i n g , to y i e l d the next two t h e o r e m s . T h e o r e m 1.24: g e n u s ( K 9 m 9 X K ) = 1 + m((m - 2 ) ( r + s) + r s ) , r < 2m, s < 2m. Ill y III L j O * G e n e r a l i z e the n - cube as f o l l o w s . L e t Q(s)^ = K g g and r e c u r s i v e l y d e f i n e Q ( s ) n = Q ( s ) n _ i x K s s f o r n — 2 ' Theorem 1.25: genus( Q(2m) n ) = 1 + 2 2 n _ 2 m n ( m n - 2). Because e v e r y even c y c l e i s a b i p a r t i t e g r a p h , though not c o m p l e t e , W h i t e was a b l e to e x t e n d h i s methods to C a r t e s i a n p r o d u c t s o f even c y c l e s . We d e f i n e the graph J n r e c u r s i v e l y as f o l l o w s . L e t = C 2 m ( l ) ' t h e c v c l e o n 2 m ( D v e r t i c e s . L e t J R = J n - 1 X C 2 m ( n ) where e a c h m ( i ) .> 2. L e t M ^ n ^ d e n o t e t h e p r o d u c t m( 1) m( 2 ) . . .m( n ) . I n t h e s p e c i a l c a s e when m ( i ) = m f o r i = l , 2 , . . . , n we w i l l d e n o t e t h e g r a p h J n as J n ( m ) . T h e o r e m 1 .26 : g e n u s ( J n ) = 1 + 2 n - 2 ( , n - 2 ) M ( n ) , n > 2. The C a y l e y Genus o f a Group R e c a l l t h a t genus(Kp(G)) d e n o t e s the genus o f the C a y l e y c o l o r g r a p h Kp(G) f o r the group G. T h e n the C a y l e y genus o f a g roup G i s the minimum o f genus(Kp(G)) o v e r a l l p r e s e n t a t i o n s P o f the group G, and i s d e n o t e d genus^,(G). I t i s not c l e a r who f i r s t f o r m u l a t e d t h i s d e f i n i t i o n o f the genus o f a g r o u p , and we do not s u g g e s t by t h i s t e r m i n o l o g y t h a t the d e f i n i t i o n o r i g i n a t e d w i t h C a y l e y . A l l t h a t i s known i s t h a t the 21 e a r l i e s t s i g n i f i c a n t work on the C a y l e y genus was done by H. Maschke [21] i n 1896. He c l a s s i f i e d , as f o l l o w s , a l l f i n i t e g r o u p s o f genus 0. 1) G r o u p s o f genus 0, 1, 2: M a s c h k e , P r o u l x , and T u c k e r Theorem 1.27: ( M a s c h k e , [21]) A f i n i t e g r o u p G has g e n u s ^ ( G ) = 0 i f , and o n l y i f , G = G ^ X G 2 , w h e r e G j = Z ^ o r Z 2 a n d G 2 = Z n , D n , S ^ , A ^ , o r A 5 . Z n i s t h e g r o u p o f i n t e g e r s mod n , D n i s the d i h e d r a l group o f o r d e r 2n, S n i s the s y m m e t r i c g r o u p , and A n i s the a l t e r n a t i n g g r o u p . In a 1978 paper ( [24] ) e n t i t l e d " C l a s s i f t c a t i o n o f the T o r o i d a l G r o u p s , " V. K. P r o u l x c o n t i n u t e d the r e s e a r c h program begun by Maschke 82 y e a r s e a r l i e r by a n n o u n c i n g the p r e s e n t a t i o n t y p e s o f a l l g r o u p s o f C a y l e y genus 1. T h i s i s work i n the same s p i r i t as M a s c h k e ' s d e t e r m i n a t i o n o f the g r o u p s o f C a y l e y genus 0, though c o n s i d e r a b l y more advanced i n d i f f i c u l t y , i n v o l v i n g , a c c o r d i n g to T u c k e r , who has s t u d i e d h e r w o r k , o v e r one h u n d r e d pages o f i n t r i c a t e m a n i p u l a t i o n s o f g e n e r a t o r s and r e l a t i o n s . P r o u l x has d e t e r m i n e d a n e c e s s a r y c o n d i t i o n f o r a f i n i t e group G to have C a y l e y genus 1. Theorem 1.28: L e t a p r e s e n t a t i o n o f the f i n i t e group G have k g e n e r a t o r s o f o r d e r 2, m g e n e r a t o r s o f o r d e r 3, and n g e n e r a t o r s o f o r d e r g r e a t e r t h a n 3. Then f o r G to be t o r o i d a l i t i s n e c e s s a r y t h a t 3 < k + 5 m / 3 + 2n < 4. 22 C o r o l l a r y 1.29: E v e r y t o r o i d a l group has a p r e s e n t a t i o n w i t h two, t h r e e , o r f o u r g e n e r a t o r s . Theorem 1.30: ( P r o u l x , [ 2 4 ] ) L e t the f i n i t e group G, w i t h genuS(-,(G) > 0, h a v e a p r e s e n t a t i o n G = < R, S: . . . >. T h e n G i s t o r o i d a l i f , and o n l y i f , a t l e a s t one o f the f o l l o w i n g c a s e s i s s a t i s f i e d . The c a s e s are not m u t u a l l y e x c l u s i v e . 1) G = < R, S: S 2 = R 3 S R S = 1 > 2) G = < R, S: S 2 = R 3 S R S = 1 > 3) G = < R, S: s 2 = ( R 2 S ! I 2 = 1, R a n d RS h a v e e v e n 4) G = < R, S: s 2 = R 2SR~ ' 2S = 1, R has even o r d e r , 5) G = < RJ S: R 6 = S 2 = ( R S ) 3 , ... > 6) G = < R, S: R 4 = S 2 = ( R S ) 4 = 1, ... > 7) G = < R> S: R 4 = S 2 - ( R S ) 2 ( R _ 1 S ) 2 = 1, . . . . > 8) G = < R, S: R 3 = s 2 = ( R S R S R - 1 S ) 2 = 1, . . . > 9) G = < R, S: R 3 = s 2 = ( R S ) 3 ( R _ 1 S ) , 3 = 1, . 10) G = < R, S: R 4 = s 2 = ( R S R _ 1 S ) 2 = 1, ... > 11) G = < R) S: R 3 = s 2 = ( R S R _ 1 S ) 3 = 1 , ... > 12) G = z m x m Z n , g - c . d o f m,n > 3 13) G = < R, S: R 2 P , 5 = R S R _ 1 S = 1 f o r some i n t e g e r s p > 1, q > 2; ... > 14) G = < R, S: R 2 P = s2<* = R 2 S 2 = 1 f o r some i n t e g e r s p , q > 1, ... > 15) G = < R> S: R 4 = s 4 = ( R S ) 2 = 1, ... > 16 ) G = < R, S: R 2 P , = s2* = ( R S ) 2 = ( R _ 1 S ) 2 = 1 f o r i n t e g e r s p , q > 1; . . . > 2 3 17) G = < R, S: R 3 = S 6 = ( R S ) 2 = 1, . . . > 18) G = < R, S: R 3 = S 3 = ( R S ) 3 = 1, . . . > 19) G = < R, S : R 3 = S 3 = R S R S - 1 R _ 1 S - 1 = 1 >. Theorem 1.31: ( P r o u l x , [24]) L e t the f i n i t e g r o u p G , w i t h g e n u s c ( G ) > 0, h a v e a p r e s e n t a t i o n G = < R, S , T : . . . >. T h e n G i s t o r o i d a l i f , and o n l y i f , a t l e a s t one o f the f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d . 1) G = < R, S , T : R 2 = S 2 = T 2 = ( R S ) 3 = ( R T ) 3 = ( S T ) 3 = 1, a l l o t h e r r e d u c e d r e l a t i o n s are o f even l e n g t h , . . . > 2) G = < R, S, T : R 2 = S 2 = T 2 = ( R S T ) 2 = 1, . . . > 3) G = < R, S, T : R 2 = S 2 = T 2 = R S R T S T = ( R S ) 2 k = ( T S ) 2 k = 1 f o r some k > 0, . . . > 4) G = < R, S, T : R 2 = S 2 = T 2 = ( T R T S ) 2 = ( R S ) 2 = 1 , . . . > 5) G = < R, S, T : R 2 = S 2 = T 2 = ( R T ) 2 ( S T ) 2 = ( R S ) 2 = 1 > 6) G = < R, S, T : R 2 = S 2 = T 2 = ( R S ) 2 = ( R T ) 4 = ( S T ) 4 = 1, . . . > 7) G = < R, S, T : R 2 = S 2 = T 2 = ( R S ) 2 = ( R T ) 6 = ( S T ) 3 = 1, . . . > 8) G = < R, S , T : R r = S 2 = T 2 = ( R S ) 2 = ( R T ) 2 = ( S T ) P = 1 where r > 2 and p i s r e l a t i v e l y p r i m e to r , . . . > 9) G = < R, S , T : R 2 p = S 2 = T 2 = ( R S ) 2 = R T R - 1 T = ( S T ) 2 q = 1 f o r some i n t e g e r s p , q > 1; . . . > 24 10) G = < R, S, T : R r = S 2 = T 2 = RSR L S = RTR * T = ( S T ) P = 1 where r > 2 and p i s r e l a t i v e l y p r i m e to r , . . . > Theorem 1.32: ( P r o u l x , [24]) L e t the f i n i t e group G , w i t h g e n u s c ( G ) > 0, h a v e a p r e s e n t a t i o n G = < R, S, T , U : . . . >. T h e n G i s t o r o i d a l i f , a n d o n l y i f , G = < R, S, T , U : R 2 = S 2 = T 2 = U 2 = ( R S ) 2 = ( S T ) 2 = ( T U ) 2 = ( U R ) 2 = ( R T ) P = ( S U ) q = 1 w h e r e p = 2 a n d q i s e v e n o r p , q > 2 >. T h e g r o u p G i s t h e d i r e c t p r o d u c t o f two d i h e d r a l g r o u p s , G = X D ^ . N o t i c e t h a t i n some c a s e s above a s i n g l e group i s d e f i n e d , w h i l e i n o t h e r c a s e s i t i s an i n f i n i t e c l a s s o f g r o u p s . I t s h o u l d a l s o be n o t e d t h a t f o r some o f t h e s e c a s e s P r o u l x i n c l u d e d a d d i t i o n a l r e s t r i c t i o n s c o n c e r n i n g the n o r m a l i t y and i n d e x o f c e r t a i n s u b g r o u p s . H o w e v e r , T u c k e r [31,] has shown t h a t a l l such s u b g r o u p r e s t r i c t i o n s are u n n e c e s s a r y , so we o m i t them. In a d d i t i o n , T u c k e r [29] has shown the f o l l o w i n g : Theorem 1.33: T h e r e i s e x a c t l y one g r o u p G w i t h genus (-,(G) = 2. The o r d e r o f G i s 96 and a p r e s e n t a t i o n i s G = < x , y , z : x 2 = y 2 = z 2 = ( x y ) 2 = ( y z ) 3 = ( x z ) 8 = y ( x z ) 4 y ( x z ) 4 = 1>. I t h i n k i t i s f a i r to say t h a t the l o g i c a l b e g i n n i n g , and o f t e n the h i s t o r i c a l b e g i n n i n g , o f r e s e a r c h on a d e f i n i t i o n o f g r o u p genus i s the a t t e m p t to f i n d a l l g r o u p s f o r a g i v e n s m a l l v a l u e o f the g e n u s , as M a s c h k e , P r o u l x , and T u c k e r have d o n e . T h i s a p p r o a c h soon becomes c o m p l e t e l y i m p r a c t i c a l , h o w e v e r , f o r e v e n m o d e r a t e v a l u e s o f 25 the genus , and r e s e a r c h q u i c k l y t a k e s the f o r m o f s e a r c h f o r the genus o f each member o f a c l a s s o f g r o u p s . R e c e n t r e s e a r c h , w i t h the e x c e p t i o n o f P r o u l x ' s work and T u c k e r ' s i s o l a t i o n o f the one g r o u p o f C a y l e y genus 2, tends t o be o f t h i s t y p e . L e t us c o n s i d e r now some o f the few genus f o r m u l a s f o r c l a s s e s o f g r o u p s t h a t a r e known. We b e g i n by l o o k i n g a t e l e m e n t a r y r e s u l t s and the i d e a s t h a t a r e i n v o l v e d i n t h e i r p r o o f . The f o l l o w i n g t h e o r e m w i l l be a u s e f u l t o o l . Theorem 1.34: L e t G be a f i n i t e group whose o r d e r i s not a m u l t i p l e o f 3. L e t P be a m i n i m a l p r e s e n t a t i o n f o r G. Then K p ( G ) c o n t a i n s no t r i a n g l e s . p r o o f : I f Kp(G) c o n t a i n s a t r i a n g l e , t h e n t h e r e i s a c o r r e s p o n d i n g r e l a t i o n g j ^ ^ g 2 ^ ^ g 3 ^ " ^ = 1> where each g £ i s a g e n e r a t o r a n d e a c h y ( i ) i s +1 o r - 1 . I f any t w o o f t h e g ^ a r e d i s t i n c t , t h e n one o f t h e m i s r e d u n d a n t . H e n c e g^ = g 2 = g^ = g . B u t now y ( l ) = y ( 2 ) = y ( 3 ) , o r e l s e g i s t r i v i a l . T h e r e f o r e g o r i t s i n v e r s e has o r d e r 3, and 3 d i v i d e s | G | , a c o n t r a d i c t i o n . # S i n c e p r e s e n t a t i o n s o f a g r o u p p l a y a c r i t i c a l r o l e i n the d e f i n i t i o n o f the C a y l e y genus o f a g r o u p , i t i s i m p o r t a n t to n o t e t h a t i f a p r e s e n t a t i o n P f o r a g r o u p G g i v e s genuS(-,(G), t h e n P may be assumed to have no r e d u n d a n t g e n e r a t o r s , b e c a u s e s u c h a p r e s e n t a t i o n g i v i n g a minimum genus a l w a y s e x i s t s . In a r g u m e n t s a b o u t the C a y l e y genus o f a g r o u p i t t h e r e f o r e does no harm to assume t h a t the genus p r e s e n t a t i o n P i s m i n i m a l . 26 2) The Q u a t e r n i o n Group Q L e t us now d e t e r m i n e the C a y l e y genus o f the q u a t e r n i o n g r o u p Q, a n o n - a b e l i a n group o f o r d e r 8. L e t P be a p r e s e n t a t i o n o f Q s u c h t h a t genus^CQ) = g e n u s ( K p ( Q ) ) . We may assume t h a t P i s m i n i m a l . T h e r e f o r e , K p ( Q ) has no t r i a n g l e s , by the l a s t t h e o r e m . I t i s c l e a r t h a t P must have at l e a s t two g e n e r a t o r s . I t c a n r e a d i l y be shown t h a t i f P has p r e c i s e l y two g e n e r a t o r s t h e n n e i t h e r i s o f o r d e r 2. P cannot have t h r e e g e n e r a t o r s o f o r d e r 2, b e c a u s e Q has a u n i q u e e l e m e n t of o r d e r 2. T h i s means t h a t the C a y l e y g r a p h K p ( Q ) i s r e g u l a r o f d e g r e e a t l e a s t 4. T h e r e f o r e 2E >_ 4V = 32 . By c o r o l l a r y 1.15 f o r graphs we have t h a t genus(Q) > E / 4 - V / 2 + 1 > 1. One o f the s t a n d a r d p r e s e n t a t i o n s o f the q u a t e r n i o n g r o u p Q i s 9 9 9 Q = < x, y : x = y = ( x y ) >. A C a y l e y c o l o r g r a p h f o r t h i s p r e s e n t a t i o n i s shown b e l o w . I t i s ^ . S i n c e i t c o n t a i n s ^ a s a s u b g r a p h , K u r a t o w s k i ' s t h e o r e m v e r i f i e s the l a s t i n e q u a l i t y : Q i s n o n - p l a n a r . 1 The d i a g r a m on the next page shows a C a y l e y c o l o r g r a p h o f Q, w i t h the same p r e s e n t a t i o n , embedded i n the t o r u s . 2 7 Theorem 1.35: L e t Q be the q u a t e r n i o n • g r o u p . Then the C a y l e y g e n u s o f Q i s 1, g e n u s c ( Q ) = 1. S i n c e Q i s a t o r o i d a l group i t must be i n c l u d e d i n P r o u l x ' s l i s t . In f a c t , a p r e s e n t a t i o n o f t y p e 14 may be o b t a i n e d f r o m the g i v e n p r e s e n t a t i o n f o r Q by the s u b s t i t u t i o n R = x and S = y ^ , s i n c e x and y b o t h have o r d e r 4, as c a n be s e e n f r o m the C a y l e y g r a p h above . 3) The Groups G = ( Z 2 m ) n The next t h e o r e m s , due to A. T. W h i t e , g i v e the f i r s t d e c i d e d l y n o n - t r i v i a l genus f o r m u l a s f o r c e r t a i n i n f i n i t e c l a s s e s o f g r o u p s . Theorem 1.36: ( W h i t e , [34]) L e t G Q = ( Z 2 ) n . Then g e n u s c ( G n ) = 1 + 2 n _ 3 ( n - 4 ) , n > 2. p r o o f : I f G n i s d e f i n e d r e c u r s i v e l y as G^ = Z 2 and G n = Z 2 X G n _ p and i f G n has the s t a n d a r d p r e s e n t a t i o n P, t h e n the C a y l e y c o l o r g r a p h K p ( G n ) i s Q n > t h e n - c u b e . T h e r e f o r e , u s i n g 28 R i r t g e l ' s r e s u l t on the genus o f the n - c u b e r e p o r t e d i n Theorem 1.21, we have g e n u s ( G n ) < g e n u s ( Q n ) = 1 + 2 n - 3 ( n - 4) . On the o t h e r h a n d , any p r e s e n t a t i o n P f o r G n must have at l e a s t n g e n e r a t o r s , and so 2E ^ nV = n 2 n i n the g r a p h K p ( G n ) « We may assume P i s m i n i m a l , and b e c a u s e 3 does not d i v i d e | G | = 2 n , K p ( G n ) has no t r i a n g l e s . T h e r e f o r e by the g r a p h t h e o r e t i c r e s u l t 1.15 f rom the l a s t s e c t i o n , g e n u s c ( G n ) > E / 4 - V / 2 + 1 > n 2 n - 3 - 2 n _ 1 + 1 = 1 + 2 n _ 3 ( n - 4 ) . # W h i t e has g e n e r a l i z e d t h i s r e s u l t as f o l l o w s . R e c a l l t h a t the g r a p h J n ^ m ^ has a l r e a d y been d e f i n e d as the C a r t e s i a n p r o d u c t o f n c o p i e s o f the c y c l e C 2 r a - L e t us denote by G n ^ m ^ the a b e l i a n g r o u p w i t h m i n i m a l C a y l e y c o l o r g r a p h J n ^ m ^ . T h a t i s , G ^ m ^ = and G n ^ = G ^ ^ " 1 ^ X Z 2 m . Then W h i t e g e t s the n e x t t h e o r e m . Theorem 1.37: ( W h i t e , [34]) g e n u s c ( G n ( m ) ) = 1 + 2 n _ 2 ( n - 2 )m n 4) H a m i l t o n i a n G r o u p s L e t ' s l o o k now at a few known r e s u l t s c o n c e r n i n g a n o t h e r f a m i l y o f f i n i t e g r o u p s , the H a m i l t o n i a n g r o u p s . A n o n - a b e l i a n g r o u p G i s H a m i l t o n i a n i f e v e r y s u b g r o u p o f G i s n o r m a l i n G. H a m i l t o n i a n g r o u p s a r e c o m p l e t e l y c h a r a c t e r i z e d by the n e x t t h e o r e m . Theorem 1.38: ( C o x e t e r and M o s e r ) G i s a H a m i l t o n i a n g r o u p i f , a n d o n l y i f , G = Q X A j X A 2 , w h e r e Q i s t h e g r o u p o f q u a t e r n i o n s , A^ i s a b e l i a n o f odd o r d e r , and A 2 i s a group f o r w h i c h a 2 = 1 f o r a l l a i n A 2 - A 2 i s t h e n ( Z 2 ) n . • 29 W h i t e g i v e s c r e d i t to P. H i m e l w r i g h t , a p p a r e n t l y a s t u d e n t o f h i s a t W e s t e r n M i c h i g a n U n i v e r s i t y , f o r p r o v i n g the next few r e s u l t s a b o u t H a m i l t o n i a n groups i n an u n p u b l i s h e d Ph.D. t h e s i s . T h e o r e m 1 .39 : g e n u s c ( Q X ( Z 2 ) n ) = n ( 2 n ) + 1. T h e o r e m 1 .40 : g e n u s c ( Q X Z m X ( Z 2 ) n ) = ( m ) ( n ) ( 2 n ) + 1, m o d d . C o r o l l a r y 1.41: The g r o u p s Q X Z m X ( Z 2 ) 8 , m o d d , have C a y l e y genus a s y m p t o t i c to t h e i r o r d e r . O b v i o u s l y , i f G i s a H a m i l t o n i a n g r o u p , t h e n G = Q X Z m ( i ) X Z m ( 2 ) X X Z m ( k ) X ( Z 2 ) n , w h e r e e a c h m ( i ) i s o d d a n d m ( i - l ) d i v i d e s m ( i ) . Theorem 1.42: I f G i s the H a m i l t o n i a n group G = Q X Z m Q ) X Z m ( 2 ) X * " X Z m ( k ) X ^ 2 ^ ' t n e n g e n u s c ( G ) i s a s y m p t o t i c t o 2 n ( k + n - l ) ( m ( l ) m ( 2 ) . . . m ( k ) ) i f 1 < k < n - 1 . 5) T h e o r e t i c a l R e s u l t s L e t us now c o n s i d e r some t h e o r e t i c a l r e s u l t s on the C a y l e y genus o f a g r o u p . I m p o r t a n t bounds on genus^(G) c a n be o b t a i n e d by e x p l i c i t c o n s t r u c t i o n o f erabeddings o f C a y l e y c o l o r graphs f o r g r o u p s w i t h c e r t a i n k i n d s o f p r e s e n t a t i o n . The method o f c o n s t r u c t i o n uses s o - c a l l e d v o l t a g e g r a p h s and i s due t o G r o s s and A l p e r t ( [ 1 1 ] ) . The p a r t i c u l a r method we w i l l use h e r e i s known as the o r d i n a r y v o l t a g e g r a p h c o n s t r u c t i o n , as o p p o s e d t o the p e r m u t a t i o n v o l t a g e g r a p h c o n t r u c t i o n , and i t a p p e a r s i n a p a p e r ( [12] ) e n t i t l e d " G e n e r a t i n g A l l G r a p h C o v e r i n g s by P e r m u t a t i o n V o l t a g e A s s i g n m e n t s " by J . L . G r o s s a n d T . W. T u c k e r . 30 L e t K be a d i r e c t e d g r a p h and G a f i n i t e g r o u p . A mapping f : E ( K ) -> G f r o m t h e e d g e s e t o f K t o G i s s a i d t o be a v o l t a g e  a s s i g n m e n t i n G f o r K, and the p a i r ( K , f ) an o r d i n a r y v o l t a g e  g r a p h . A s e c o n d g r a p h , the d e r i v e d g r a p h w i l l now be c o n s t r u c t e d as f o l l o w s . T h e v e r t e x s e t V ( K f ) = V ( K ) X G a n d t h e e d g e s e t E ( K f ) = E ( K ) X G . A v e r t e x i s t h e n d e n o t e d as ( v , g ) a n d an e d g e as ( k , g ) . T h e n i f e d g e k o f K r u n s f r o m v e r t e x u t o v e r t e x v , l e t e d g e ( k , g ) o f r u n f r o m v e r t e x ( u , g ) t o v e r t e x ( v , g b ) , w h e r e b = f ( k ) . In t h i s s i t u a t i o n d e f i n e the p r o j e c t i o n p: -> K by the r u l e s p ( ( v , g ) ) = v f o r v e r t i c e s ( v , g ) and p ( ( k , g ) ) = k f o r e d g e s ( k , g ) . Theorem 1.43: The p r o j e c t i o n p: K f -> K a s s o c i a t e d w i t h an o r d i n a r y v o l t a g e g r a p h ( K , f ) w i t h v o l t a g e s i n g r o u p G h a v i n g | G | = n , i s a r e g u l a r n - s h e e t e d c o v e r i n g , and G a c t s on by c o v e r i n g t r a n s f o r m a t i o n s . We w i l l a l s o want the f o l l o w i n g c o n n e c t i o n w i t h C a y l e y g r a p h s . Theorem 1.44: The c o n n e c t e d g r a p h K i s a C a y l e y g r a p h f o r a group G i f , and o n l y i f , K = f o r some bouquet B o f c i r c l e s and some v o l t a g e a s s i g n m e n t f : E(B) -> G. p r o o f : I f G h a s g e n e r a t o r s g p . . . , g s , t a k e a b o u q u e t B o f s c i r c l e s k i , . . . , k g . L e t f ( k ^ ) = g - be a v o l t a g e a s s i g n m e n t . T h e n K = B ^ , t h e d e r i v e d g r a p h . # E v e r y C a y l e y g r a p h c l e a r l y a r i s e s i n t h i s way. The next t h e o r e m i s the r e s u l t we r e a l l y want f r o m the t h e o r y 3 1 o f v o l t a g e g r a p h s . I t shows t h a t the C a y l e y g r a p h f o r a g r o u p G h a v i n g a c e r t a i n p r e s e n t a t i o n embeds i n a s u r f a c e o f genus t h a t c a n be c a l c u l a t e d f r o m the p r e s e n t a t i o n o f G. The C a y l e y g r a p h i s r e a l i z e d as the d e r i v e d g r a p h a s s o c i a t e d w i t h a v o l t a g e a s s i g n m e n t i n G f o r a b o u q u e t o f c i r c l e s . Theorem 1.45: L e t G be a g r o u p , w i t h |G| = n , g e n e r a t e d by e l e m e n t s g j , . . . , g g where the o r d e r o f g^ i s d^ and the o r d e r o f g ^ g 2 . . . g s i s d g + j . Then the C a y l e y graph K f o r G c a n be embedded i n a s u r f a c e o f genus 1 + ( n / 2 ) ( s - 1 - ( 1 / d j ) - . . . - ( l / d s ) - ( l / d g + 1 ) ) . p r o o f : Suppose B i s a b o u q u e t o f s l o o p s embedded i n the 2-s p h e r e so t h a t no l o o p i n t e r s e c t s a n o t h e r and e a c h l o o p bounds a 2-c e l l . T h e r e a r e s+1 f a c e s f o r t h i s e m b e d d i n g . The C a y l e y g r a p h K f o r G i s i d e n t i c a l to the d e r i v e d graph B^ f o r the v o l t a g e a s s i g n m e n t f o f the s g e n e r a t o r s g p . . . , g g to the s l o o p s . The g r a p h K = B^ h a s n v e r t i c e s a n d ns e d g e s . , We c o n s t r u c t the e m b e d d i n g as f o l l o w s . In the g r a p h K = B^ b e g i n at any v e r t e x , say ( v , g ) . An edge runs f r o m t h i s v e r t e x to ( v 388^> a n o t h e r edge c o n t i n u e s f r o m h e r e and goes to ( v , g g ^ g ^ ) , and so on . A f t e r d^ edges have been t r a v e r s e d , the o r d e r o f g ^ , we a r e back a t v e r t e x ( v , g ) . A t t a c h a 2 - c e l l w i t h d^ edges to t h i s c i r c u i t . S t a r t a g a i n a t a v e r t e x (v ,h) not i n c l u d e d i n the l a s t c i r c u i t , and r e p e a t the p r o c e d u r e . G e n e r a t o r g^ w i l l c l e a r l y p r o d u c e n / d ^ 2 - c e l l s by t h i s c o n s t r u c t i o n , g e n e r a t o r g 2 w i l l p r o d u c e n / d 9 2 - c e l l s , and so on f o r the s g e n e r a t o r s . F i n a l l y , 32 l e t h = g^g2—.gg. U s i n g the same c o n s t r u c t i o n w i t h h t h a t we used w i t h the g e n e r a t o r s , we o b t a i n n / d g + ^ a d d i t i o n a l 2 - c e l l s . The p r o j e c t i o n p: -> B t a k e s the b o u n d a r y o f e a c h o f the 2 - c e l l s to one o f the l o o p s o f B , a l l n / d ^ c i r c u i t s p r o d u c e d by g e n e r a t o r g^ mapping to one l o o p , and so on . Now p can be e x t e n d e d t o map the i n t e r i o r o f any d - s i d e d 2 - c e l l o n t o the i n t e r i o r o f the l o o p u s i n g the c a n o n i c a l example z -> z d . Then p i s a homeo-m o r p h i s m on s m a l l enough n e i g h b o r h o o d s o f p o i n t s i n the 2 - c e l l s , and s i n c e B i s embedded i n a s u r f a c e , B^ must a l s o be embedded i n a s u r f a c e . T h e s u r f a c e s o c o n s t r u c t e d has ( n / d ^ ) + . . . + ( n / d s ) + ( n / d g + p f a c e s , n v e r t i c e s , and ns e d g e s . The genus f o r m u l a f o l l o w s . # I n f a c t , i f S i s the s u r f a c e i n w h i c h K = B^ i s embedded and B 2 • 2 i s embedded i n the 2 - s p h e r e S , the n a t u r a l p r o j e c t i o n f r o m S to S i s a b r a n c h e d c o v e r i n g w i t h s+1 b r a n c h p o i n t s h a v i n g o r d e r s d l » * " > d s + l * L e t G = < x , y , z : . . . > be a g r o u p . T h e g e n e r a t i n g s e t 9 7 9 < x , y , z > i s c a l l e d a ( p , q , r ) g e n e r a t i n g s e t i f x = y = z = 1 a n d the o r d e r s o f xy, y z , and xz a r e p , q , and r r e s p e c t i v e l y . W i t h o u t f u r t h e r r e l a t i o n s , G i s the so c a l l e d f u l l t r i a n g l e group T ( p , q , r ) , w h i c h we w i l l d e s c r i b e i n d e t a i l l a t e r . L e t G = < x , y : . . . > be a g r o u p . T h e n t h e g e n e r a t i n g s e t <x ,y } i s c a l l e d a ( p , q ; r ) ° g e n e r a t i n g s e t i f x a n d y h a v e o r d e r s p and q r e s p e c t i v e l y , and e i t h e r xy or x ^y has o r d e r r . W i t h o u t f u r t h e r r e l a t i o n s , G i s the g r o u p T ° ( p , q , r ) , the o r i e n t a t i o n 33 p r e s e r v i n g s u b g r o u p o f the f u l l t r i a n g l e group T ( p , q , r ) . L e t G = < x , y : . . . > be a g r o u p . T h e s e t { x , y } i s c a l l e d a ( p » P ; q ) C g e n e r a t i n g s e t i f t h e o r d e r s o f x, y , a n d y x y x ~ * a r e p , 2, and q r e s p e c t i v e l y . N o t i c e t h a t yxyx ^ i s the c o m m u t a t o r o f x and y. T u c k e r ( [31 ] ) uses t h i s n o t a t i o n , b e c a u s e , he s a y s , " a ( p , p ; q ) c g e n e r a t i n g s e t behaves i n many ways l i k e a ( p , p , q ) g e n e r a t i n g s e t . " T u c k e r ( [31 ] ) makes the f o l l o w i n g o b s e r v a t i o n s and d e f i n i t i o n s . A g r o u p G w i t h a ( p , q , r ) g e n e r a t i n g s e t i s an i m a g e o f T ( p , q , r ) . I f the image o f T ° ( p , q , r ) i s an i n d e x 2 s u b g r o u p o f G, the g e n e r a t i n g s e t i s s a i d t o be p r o p e r 1 y ( p , q , r ) . A g r o u p G w i t h a ( p , p ; q ) c g e n e r a t i n g s e t i s an image o f T ( p , p , q ) . I f the image o f T ° ( p , p , q ) i s an i n d e x 2 subgroup o f G , the g e n e r a t i n g s e t i s s a i d to be p r o p e r l y ( p , p ; q ) c . Theorem 1.46: ( T u c k e r , [31]) L e t the g e n e r a t i n g s e t { x , y , z > f o r a f i n i t e group G be p r o p e r l y ( p , q , r ) . Then g e n u s c ( G ) < 1 + ( | G | / 4 ) ( 1 - 1 / p - 1 / q - 1 / r ) . T h e o r e m 1 .47 : ( T u c k e r , [ 3 1 ] ) L e t < x , y } be a ( p , q ; r ) ° g e n e r a t i n g s e t f o r the f i n i t e group G. Then g e n u s c ( G ) < 1 + ( | G | / 2 ) ( 1 - 1 / p - 1 / q - 1 / r ) . Theorem 1.48: ( T u c k e r ) L e t the g e n e r a t i n g s e t { x , y } f o r a f i n i t e group G be p r o p e r l y ( p , p ; q ) c . T h e n g e n u s c ( G ) < 1 + ( | G | / 4 ) ( 1 - 2 / p - 1 / q ) . 6) A p p l i c a t i o n s t o S n and A R T h i s t h e o r y w i l l a l l o w us to c o n s i d e r a few r e s u l t s c o n c e r n i n g 34 the C a y l e y genus o f the g r o u p s S n and A n , the s y m m e t r i c and a l t e r n a t i n g g r o u p s on n l e t t e r s . These r e s u l t s a r e c i t e d by W h i t e , who used the e m b e d d i n g o f T h e o r e m 1.45, w h i c h he o b t a i n e d by a s l i g h t l y d i f f e r e n t method. T h e o r e m 1 .49 : g e n u s c ( S n ) < 1 + ( ( n - 2 ) ! / 4 ) ( n 2 - 3n + 1) , n>2. p r o o f : L e t x = ( l , 2 , . . . , n ) a n d y = ( 1 , 2 ) . I t i s w e l l k n o w n t h a t x and y g e n e r a t e S n . S i n c e x n = y 2 = ( x y ) n _ * = 1, t h e r e s u l t f o l l o w s from Theorem 1 .47 . # Theorem 1 .50 : (1) g e n u s c ( A n ) < 1 + ( ( n - 1 ) ( n - 3 ) ! / 8 ) ( n 2 - 4n + 2 ) ; n o d d , n>3 (2) g e n u s c ( A n ) < 1 + ( ( n - 2 ) ! / 8 ) ( n 2 - 3n + 1 ) ; n e v e n , n>4. p r o o f : I f n i s o d d , n >3 , l e t x = ( 1, 2 , . . . , n - 2 ) a n d y = ( l , n - l ) ( 2 , n ) . Then x and y g e n e r a t e A Q and x 1 1 - 2 = y 2 = ( x y ) n = 1. I f n i s e v e n , n>4, l e t x = ( 1 , 2 , . . . , n - l ) ( 2 , n ) a n d y = ( l , 2 ) ( 3 , n ) . T h e n x and y g e n e r a t e A n and x n = y 2 = ( x y ) n _ 1 = 1. The r e s u l t f o l l o w s from Theorem 1 . 4 7 . # These r e s u l t s o f W h i t e have been s h a r p e n e d c o n s i d e r a b l y by T u c k e r , as we w i l l soon s e e . T u c k e r has a c t u a l l y d e t e r m i n e d the e x a c t C a y l e y genus o f S n and has worked out a l o w e r bound f o r the C a y l e y g e n u s o f A n , i n b o t h c a s e s f o r n >^  168. We have a l r e a d y a l l u d e d t o the 1980 paper [ 2 8 ] , "The Number o f G r o u p s o f a G i v e n G e n u s , " by T u c k e r , b u t i t s f u l l s i g n i f i c a n c e i s r e v e a l e d i n the f o l l o w i n g t h e o r e m and i t s i m p l i c a t i o n s . T h e o r e m 1 .51 : I f 1 < g < | G | / 1 6 8 + 1, a n d i f K p ( G ) i s a C a y l e y c o l o r g r a p h f o r G w i t h an i r r e d u n d a n t g e n e r a t i n g s e t t h a t embeds i n 35 a s u r f a c e o f genus g , t h e n genus^CG) , the C a y l e y genus o f the group G , i s 0 o r 1. C o r o l l a r y 1 .52 : I f g e n u s c ( G ) > 1 t h e n g e n u s c ( G ) > | G | / 1 6 8 + 1. C o r o l l a r y 1 .53 : I f g e n u s c ( G ) > 1 t h e n | G | < 168 (g - 1) . C o r o l l a r y 1.54: The number o f groups o f a g i v e n C a y l e y g e n u s > 1 i s f i n i t e . T u c k e r a r r i v e s at t h i s t h e o r e m t h r o u g h e x c e e d i n g l y i n t r i c a t e and d i f f i c u l t m a n i p u l a t i o n s o f g e n e r a t o r s and r e l a t i o n s i n a l a b o r i o u s c a s e by c a s e d e r i v a t i o n . We w i l l not a t t e m p t to r e p r o d u c e i t h e r e . The most u s e f u l f o r m u l a t i o n o f T u c k e r ' s r e s u l t i s p r o b a b l y t h a t o f c o r o l l a r y 1.53, i n w h i c h the o r d e r o f the group G i s seen to be bounded above by 168(g - 1), g b e i n g the genus o f a s u r f a c e i n w h i c h some C a y l e y c o l o r g r a p h Kp(G) embeds. We w i l l say t h a t 168(g - 1) i s a q u a s i - H u r w i t z bound on | G | . The theorem i s an a n a l o g u e o f an i m p o r t a n t t h e o r e m o f H u r w i t z i n the n i n e t e e n t h c e n t u r y c o n c e r n i n g the s e c o n d d e f i n i t i o n o f group g e n u s , w h i c h w i l l be d e a l t w i t h i n the second h a l f o f t h i s p a p e r . But t h e r e i s a n o t h e r f a r r e a c h i n g c o n s e q u e n c e o f T u c k e r ' s t h e o r e m 1.51. The f o r m u l a t i o n i n the f i r s t c o r o l l a r y p l a c e s a l o w e r bound o f 1 + |G|/168 on the C a y l e y genus o f a group whose genus i s g r e a t e r than one . But T u c k e r has shown, as we have s e e n i n Theorem 1.46, t h a t a group w i t h a p r o p e r ( p , q , r ) g e n e r a t i n g s e t has an u p p e r bound on i t s C a y l e y genus o f 1 + (|G|/4)(1 - 1/p - 1 /q - 1 / r ) . T h e r e f o r e , we have the f o l l o w i n g r e s u l t f o r the c a s e when t h e s e two b o u n d s c o i n c i d e , f o r ( p , q , r ) = ( 2 , 3 , 7 ) . 36 Theorem 1.55: L e t the f i n i t e group G have a p r o p e r (2 ,3 ,7 ) g e n e r a t i n g s e t . Then g e n u s c ( G ) = |G|/168 + 1. M. D. E . Conder showed i n 1980 i n a paper e n t i t l e d " G e n e r a t o r s f o r A l t e r n a t i n g and S y m m e t r i c G r o u p s " ( [4 ] ) t h a t the s y m m e t r i c group S n i s an i m a g e o f T ( 2 , 3 , 7 ) f o r a l l n > 168. T u c k e r ( [ 2 9 ] ) o b s e r v e s t h a t the image o f the subgroup T ° ( 2 , 3 , 7 ) i s A n , a subgroup o f i n d e x 2 i n S n > Theorem 1.55 then g i v e s the most s p e c t a c u l a r o f C a y l e y genus f o r m u l a s . Theorem 1.56 ( T u c k e r [ 29 ] ) : g e n u s c ( S n ) = n ! /168 + 1, n>168. We w i l l d i s c u s s the p r o b l e m o f d e t e r m i n i n g the C a y l e y genus o f the a l t e r n a t i n g groups A R i n the c o n c l u s i o n s e c t i o n o f t h i s p a p e r . 7) The Q u a s i - H u r w i t z Bound T u c k e r has i m p r o v e d h i s q u a s i - H u r w i t z upper bound on the o r d e r o f a g r o u p i n a l o n g paper e n t i t l e d " A R e f i n e d H u r w i t z Theorem f o r I m b e d d i n g s o f I r r e d u n d a n t C a y l e y G r a p h s " [31 ] . Here he i s a b l e to show t h a t g r o u p s whose o r d e r i s near the 168(g - 1) u p p e r b o u n d , s p e c i f i c a l l y g r e a t e r than 24(g - 1), must have p r e s e n t a t i o n s as q u o t i e n t s o f c e r t a i n k i n d s o f t r i a n g l e g r o u p s . We w i l l now c o n s i d e r t h o s e r e s u l t s i n d e t a i l . Case I : K p ( G ) has d e g r e e 4 . L e t Kp(G) be an i r r e d u n d a n t C a y l e y g r a p h f o r group G h a v i n g d e g r e e 4 and genus^(G) = g > 1 embedded i n an o r i e n t a b l e s u r f a c e o f genus g. L e t X be the g e n e r a t i n g s e t o f the p r e s e n t a t i o n P. 37 Theorem 1.57: Suppose X has no i n v o l u t i o n s . Then i f | G | > 2 4 ( g - 1 ) , X i s e i t h e r ( 3 , r ; 2 ) ° , 7 < r < 11 , w i t h 2g - 2 = | G | ( r - 6 ) / 6 r , o r X i s ( 4 , 5 ; 2 ) ° w i t h 2g - 2 = | G | / 20 . Theorem 1.58: I f X has two or f o u r i n v o l u t i o n s t h e n | G | £ 2 4 ( g - 1) . N o t i c e t h a t i n a l l c a s e s h e r e , when the C a y l e y g r a p h has d e g r e e 4 and | G | > 24(g - 1), | G | i s a c t u a l l y l e s s t h a n o r e q u a l t o 84(g - 1), h a l f o f the q u a s i - H u r w i t z upper bound t h a t T u c k e r has a l r e a d y p r o v e d . Case I I : K p ( G ) has d e g r e e 3 . Now l e t K p ( G ) be an i r r e d u n d a n t C a y l e y g r a p h f o r a group G h a v i n g d e g r e e 3 and g e n u s ^ C G ) = g > 1 embedded i n an o r i e n t a b l e s u r f a c e o f genus g. L e t X be the g e n e r a t i n g s e t o f the p r e s e n t a t i o n . Theorem 1.59: Suppose X has one i n v o l u t i o n . I f | G | > 24(g - 1) t h e n one o f the f o l l o w i n g h o l d s : , a) X i s ( 5 , 2 ; 4 ) ° or ( 4 , 2 ; 5 ) ° and 2g - 2 = | G | / 2 0 b) X i s ( 5 , 5 ; 2 ) c and 2 g - 2 >_ | G | / 4 0 , w i t h e q u a l i t y i f , and o n l y i f , X i s p r o p e r l y ( 5 , 5 ; 2 ) c c ) X i s ( r , 2 ; 3 ) ° a n d 2g - 2 = | G | ( r - 6 ) / 6 r d) X i s ( 3 , 2 ; r ) ° and t h e r e i s a r e d u c e d r e l a t o r i n v o l v i n g b o t h g e n e r a t o r s o f l e n g t h l e s s t h a n 24 but none o f l e n g t h l e s s t h a n 14. T h e o r e m 1 .60 : I f | G | > 4 8 ( g - 1) t h e n X i s ( 3 , 2 ; 7 ) ° o r ( 7 , 2 ; 3 ) ° a n d 2g - 2 = | G | / 4 2 . 38 Theorem 1.61: Suppose X c o n s i s t s o f t h r e e i n v o l u t i o n s . I f | G | > 2 4 ( g - 1) t h e n X i s ( 2 , p , q ) , 3 < p < 5, o r X i s ( 3 , 3 , r ) w i t h r = 4 , 5 . I f | G | > 4 8 ( g - 1) t h e n X i s ( 2 , 4 , 5 ) and 2g - 2 = | G | / 4 0 , o r X i s ( 3 , 2 , r ) , 7 < r < 11 , w i t h 2 g - 2 = | G | ( r - 6 ) / 1 2 r . E q u a l i t y h o l d s i n t h i s t h e o r e m i f , and o n l y i f , X i s p r o p e r l y ( 2 , 4 , 5 ) o r ( 3 , 2 , r ) . In a l l o f t h e s e c a s e s , when the C a y l e y g r a p h has d e g r e e 3 and | G | > 24(g - 1), | G | i s l e s s t h a n or e q u a l to the q u a s i - H u r w i t z bound 168(g - 1), as e x p e c t e d . In a t t e m p t i n g to d e t e r m i n e t h e s e p a r t i a l p r e s e n t a t i o n s f o r g r o u p s " n e a r " the upper b o u n d , w i t h |G| > 24(g - 1), i t i s u n n e c e s s a r y to c o n s i d e r C a y l e y g r a p h s of d e g r e e g r e a t e r t h a n 4, b e c a u s e f o r t h e s e , the f u n d a m e n t a l i n e q u a l i t y o f Theorem 1.17 , w i t h n = 4 , g i v e s | G | _< 24(g - 1) . 39 C h a p t e r Two: The B u r n s i d e Genus o f a G r o u p . The F u n d a m e n t a l Group We now need to r e v i e w the i d e a o f the f u n d a m e n t a l group o f a t o p o l o g i c a l s p a c e . T h i s group i s an i m p o r t a n t a l g e b r a i c i n v a r i a n t o f the s p a c e . We make the f o l l o w i n g p r e l i m i n a r y d e f i n i t i o n s . A p a t h i n a t o p o l o g i c a l s p a c e Y i s a c o n t i n u o u s map f : [0 ,1] -> Y o f the c l o s e d u n i t i n t e r v a l i n t o Y. A c l o s e d p a t h i s a p a t h w i t h f ( 0 ) = f ( l ) = a i n Y. S u c h a p a t h i s s a i d t o be b a s e d a t a . T h e i n v e r s e o f a p a t h f i s f ^ (s ) = f ( l - s ) . I f f a n d g a r e p a t h s i n Y w i t h f ( l ) = g ( 0 ) , t h e n t h e p r o d u c t o f f a n d g ( f ( 2 s ) 0 < s < 0 .5 i s the p a t h f g = I l g ( 2 s - 1) 0 .5 < s < 1 The i d e n t i t y p a t h i s the c o n s t a n t p a t h i , d e f i n e d by i ( s ) = a f o r a l l 0 < s < 1. Two p a t h s f and g, w i t h f(0)=g(0) and f ( l ) = g ( l ) , a re h o m o t o p i c , w r i t t e n f ~ g, i f t h e r e e x i s t s a c o n t i n u o u s map h : [0,1] X [0,1] -> Y s u c h t h a t h ( s , 0 ) = f ( s ) a n d h ( s , l ) = g ( s ) f o r a l l s i n [ 0 , 1 ] . I f two p a t h s are h o m o t o p i c , one c a n be c o n t i n u o u s l y d e f o r m e d i n t o the o t h e r . T h i s i s an e q u i v a l e n c e r e l a t i o n among c l o s e d p a t h s b a s e d a t a p o i n t . We w i l l d e n o t e the e q u i v a l e n c e c l a s s c o n t a i n i n g t h e p a t h f as [ f ] . We d e f i n e a b i n a r y o p e r a t i o n among the e q u i v a l e n c e c l a s s e s o f p a t h s b a s e d a t a i n Y by l e t t i n g [ f ] [g ] = [h ] i f , a n d o n l y i f , f g ~ h . The e q u i v a l e n c e c l a s s e s , w i t h t h i s o p e r a t i o n , f o r m a g r o u p . 40 The i d e n t i t y i s [ i ] and [ f ] _ i i s [ f ] . The group i s c a l l e d the f u n d a m e n t a l group o f the s p a c e Y, w i t h b a s e p o i n t a , a n d i s d e n o t e d I T C Y j a ) . We w i l l now r e c o r d a few o f the b a s i c f a c t s about f u n d a m e n t a l g r o u p s and c o n s i d e r some o f the most i m p o r t a n t e x a m p l e s . Theorem 2.1: I f the t o p o l o g i c a l space Y i s p a t h w i s e c o n n e c t e d , the groups |T (Y ,a ) and f T ( Y , b ) a r e i s o m o r p h i c f o r any two p o i n t s a ,b i n Y . p r o o f : L e t f be a c l o s e d p a t h b a s e d a t a , a n d l e t g be a p a t h f r o m b to a. Then the map [ f ] -> [ g f g d e f i n e s an i s o m o r p h i s m b e t w e e n " f T ( Y , a ) and f T ( Y , b ) . Because o f t h i s theorem we w i l l u s u a l l y s u p p r e s s the base p o i n t i n o u r n o t a t i o n and s i m p l y w r i t e the f u n d a m e n t a l g r o u p o f Y as f T ( Y ) . The next two r e s u l t s a r e o f p r i m a r y i m p o r t a n c e . Theorem 2.2: I f t o p o l o g i c a l s p a c e s X, and Y a r e h o m e o m o r p h i c , t h e n IT(X) i s i s o m o r p h i c to f T ( Y ) . Theorem 2.3: The f u n d a m e n t a l group o f a p r o d u c t space Y X W, TT (Y X W), i s i s o m o r p h i c to the d i r e c t p r o d u c t o f f u n d a m e n t a l g r o u p s , IT(Y) X fT(W). L e t ' s l o o k a t a few e x a m p l e s o f f u n d a m e n t a l g r o u p s o f t o p o l o g i c a l s p a c e s , w i t h most a t t e n t i o n g i v e n to t h o s e o f the c l o s e d o r i e n t a b l e s u r f a c e s . Theorem 2.4: The f u n d a m e n t a l group o f the c i r c l e i s i n f i n i t e c y c l i c and hence i s o r m o r p h i c t o Z , the i n t e g e r s u n d e r a d d i t i o n . The p u n c t u r e d p l a n e has the same f u n d a m e n t a l g r o u p . 41 Theorem 2.5: The f u n d a m e n t a l group o f the s p h e r e , the compact o r i e n t a b l e s u r f a c e o f genus 0, i s the t r i v i a l g r o u p . In f a c t , f o r any s i m p l y c o n n e c t e d space e v e r y c l o s e d p a t h c a n be c o n t i n u o u s l y d e f o r m e d to a p o i n t , and so e v e r y s u c h space has a t r i v i a l f u n d a m e n t a l g r o u p . Theorem 2.6: The f u n d a m e n t a l group o f the t o r u s , the compact o r i e n t a b l e s u r f a c e o f genus 1, i s the f r e e a b e l i a n group on two g e n e r a t o r s and hence i s i s o m o r p h i c to the d i r e c t p r o d u c t Z X Z. T h i s f o l l o w s because the t o r u s i s the C a r t e s i a n p r o d u c t o f two c i r c l e s . Theorem 2.7: A c l o s e d , o r i e n t a b l e s u r f a c e S o f genus g, the c o n n e c t e d sum o f g t o r i , has a f u n d a m e n t a l g r o u p w i t h 2g g e n e r a t o r s and the f o l l o w i n g p r e s e n t a t i o n : -TT (S) = < a 1 , b 1 , . . . , a g , b g : a {b *b x ~ 1 - - - a g b g a g ~ l b g ~ 1 = 1 >• The f u n d a m e n t a l groups o f the s p h e r e and the t o r u s do have t h i s f o r m , v a c u o u s l y so i n the c a s e o f the s p h e r e . In the c a s e o f the t o r u s , the p r e s e n t a t i o n f T ( S ) = < a , b : a b a - 1 b _ 1 = 1 > c o n f i r m s t h a t the group i s f r e e a b e l i a n w i t h two g e n e r a t o r s , t h a t i s Z X Z . F o r a c l o s e d o r i e n t a b l e s u r f a c e o f g e n u s g > 1, t h e f u n d a m e n t a l group i s non - a b e l i a n . F u n d a m e n t a l P o l y g o n s A t o r u s i s o f t e n s e e n as a s q u a r e w i t h o p p o s i t e edges i d e n t i f i e d , as i n the d i a g r a m b e l o w . Edges t o be i d e n t i f i e d are marked w i t h the same v e c t o r . The two edges a and b c o r r e s p o n d to 42 the two g e n e r a t o r s o f the f u n d a m e n t a l group o f the t o r u s , and one c i r c u i t a r o u n d the square c o r r e s p o n d s t o the r e l a t o r aba ^b The s u r f a c e o f genus two , the c o n n e c t e d sum o f two t o r i , i s s e e n as the o c t a g o n shown b e l o w , w i t h the same m a r k i n g scheme s h o w i n g w h i c h p a i r s o f edges are to be i d e n t i f i e d . A g a i n the d i s t i n c t edges c o r r e s p o n d to the f o u r g e n e r a t o r s o f the f u n d a m e n t a l group o f t h i s s u r f a c e , and one c i r c u i t a r o u n d the o c t a g o n r e p r e s e n t s the d e f i n i n g r e l a t o r f o r t h a t g r o u p . In the same way, a c l o s e d , o r i e n t a b l e s u r f a c e o f genus g i s p i c t u r e d as a 4g - s i d e d p o l y g o n w i t h the edges i d e n t i f i e d i n p a i r s . T h e r e i s much more to be s a i d a b o u t t h i s way o f u n d e r s t a n d i n g a 43 s u r f a c e o f genus g. The way i n w h i c h the edges c o r r e s p o n d to g e n e r a t o r s o f the f u n d a m e n t a l g r o u p w i l l be c l a r i f i e d when we a r e a b l e to i n t e r p r e t the f u n d a m e n t a l group as a g r o u p o f m o t i o n s o f the c o m p l e x or h y p e r b o l i c p l a n e , m o t i o n s t h a t e f f e c t the i d e n t i f i c a t i o n o f e d g e s . But t h a t must w a i t . C o v e r i n g Spaces The t h e o r y o f c o v e r i n g s p a c e s , and o f b r a n c h e d c o v e r i n g s p a c e s , has a g r e a t d e a l to do w i t h the s e c o n d d e f i n i t i o n o f the genus o f a g r o u p . Some o f the s a l i e n t f a c t s about c o v e r i n g s p a c e s t h a t w i l l be r e l e v a n t to the work on group genus w i l l now be g i v e n . In t h i s p a p e r we are c o n c e r n e d o n l y w i t h the c l o s e d , o r i e n t a b l e s u r f a c e s o f genus g ^ 0, and w i t h the c o m p l e x and h y p e r b o l i c p l a n e s . T h e r e f o r e , we w i l l not a t t e m p t to p r e s e n t c o v e r i n g space t h e o r y i n i t s f u l l g e n e r a l i t y , but w i l l g i v e the r e s u l t s f o r t h e s e s u r f a c e s . L e t S be a s u r f a c e . A c o v e r i n g s p a c e o f S i s a p a i r ( S c , p ) c o n s i s t i n g o f a s u r f a c e S £ t o g e t h e r w i t h a c o n t i n u o u s map p: S c -> S, c a l l e d the p r o j e c t i o n map o r c o v e r i n g map, s u c h t h a t f o r e a c h p o i n t a i n S t h e r e e x i s t s an open c o n n e c t e d n e i g h b o r h o o d N o f a s u c h t h a t e a c h component o f p ^(N) i s mapped h o m e o m o r p h i c a l l y o n t o N by p . We w i l l s o m e t i m e s c a l l S c the c o v e r i n g s p a c e and S the base s p a c e . We c o n s i d e r o n l y the c a s e o f a c o n n e c t e d c o v e r i n g space i n a l l t h a t f o l l o w s . I f V i s a c o n n e c t e d component o f p ^ ( N ) , t h e n p ^ ( a ) f | V c o n s i s t s a s i n g l e p o i n t f o r any a i n N. T h e r e f o r e , |p *(a)| = n i s the same 4 4 f o r any a i n N and f o r any open n e i g h b o r h o o d N o f a , and e q u a l s the number o f c o n n e c t e d components o f p ^(N). We w i l l say t h a t S c i s an n - s h e e t e d c o v e r i n g o f S. L e t ( S c , p ) be a c o v e r i n g s p a c e o f S. A c o v e r i n g t r a n s f o r m a t i o n o f ( S c , p ) i s a h o m e o m o r p h i s m f : S c -> S c s u c h t h a t p ( b ) = p ( f ( b ) ) f o r a l l b i n S.,. In o t h e r w o r d s , f maps the s e t p - ^ ( a ) to i t s e l f f o r a 11 a i n S. C o v e r i n g t r a n s f o r m a t i o n s a r e o f t e n c a l l e d a u t o m o r p h i s m s . A l l a u t o m o r p h i s m s o f a c o v e r i n g s p a c e f o r m a group under c o m p o s i t i o n t h a t w i l l be d e n o t e d as A ( S c ) , where the map p i s u n d e r s t o o d . We now want to i n t r o d u c e the c o n v e n i e n t and v e r y g e n e r a l i d e a o f a group a c t i n g on a s e t . A g r o u p o f homeomorphisms o f a t o p o l o g i c a l s p a c e i s an e x a m p l e , and p r o b a b l y the one m o t i v a t i n g the next d e f i n i t i o n . A g r o u p G ac t s on a s e t S i f t h e r e i s a m a p p i n g m: G X S -> S w i t h the f o l l o w i n g p r o p e r t i e s . We use the n o t a t i o n m(g,s) = g(s) f o r c o n v e n i e n c e , and r e q u i r e : (1) g h ( s ) = g ( h ( s ) ) f o r a l l g , h i n G ; a n d (2) g ( g _ 1 ( s ) ) = g _ 1 ( g ( s ) ) = s f o r a l l g i n G . A group G a c t s t r a n s i t i v e l y on S i f f o r e a c h p a i r o f p o i n t s a, b i n S, t h e r e i s a g i n G s u c h t h a t g ( a ) = b . L e t G be a group o f h o m e o m o r p h i s m s o f a t o p o l o g i c a l space X. Then G i s s a i d to a c t p r o p e r l y d i s c o n t i n u o u s l y on X i f e a c h p o i n t x has a n e i g h b o r h o o d N such t h a t g j ( N ) f \ g 2 ( N ) i s empty f o r e a c h d i s t i n c t p a i r g p g 2 i n G . 45 Theorem 2.8: I f S c i s a c o v e r i n g space o f S, t h e n the g r o u p A ( S c ) o f c o v e r i n g t r a n s f o r m a t i o n s a c t s p r o p e r l y d i s c o n t i n u o u s l y on S c . Hence , i f g i s a n o n t r i v i a l a u t o m o r p h i s m , i t has no f i x e d p o i n t s . The c r i t e r i o n o f p r o p e r d i s c o n t i n u i t y w i l l a l l o w us to o b t a i n a c o n v e r s e o f the l a s t t h e o r e m . R e c a l l f i r s t t h a t i f G i s a group o f homeomorphisms o f a t o p o l o g i c a l space X, the o r b i t o f a p o i n t a i n X i s the s e t d e n o t e d as Ga a n d d e f i n e d as Ga = { g ( a ) : g i n G }. T h e s e t o f o r b i t s f o r m the q u o t i e n t space X / G , and the p r o j e c t i o n map p : X -> X / G i s g i v e n by p(a) = Ga . The space X / G has a t o p o l o g y d e f i n e d by the r u l e t h a t a s e t A i n X / G i s open i f , and o n l y i f , p * (A) i s open i n X . ( S c , p ) i s a r e g u l a r c o v e r i n g space of S i f the group A ( S c ) o f c o v e r i n g t r a n s f o r m a t i o n s a c t s t r a n s i t i v e l y on the s e t p ^(a) , s o m e t i m e s c a l l e d the f i b e r o v e r a, f o r each a i n S. T h i s i m p o r t a n t c l a s s o f c o v e r i n g spaces i s the one t h a t r e a l l y c o n c e r n s us i n t h i s p a p e r . F o r such s p a c e s , the e l e m e n t s o f the f i b e r a r e i n t e r c h a n g e a b l e , e q u i v a l e n t , or " l o o k a l i k e . " T h e i r s i g n i f i c a n c e comes from the next t h e o r e m . Theorem 2.9: I f ( S c , p ) i s a r e g u l a r c o v e r i n g s p a c e o f S, w i t h a u t o m o r p h i s m group A ( S C ) , the q u o t i e n t space S c / A ( S c ) i s h o m e o m o r p h i c to S, w i t h the homeomorphism p a i r i n g a p o i n t i n S and i t s o r b i t i n S c « Now f o r the c o n v e r s e o f T h e o r e m 2.9 a b o v e . T h i s r e s u l t shows t h a t g r o u p s o f p r o p e r l y d i s c o n t i n u o u s homeomorphisms have the 46 " r i g h t " p r o p e r t i e s . Theorem 2.10: L e t X be a t o p o l o g i c a l space t h a t i s c o n n e c t e d and l o c a l l y p a t h w i s e c o n n e c t e d , and suppose t h a t G i s a p r o p e r l y d i s c o n t i n u o u s group o f homeomorphisms o f X. Then (X,p) i s a c o v e r i n g s p a c e o f X / G , where p i s the n a t u r a l p r o j e c t i o n p : X -> X / G ; a n d , i n a d d i t i o n , A ( X , p ) = G . F u n d a m e n t a l G r o u p s and C o v e r i n g Spaces Suppose ( S c , p ) i s a c o v e r i n g s u r f a c e o f S w i t h p r o j e c t i o n p. T h e r e i s an i n d u c e d map p * : TT(S c,b) -> T^CSjpCb)) between f u n d a m e n t a l g r o u p s of the c o v e r and the base d e f i n e d by p ^ X t f ] ) = [ p o f ] . T h i s m a p p i n g can i n f a c t be shown to be a group homomorphism. ' T h e o r e m 2 . 1 1 : L e t ( S c , p ) be a c o v e r i n g s p a c e o f S a n d l e t b be a p o i n t i n S c w i t h p(b) = a. Then the homomorphism p ^ : 'fP ( S c , b ) -> '(7~(S,a) i s a c t u a l l y a monomorphisra . T h u s , we may t h i n k o f the f u n d a m e n t a l g r o u p o f the c o v e r i n g s u r f a c e as a subgroup o f the f u n d a m e n t a l group o f the base . When S c i s s i m p l y c o n n e c t e d and f T ( S c , b ) i s t r i v i a l , we say t h a t S c i s a u n i v e r s a 1 c o v e r i n g s p a c e . A s e c o n d , e q u i v a l e n t d e f i n i t i o n o f a r e g u l a r c o v e r i n g i n t e r m s o f the two f u n d a m e n t a l g r o u p s c a n now be g i v e n . A c o v e r i n g space ( S c , b ) i s c a l l e d r e g u l a r i f the group p*(7T (S c ,b)) i s a n o r m a l s u b g r o u p o f fT(S,a). The number o f s h e e t s o f the c o v e r i n g i s r e l a t e d t o the 47 f u n d a m e n t a l groups f T ( S c , b ) and ^ ( S j a ) i n the f o l l o w i n g r e s u l t . Theorem 2.12: L e t ( S c , b ) be a r e g u l a r c o v e r i n g space o f S w i t h p(b) = a. Then the s e t p~^(a) i s i n b i j e c t i v e c o r r e s p o n d -ence w i t h the e l e m e n t s o f the g r o u p f r ( S ) a ) / p ^ ( y T r ( S c , b ) . The number o f s h e e t s o f the c o v e r i n g i s t h e n e q u a l to the o r d e r o f t h i s q u o t i e n t g r o u p . Theorem 2.13: I f ( S c , b ) i s a r e g u l a r c o v e r i n g s u r f a c e o f S, t h e n A ( S c , b ) i s i s o m o r p h i c to the f a c t o r g r o u p 7r(S,a)/p*(TT(S,,b)) f o r any a i n S a n d any b i n p ^ ( a ) . Theorem 2.14: I f ( S c , b ) i s a u n i v e r s a l c o v e r i n g space o f S, t h e n A ( S c , b ) i s i s o m o r p h i c to 4 r ( S , a ) , and the number o f s h e e t s o f the c o v e r i n g space i s e q u a l to the o r d e r o f ^T(S,a). E v e n i f a c o v e r i n g p : S c -> S i s not r e g u l a r , the a u t o m o r p h i s m group A ( S c , p ) i s r e l a t e d i n a s i m p l e way to the f u n d a m e n t a l g r o u p s o f S c and S. R e c a l l t h a t i f H i s a s u b g r o u p o f G , t h e n t h e n o r m a l i z e r o f H i n G , N [ H ] , i s the l a r g e s t s u b g r o u p o f G i n w h i c h H i s n o r m a l . Theorem 2.15: The a u t o m o r p h i s m group A ( S c , b ) i s i s o m o r p h i c to the q u o t i e n t group N [ p ^ ( / J ? ( S c , b ) ) ] / p ^ C 1f"(S , b ) ) , where the group N [ p ^ ( / l f " ( S c ) b ) ) ] i s the n o r m a l i z e r i n the g r o u p -fPCSja). e x a m p l e : L e t C be the c o m p l e x p l a n e . The t r a n s l a t i o n s t : z -> z + 1 a n d u : z -> z + i g e n e r a t e a g r o u p t h a t a c t s p r o p e r l y d i s c o n t i n u o u s l y on C and i s i s o m o r p h i c to Z X Z. The q u o t i e n t s p a c e C / ( Z X Z) i s the t o r u s , w i t h f u n d a m e n t a l group Z X Z . In the d i a g r a m b e l o w we see t h a t the i d e n t i f i c a t i o n o f o p p o s i t e edges i n 48 the f u n d a m e n t a l s q u a r e f o r the t o r u s i s e f f e c t e d by the g e n e r a t o r s o f i t s f u n d a m e n t a l g r o u p . F u r t h e r m o r e p: C -> C / ( Z X Z) i s a r e g u l a r c o v e r i n g o f the t o r u s , and i n f a c t a u n i v e r s a l c o v e r i n g . 2-<l I < * I * 1 * 1 "] 4 | 1 t ( | r B r a n c h e d C o v e r i n g Spaces L e t us suppose t h a t a group G a c t s o n - a t o p o l o g i c a l space S as a group o f h o m e o m o r p h i s m s . S w i l l be a s u r f a c e i n t h i s p a p e r . I f G i s not p r o p e r l y d i s c o n t i n u o u s b u t has a manageable s e t o f f i x e d p o i n t s , S / G w i l l s t i l l be a s u r f a c e and p: S -> S / G w i l l be a b r a n c h e d c o v e r i n g . In t h i s s i t u a t i o n the f o l l o w i n g t e r m i n o l o g y a p p l i e s . T h e s t a b i l i z e r o f a p o i n t s i n S i s t h e g r o u p G g = { g i n G : g ( s ) = s }, a n d t h e s e t o f f i x e d p o i n t s i s F i x ( G ) = { s i n S : G g i s n o t t r i v i a l }. T h e a c t i o n o f G on S i s f r e e i f F i x ( G ) i s e m p t y . I f G a c t s on a s u r f a c e S i n s u c h a way t h a t F i x ( G ) i s n o n - e m p t y 4 9 but d i s c r e t e , and |G S I i s f i n i t e f o r a l l s, t h e n S / G i s a s u r f a c e . I f p : S -> S / G i s the n a t u r a l p r o j e c t i o n , t h e n (S ,p) i s c a l l e d a b r a n c h e d c o v e r i n g o f S / G , and p ( F i x ( g ) ) c o n s i s t s o f the b r a n c h  p o i n t s . Under the same c o n d i t i o n s , i f a p o i n t i n S / G i s not a b r a n c h p o i n t , t h e n i t has an open n e i g h b o r h o o d N s u c h t h a t each c o n n e c t e d component o f p~^(N) i s mapped h o m e o m o r p h i c a l l y o n t o N by p, as i n the c a s e o f an o r d i n a r y c o v e r i n g . I f a p o i n t b i s a b r a n c h p o i n t t h e n t h e r e i s no s u c h n e i g h b o r h o o d , and the m a p p i n g p i s l o c a l l y many - t o - o n e . In f a c t , i f p ( s ) = b , i t i s | G g | - t o - one i n a n e i g h b o r h o o d o f s. The o r d e r o f the s t a b i l i z e r , 1GsI, i s the same f o r e a c h s i n p~^(b) and i s c a l l e d the o r d e r o f the b r a n c h p o i n t b , w h i c h we w i l l d e n o t e as m^. I t i s t h e n e v i d e n t t h a t the c a r d i n a l i t y o f the f i b e r p _ 1 ( b ) i s | G | / m b . The R i e m a n n - H u r w i t z F o r m u l a Suppose t h a t a f i n i t e group G a c t s on the s u r f a c e S and t h a t (S ,p ) i s a b r a n c h e d c o v e r i n g o f the s u r f a c e S / G , h a v i n g b r a n c h p o i n t s b ^ , . . . , b k w i t h o r d e r s m ^ , . . . , ! ! ^ . T h e n t h e g e n u s o f S c a n be d e t e r m i n e d f r o m the genus o f S / G , the o r d e r s o f the b r a n c h p o i n t s , and the o r d e r o f G by the R i e m a n n - H u r w i t z f o r m u l a . Suppose a t r i a n g u l a t i o n o f S / G has V v e r t i c e s , E e d g e s , and F f a c e s , w i t h e a c h b r a n c h p o i n t a v e r t e x . We l i f t the t r i a n g u l a t i o n t o S. The edges and f a c e s a r e t h e n r e p l i c a t e d | G | t i m e s , once f o r e a c h 50 s h e e t , as a r e the v e r t i c e s not a t b r a n c h p o i n t s . A t each b r a n c h p o i n t b^ o f o r d e r m^, l i f t i n g the t r i a n g u l a t i o n t o S r e p l i c a t e s t h e v e r t e x o n l y |G1/m^ t i m e s . L e t the s u r f a c e S / G have genus j and E u l e r c h a r a c t e r i s t i c 2 - 2 j , and l e t S have genus g and E u l e r c h a r a c t e r i s t i c 2 - 2g. Then (2 - 2g) e q u a l s | G | t i m e s (2 - 2 j ) minus the sum o f the o v e r c o u n t s ( | G | - | G | / m ^ ) at each b r a n c h p o i n t . T h i s g i v e s the c l a s s i c a l Riemann - H u r w i t z f o r m u l a : 2 - 2g = |G|(2 - 2j - ( ( l - l / r a 1 ) + . . . + ( l - l / m k ) ) ) . Riemann S u r f a c e s The t h e o r y o f b r a n c h e d c o v e r i n g s a r i s e s i n the c o n s t r u c t i o n o f R i e m a n n s u r f a c e s f o r c e r t a i n m u l t i p l e - v a l u e d f u n c t i o n s i n c o m p l e x a n a l y s i s . F o r e x a m p l e , c o n s i d e r the e q u a t i o n : 9 w = (z - a)(z - b ) ( z - c ) w i t h a , b , and c d i s t i n c t c o m p l e x numbers . S i n c e the c o r r e s p o n d e n c e b e t w e e n z a n d w i s 1 - 2, e x c e p t f o r z = a , b , c , o r o O ; t h e c o m p l e x s p h e r e S i s r e p l a c e d by a two - s h e e t e d s u r f a c e t o e n l a r g e the d o m a i n f o r z and p r o d u c e the d e s i r e d 1 - 1 c o r r e s p o n d e n c e . T h i s two - s h e e t e d s u r f a c e i s the R i e m a n n s u r f a c e f o r t h e e q u a t i o n . T h i s e q u a t i o n i n w and z i s s a i d t o have a b r a n c h p o i n t a t z = 51 ZQ i f , (1) w h a s a s i n g l e v a l u e WQ a t Z Q , a n d (2) as z d e s c r i b e s a s m a l l c i r c l e a r o u n d ZQ o n c e , w does not d e s c r i b e a c l o s e d c u r v e a r o u n d WQ. In o u r e x a m p l e , the c a n d i d a t e s f o r b r a n c h p o i n t s a r e a, b , c, and i n f i n i t y . U s i n g the a b b r e v i a t i o n c i s ( 9 ) = c o s ( 0 ) + i s i n ( 0 ) , we now l e t : z - a = r j c i s ( 0 ^ ) z - b = r 2 c i s ( 9 2 ) z - c = r 3 c i s ( 0 3 ) . T h e n , on the two s h e e t s , we h a v e : w l = Y / r i r 2 r 3 c i s ^ 6 l + e 2 + e 3 ^ 2 ^ w 2 = V r l r 2 r 3 c i s ^ e l + 9 2 + 9 3 ^ 2 + ^ ^ (1) L e t z d e s c r i b e a s m a l l c i r c l e a b o u t a ( o r b , o r c ) . T h e n 6 j -> © j + 2 ( ' f T ) , 02 - > 02> ^3 © 3 ; and h e n c e w c h a n g e s s h e e t s . T h e r e f o r e a, b , and c are b r a n c h p o i n t s . (2) L e t z d e s c r i b e a c i r c l e a b o u t a a n d b ( o r a and c , o r b a n d c ) . T h e n 0 1 -> 0 | + 2("7? ) , 0 2 -> 0 £ + 2(7? ), 0 3 -> © 3 . H e n c e w does not change s h e e t s , o r the c i r c l e c u t s b r a n c h l i n e s an even number of t i m e s . (3) L e t z d e s c r i b e a c i r c l e a b o u t a , b , a n d c . T h e n ©x -> Sl + 2 ( f T ) , © 2 -> 0 2 + 2(-7T), 0 3 -> 0 3 + 2 ( 1 ? ) . H e n c e w 52 changes s h e e t s . Thus i n f i n i t y i s a b r a n c h p o i n t . One c a n c o n s t r u c t the s h e e t s o f the c o v e r i n g by a d d i n g b r a n c h l i n e s . A s t a n d a r d s c h e m a t i c d i a g r a m o f the R i e m a n n s u r f a c e f o r o u r e q u a t i o n i s shown b e l o w . T h i s R i e m a n n s u r f a c e R i s thus R = { ( z , w ) : w 2 = (z - a ) ( z - b ) ( z - c ) }. T o g e t h e r w i t h p r o j e c t i o n p: R -> S d e f i n e d by p(z ,w) = z , the p a i r (R,p) i s a b r a n c h e d c o v e r i n g o f S as we have d e f i n e d i t i n the g e n e r a l t h e o r y . The o r d e r o f e a c h b r a n c h p o i n t i s 2, and the group G a c t i n g on R i s Z 2 , whose a c t i o n i n t e r c h a n g e s the s h e e t s . By c o n s t r u c t i o n t h i s R i e m a n n s u r f a c e i s c o m p a c t , o r i e n t a b l e , and w i t h o u t b o u n d a r y . I t i s w e l l known i n t o p o l o g y t h a t s u c h a s u r f a c e must be a s p h e r e w i t h g h a n d l e s . Thus the genus o f R c a n be d e t e r m i n e d f r o m the Riemann - H u r w i t z f o r m u l a . A c a l c u l a t i o n r e v e a l s t h a t the genus o f R i s 1, and so R i s a t o r u s . A b s t r a c t Riemann S u r f a c e s The Riemann s u r f a c e s t h a t a r i s e i n c o m p l e x a n a l y s i s m o t i v a t e 53 the d e f i n i t i o n o f an a b s t r a c t Riemann s u r f a c e . The key f e a t u r e o f s u c h a s u r f a c e i s t h a t i t c a r r i e s a c o m p l e x c o o r d i n a t e s y s t e m t h a t makes i s l o c a l l y i d e n t i c a l to open s u b s e t s o f the complex p l a n e . A Riemann s u r f a c e i s a c o n n e c t e d H a u s d o r f f t o p o l o g i c a l s p a c e R t o g e t h e r w i t h a f a m i l y { 0^ } o f open s u b s e t s o f R and maps * f i : ° i _ > C * s u c h t h a t : 1) U 0i = R 2) each f^ i s a homeomorphism o f 0^ o n t o an open s u b s e t o f the c o m p l e x p l a n e C 3) i f 0— = O ^ D 0j i s not empty then the c o m p o s i t i o n f ^ f j - 1 ) : f j ( ° i j ) _ > f i ( ° i j ) i s a n a n a l y t i c map b e t w e e n t h e s e s u b s e t s o f the c o m p l e x p l a n e . Each p a i r (0^, f^) i s c a l l e d a c h a r t and the e n t i r e c o l l e c t i o n o f c h a r t s i s c a l l e d an a t l a s f o r the space R. C l e a r l y , i t i s the a t l a s t h a t endows R w i t h l o c a l c o m p l e x c o o r d i n a t e s y s t e m s . e x a m p l e : A s i m p l e e x a m p l e o f a Riemann s u r f a c e w i t h i t s a t l a s i s t h e s p h e r e S 2 = C U {<*>} w i t h t h e t w o c h a r t s ( C . f j ) , w h e r e f j ( z ) = z ; and (S - { 0 } , f 2 ) » where £2^^ = l / z * T n e s p h e r e i s o b v i o u s l y c o v e r e d by the open s e t s C and S -{0} and t h e c o m p o s i t i o n s f ^ ( f 2 ) and f j ^ r 1 ) a re a n a l y t i c f u n c t i o n s d e f i n e d on C - {0} . We w i l l need the f o l l o w i n g i m p o r t a n t c l a s s i f i c a t i o n t h e o r e m f o r s i m p l y c o n n e c t e d Riemann s u r f a c e s . T h i s i s c a l l e d the U n i f o r m i z a t i o n T h e o r e m . Theorem 2.16: L e t R be a s i m p l y c o n n e c t e d Riemann s u r f a c e . T h e n R i s c o n f o r m a l l y e q u i v a l e n t to e x a c t l y one o f the f o l l o w i n g t h r e e : 54 (1) t h e s p h e r e C U { oO } , (2) the c o m p l e x p l a n e C , o r (3) t h e u n i t d i s k D = < z i n C : | z | < 1 }. T o p o l o g i c a l l y , the compact Riemann s u r f a c e s t h a t a re not s i m p l y c o n n e c t e d a r e s p h e r e s w i t h h a n d l e s , our f a m i l i a r c l o s e d , o r i e n t a b l e s u r f a c e s o f f i n i t e genus . We w i l l have more to say about t h i s . A u t o m o r p h i s m s o f a_ Riemann S u r f a c e I t i s now p o s s i b l e to d e f i n e a n a l y t i c o r c o n f o r m a l maps b e t w e e n R i e m a n n s u r f a c e s . L e t R be a R i e m a n n s u r f a c e w i t h an a t l a s { ( O ^ f p } and S a Riemann s u r f a c e w i t h an a t l a s { ( O j ' . f j 1 ) } . A c o n t i n u o u s map h : R -> S i s a n a l y t i c i f each map f j 1 h f • ^ i s a n a l y t i c as a c o m p l e x f u n c t i o n on i t s d o m a i n . An a n a l y t i c f u n c t i o n h : R -> R w i t h an i n v e r s e i s c a l l e d a c o n f o r m a l a u t o m o r p h i s m o f the Riemann s u r f a c e R. C o n f o r m a l , o f c o u r s e , means a n g l e - p r e s e r v i n g , and i s synonymous w i t h 1-1 and a n a l y t i c . C o n f o r m a l mappings a r e o r i e n t a t i o n p r e s e r v i n g as w e l l . In a d d i t i o n , a c o n f o r m a l a u t o m o r p h i s m o f a Riemann s u r f a c e i s s o m e t i m e s c a l l e d a symmetry o f t h e s u r f a c e . The H u r w i t z Bound The s e t o f a l l c o n f o r m a l a u t o m o r p h i s m s o f a compact R i e m a n n s u r f a c e R c l e a r l y f o r m s a g r o u p under c o m p o s i t i o n o f f u n c t i o n s . In 1878 S c h w a r z p r o v e d t h a t s u c h a g r o u p must be f i n i t e i f the genus o f R i s g r e a t e r t h a n 1, and i n 1893 H u r w i t z e s t a b l i s h e d the u p p e r bound 55 84(g - 1) on the o r d e r o f any s u c h g r o u p . T h i s upper bound i s the b e s t p o s s i b l e , as g r o u p s G w i t h | G | = 84(g - 1) c a n be found f o r many s u r f a c e s . A group G w h i c h a t t a i n s the 84(g - 1) upper bound i s c a l l e d a H u r w i t z g r o u p . Theorem 2.17: Suppose the f i n i t e group G a c t s as a group o f c o n f o r m a l a u t o m o r p h i s m s on the compact Riemann s u r f a c e R o f genus g > 1. T h e n | G | < 8 4 ( g - 1) . p r o o f : R / G i s a s u r f a c e o f some genus p, and by the Riemann -H u r w i t z f o r m u l a we have 2g - 2 = | G | ( 2 p - 2 + ( 1 - l / m ( l ) ) + ( 1 - l / m ( 2 ) ) +...+ ( 1 - l / m ( t ) ) ) , where m ( l ) , m ( 2 ) , . . . , m(t ) a r e the o r d e r s o f the b r a n c h p o i n t s . Now g i s f i x e d , so the maximum p o s s i b l e v a l u e f o r |G| c o r r e s p o n d s t o the minimum p o s i t i v e v a l u e f o r the f a c t o r (2p - 2 + ( 1 - l / m ( D ) + ( 1 - l / m ( 2 ) ) +...+ ( 1 - l / m ( t ) ) ) . C a l l t h i s q u a n t i t y A f o r c o n v e n i e n c e . C l e a r l y p = 0 w i l l be b e s t f o r the minimum v a l u e o f A. We now c o n s i d e r the s m a l l e s t p o s i t i v e v a l u e of, A as a f u n c t i o n o f t , the number o f b r a n c h p o i n t s . We w i l l a l w a y s a s s u m e t h e o r d e r i n g : m ( l ) <. m(2) <. . . . < m ( t ) . Case I : I f t > 4 , the q u a n t i t y A >^  1 /2 , s i n c e each m ( i ) >^  2. C a s e I I : I f t = 4 , (m( 1 ) , m ( 2 ) , m ( 3 ) , m ( 4 ) ) = ( 2 , 2 , 2 , 2 ) i s i n a d m i s s i b l e , because t h e n A = 0. Hence the l e a s t p o s i t i v e v a l u e f o r A i s g i v e n by ( 2 , 2 , 2 , 3 ) , a n d t h a t v a l u e i s A = 1 / 6 . Case I I I : I f t = 3, t h e minimum p o s i t i v e v a l u e f o r A c o r r e s p o n d s to a maximum f o r l / m ( l ) + l / m ( 2 ) + l / m ( 3 ) s u b j e c t to t h e c o n d i t i o n : l / m ( l ) + l / m ( 2 ) + l / m ( 3 ) < 1. A l l t r i p l e s o f the f o r m (2 ,2 ,m) a r e t h e r e f o r e i n a d m i s s i b l e , as 56 the sum o f the r e c i p r o c a l s exceeds 1. T r i p l e s o f the form (2,3 ,m) a r e a d m i s s i b l e i f m>7, and the b e s t o f t h e s e , ( 2 , 3 , 7 ) , m a k e s A = 1 / 4 2 . T r i p l e s o f the form (2,4 ,m) a r e a c c e p t a b l e i f m>5, and the b e s t o f t h e s e , ( 2 , 4 , 5 ) , makes A = 1/20. A l l (2 ,n ,m) t r i p l e s w i t h n>5 a r e s e e n to g i v e a l a r g e r v a l u e f o r A . T r i p l e s o f the form (3,3 ,m) a r e a c c e p t a b l e i f m>4, and the b e s t o f t h e s e , ( 3 , 3 , 4 ) , makes A = 1/12. A l l (3 ,n ,m) t r i p l e s w i t h n>4 a r e s e e n to g i v e a l a r g e r v a l u e f o r A . A l l t r i p l e s o f the f o r m (p ,m,n) w i t h p>4 a r e c l e a r l y i n f e r i o r t o ( 3 , 3 , 4 ) . T h e r e f o r e , the l e a s t v a l u e f o r A, w i t h t h r e e b r a n c h p o i n t s , i s A = 1 / 4 2 . Case I V : I f t < 2, A < 0. T h i s case i s r u l e d o u t . C o m b i n i n g t h e s e r e s u l t s , we g e t t h a t t h e l e a s t p o s i t i v e v a l u e f o r A i s A = 1 / 4 2 f o r t h e ( 2 , 3 , 7 ) t r i p l e . T h i s v a l u e o f A g i v e s 84 (g - 1) as the maximum v a l u e f o r | G | . # The H u r w i t z 84(g - 1) bound does not a p p l y f o r a s u r f a c e o f genus 0 or 1. F o r e i t h e r a s p h e r e o r a t o r u s , a c y c l i c group o f a r b i t r a r y o r d e r , r e p r e s e n t i n g r o t a t i o n s about the o b v i o u s a x i s , w i l l be a g r o u p o f c o n f o r m a l a u t o m o r p h i s m s f o r some a n a l y t i c s t r u c t u r e . 5 7 The B u r n s i d e Genus o f a_ G r o u p One c o n s e q u e n c e o f the H u r w i t z bound i s t h a t f o r a compact R i e m a n n s u r f a c e o f genus g ^ 2, the number o f g r o u p s w h i c h a c t c o n f o r m a l l y on the s u r f a c e i s f i n i t e , up to i s o m o r p h i s m . We ask the f o l l o w i n g n a t u r a l q u e s t i o n . G i v e n a f i n i t e g r o u p G, what i s the l e a s t v a l u e o f g s u c h t h a t G a c t s on a s u r f a c e o f genus g as a group o f c o n f o r m a l a u t o m o r p h i s m s ? T h i s l e a s t v a l u e o f g w i l l be c a l l e d the B u r n s i d e genus o f the group G, and w i l l be d e n o t e d genusg(G) . We w i l l s o m e t i m e s a l s o r e f e r to t h i s as the a c t i o n genus o f t h e g r o u p . In o t h e r w o r d s , i f G a c t s as a group o f c o n f o r m a l a u t o m o r p h i s m s on a Riemann s u r f a c e o f genus g and does not so a c t on any s u r f a c e o f l e s s g e n u s , t h e n g e n u S g ( G ) = g . I t i s not b e i n g s u g g e s t e d by t h i s n o m e n c l a t u r e t h a t B u r n s i d e i s r e s p o n s i b l e f o r t h i s d e f i n i t i o n o f the genus o f a g r o u p . He d i d do some v e r y e a r l y work w i t h t h i s n o t i o n o f g r o u p g e n u s , and h i s c h a r a c t e r i z a t i o n o f a l l g r o u p s o f genus 0, 1, and 2 i n h i s 1896 t e x t b o o k , T h e o r y o f Groups o f F i n i t e O r d e r , i s the e a r l i e s t work on the s u b j e c t a v a i l a b l e i n E n g l i s h (see pp. 76-77) . The H y p e r b o l i c P l a n e A t t h i s p o i n t i t i s a p r o b l e m to see how the c o n f o r m a l a u t o m o r p h i s m s o f a Riemann s u r f a c e c a n be r e a l i z e d . The s o l u t i o n e x p l o i t s a deep c o n n e c t i o n w i t h h y p e r b o l i c g e o m e t r y . L e t H = { a + b i : b > 0 } d e n o t e the upper h a l f o f the c o m p l e x p l a n e , known as the h y p e r b o l i c p l a n e . H i s a model o f non -58 E u c l i d e a n h y p e r b o l i c g e o m e t r y . A n g l e s are measured as u s u a l , b u t s t r a i g h t l i n e s a r e e i t h e r r a y s p e r p e n d i c u l a r to t h e r e a l a x i s o r h a l f - c i r c l e s p e r p e n d i c u l a r to the r e a l a x i s . L i n e s a r e p a r a l l e l i f t h e y do not i n t e r s e c t . The E u c l i d e a n p a r a l l e l p o s t u l a t e f a i l s d r a m a t i c a l l y , because t h r o u g h a p o i n t P not on a g i v e n l i n e L t h e r e a r e i n f i n i t e l y many d i s t i n c t l i n e s p a r a l l e l t o L . A c c o r d i n g to the Riemann M a p p i n g T h e o r e m , any two s i m p l y c o n n e c t e d , p r o p e r subdomains o f the c o m p l e x p l a n e C are c o n f o r m a l l y e q u i v a l e n t . The upper h a l f p l a n e H, w i t h w h i c h we a r e most c o n c e r n e d , i s t h e r e f o r e c o n f o r m a l l y e q u i v a l e n t to the u n i t d i s k D = { z i n C : | z | < 1 }. T h e d i s k D i s a l s o a m o d e l o f h y p e r b o l i c g e o m e t r y , w i t h l i n e s e i t h e r d i a m e t e r s o f the d i s k o r h a l f - c i r c l e s p e r p e n d i c u l a r to the c i r c l e |z| = 1 . We w i l l o c c a s i o n a l l y use the two models o f the h y p e r b o l i c p l a n e i n t e r c h a n g e a b l y . A h y p e r b o l i c t r i a n g l e w i t h a n g l e s a, b , and c has an a n g l e sum a + b + c < ^ . I n a d d i t i o n , f o r t h r e e p o s i t i v e n u m b e r s a , b , a n d c, w i t h a + b + c < I T , t h e r e e x i s t s a h y p e r b o l i c t r i a n g l e , u n i q u e up to c o n g r u e n c e , w i t h a n g l e s a , b , and c. S i m i l a r t r i a n g l e s a r e c o n g r u e n t i n h y p e r b o l i c g e o m e t r y . The u p p e r h a l f p l a n e H has a non - E u c l i d e a n m e t r i c w h i c h y i e l d s an e l e m e n t o f a r c l e n g t h e q u a l to | d z | / y , where z = x + i y , and an 9 e l e m e n t o f a r e a g i v e n by d x d y / y . The m e t r i c m a g n i f i e s E u c l i d e a n d i s t a n c e s n e a r the r e a l a x i s and r e d u c e s them f a r f r o m i t . Under t h i s m e t r i c , the f o l l o w i n g r e m a r k a b l e t h e o r e m c a n be d e r i v e d . The r e s u l t i s u s u a l l y a t t r i b u t e d t o Gauss and B o n n e t . 59 Theorem 2.18: L e t T be a h y p e r b o l i c t r i a n g l e w i t h a n g l e s a , b , and c. The h y p e r b o l i c a r e a o f T i s IT - a - b - c . C o n s i d e r a h y p e r b o l i c t r i a n g l e w i t h a n g l e s T T / r , f T / s , '[T11 , where r , s , and t a r e i n t e g e r s >2. The Gauss - Bonnet t h e o r e m i m p l i e s t h a t 1 / r + 1/s + 1 / t < 1. I t a l s o f o l l o w s t h a t among a l l s u c h t r i a n g l e s , the one w i t h m i n i m a l a r e a has ( r , s , t ) = (2 ,3 ,7 ) . T h i s has f a r - r e a c h i n g c o n s e q u e n c e s . R e c a l l t h a t a b i l i n e a r t r a n s f o r m a t i o n i s an a n a l y t i c f u n c t i o n f : S 2 -> S 2 o f t h e f o r m f ( z ) = ( a z + b ) / ( c z + d ) . I f a , b , c , d a r e r e a l and ad - be = 1, i t i s e a s i l y shown t h a t f t a k e s the upper h a l f p l a n e H to i t s e l f . In f a c t , t h e s e b i l i n e a r t r a n s f o r m a t i o n s form the a u t o m o r p h i s m group o f H. The m a t r i x group S L ? ( R ) , the s p e c i a l l i n e a r group o v e r the r e a l numbers R c o n s i s t s o f a l l 2 X 2 m a t r i c e s I c d j w i t h a , b , c , d r e a l and ad - be = 1. The group PSL2(R) , the p r o j e c t i v e s p e c i a l l i n e a r group o v e r the r e a l s i s S L 2 ( R ) / { + I , - I ) . The b i l i n e a r t r a n s f o r m a t i o n s f o r m i n g the s y m m e t r i e s o f H a r e i n f a c t i s o m o r p h i c to PSL2(R). The c o r r e s p o n d e n c e t a k i n g the b i l i n e a r t r a n s f o r m a t i o n / a. M f ( z ) = (az + b ) / ( c z + d) t o t h e e l e m e n t ( o ^ o f P S L 2 ( R ) i s t h e i s o m o r p h i s m . T h e r e f o r e we r e g a r d PSL2(R) as the group o f c o n f o r m a l a u t o m o r p h i s m s o f H . Theorem 2.19: The f u l l g r o u p o f c o n f o r m a l a u t o m o r p h i s m s o f H i s t h e g r o u p P S L 2 ( R ) . In any group G, two e l e m e n t s f and g a r e s a i d t o be c o n j u g a t e i n G i f t h e r e i s an e l e m e n t h i n G s u c h t h a t f = h g h ^ . C o n j u g a c y 60 i s an e q u i v a l e n c e r e l a t i o n , and the e q u i v a l e n c e c l a s s e s a r e c a l l e d c o n j u g a c y c l a s s e s . The e l e m e n t s o f a group may t h e r e f o r e be c l a s s i f i e d by t h e i r c o n j u g a c y c l a s s . F o r g r o u p s w i t h a g e o m e t r i c r e a l i z a t i o n , l i k e PSL^CR), the e l e m e n t s o f a c o n j u g a c y c l a s s s h a r e s i g n i f i c a n t g e o m e t r i c p r o p e r t i e s . We w i l l g i v e the s t a n d a r d c l a s s i f i c a t i o n o f e l e m e n t s i n PSLjCR) by t h e i r c o n j u g a c y c l a s s . The c l a s s i f i c a t i o n w i l l not p l a y a g r e a t r o l e i n what f o l l o w s , b u t we i n c l u d e i t f o r c o m p l e t e n e s s and f o r the g e o m e t r i c u n d e r s t a n d i n g o f the a c t i o n o f PSI^CfO on H t h a t i t c o n v e y s . I t can be shown t h a t the c o n j u g a c y c l a s s o f a n o n - i d e n t i t y o f f , t r ( f ) = |a + d | . An e l e m e n t f i s c a l l e d e l l i p t i c i f t r z ( f ) < 4 , 9 9 p a r a b o l i c i f t r ( f ) = 4, and h y p e r b o l i c i f t r ( f ) > 4. An e l l i p t i c e l e m e n t has a s i n g l e f i x e d p o i n t i n H and i s c o n j u g a t e to a r o t a t i o n about t h a t f i x e d p o i n t . A p a r a b o l i c e l e m e n t has a s i n g l e f i x e d p o i n t and i s c o n j u g a t e to a t r a n s l a t i o n . The f i x e d p o i n t i s e i t h e r oO f o r a t r a n s l a t i o n z -> z + k , o r the p o i n t (a - d ) / 2 c on the r e a l a x i s . A h y p e r b o l i c e l e m e n t has two f i x e d p o i n t s i n H. Fuchs i a n Groups A p p a r e n t l y P o i n c a r e was the f i r s t to n o t i c e the c o n n e c t i o n b e t w e e n PSI^CR) and h y p e r b o l i c g e o m e t r y , as a r e s u l t o f h i s s y s t e m a t i c s t u d y o f the so c a l l e d F u c h s i a n g r o u p s i n 1880. We now d e f i n e t h e s e g r o u p s . o f P S L 9 ( R ) i s a l m o s t d e t e r m i n e d by the t r a c e 61 The group SL~(R) has a t o p o l o g y w h i c h i t i n h e r i t s f rom R by i d e n t i f y i n g the m a t r i x J w i t h the p o i n t ( a , b , c , d ) . The group P S L 2 ( R ) i s t h e n the q u o t i e n t s p a c e w h i c h i d e n t i f i e s the p o i n t s ( a , b , c , d ) and ( - a , - b , - c , - d ) . The f o l l o w i n g d e f i n i t i o n i s based on t h i s t o p o l o g y . A F u c h s i a n group i s a t o p o l o g i c a l l y d i s c r e t e subgroup o f P S L 2 ( R ) -T h e r e are e q u i v a l e n t d e f i n i t i o n s o f a F u c h s i a n group c h a r a c t e r i z i n g i t s a c t i o n on H. F o r e x a m p l e , a subgroup F o f P S L 2 ( R ) i s d i s c o n t i n u o u s i f , and o n l y i f , f o r each p o i n t p i n H a n d e a c h d i s c D c o n t a i n i n g p , t h e s e t < f ( p ) i n D : f i n F > i s f i n i t e . In o t h e r w o r d s , the o r b i t o f e a c h p o i n t p i n H i s a d i s c r e t e s u b s e t o f H. A F u c h s i a n g r o u p i s t h e n a d i s c o n t i n u o u s subgroup o f P S L 2 ( R ) . The s t u d y o f F u c h s i a n g r o u p s depends on the i d e a o f a f u n d a m e n t a l d o m a i n f o r s u c h a g r o u p . A c l o s e d s u b s e t D o f H i s a f u n d a m e n t a l d o m a i n f o r a F u c h s i a n g r o u p F i f (1) D has a n o n - e m p t y , c o n n e c t e d i n t e r i o r , ( 2 ) H = { U f ( D ) : f i n F }, a n d (3) f o r e a c h p o i n t p i n H , i f p i s i n f ( D ) a n d i n g(D) f o r t w o d i s t i n c t e l e m e n t s f , g o f F , t h e n p i s on the boundary o f b o t h f ( D ) a n d g ( D ) . We need to know t h a t a F u c h s i a n group does have a f u n d a m e n t a l d o m a i n . To t h i s e n d , c o n s i d e r t h e f o l l o w i n g . L e t p be a p o i n t i n H t h a t i s not f i x e d by any non - t r i v i a l e l e m e n t o f a F u c h s i a n g r o u p F . Then the D i r i c h l e t r e g i o n f o r F 62 c e n t e r e d a t p i s D p ( F ) = < q i n H : d ( q , p ) < d ( q , f ( p ) ) f o r a l l f i n F >. Theorem 2.20: I f F i s a F u c h s i a n group and the p o i n t p i n H i s not f i x e d by any n o n - t r i v i a l e l e m e n t o f F , t h e n the D i r i c h l e t r e g i o n D (F) i s a f u n d a m e n t a l d o m a i n f o r F. P We w i l l need to know the f o l l o w i n g i m p o r t a n t c l a s s i f i c a t i o n t h e o r e m f o r F u c h s i a n g r o u p s . Theorem 2.21: E v e r y F u c h s i a n group F has a p r e s e n t a t i o n o f the f o r m F - < x ^ , . . . , x k , a i , . . . , a , b ^ , . . . , b : x i m " ' ~ ••• - x j c x 1 . . . x k a 1 b 1 a 1 " 1 b 1 " 1 . . . a g b g ^ g ' ^ g " 1 = 1 >• I n t h i s p r e s e n t a t i o n k , g >^  0 a n d a l l m ( i ) >_2. I f x^ i s a p a r a b o l i c e l e m e n t , t h e n m(i) = ° 3 . E x c e p t f o r p e r m u t a t i o n s o f the e x p o n e n t s m ( i ) , g roups w i t h d i f f e r e n t p r e s e n t a t i o n s o f t h i s f o r m a r e n o n - i s o m o r p h i c g r o u p s . In c u s t o m a r y t e r m i n o l o g y , the m(i) a r e c a l l e d the p e r i o d s o f the F u c h s i a n g r o u p , and g the o r b i t genus . F e l i x K l e i n i n the 19th c e n t u r y d e n o t e d s u c h a g r o u p by t h e s y m b o l ( g , k ; m ( l ) , . . . , m ( k ) ) , w h i c h he c a l l e d the s i g n a t u r e o f the g r o u p . S i n c e the k i s r e d u n d a n t , modern a u t h o r s s o m e t i m e s use ( g ; m ( l ) , . . . , m ( k ) ) f o r the s i g n a t u r e , b u t we w i l l use K l e i n ' s n o t a t i o n . We o b s e r v e t h a t the t o r s i o n f r e e F u c h s i a n group w i t h s i g n a -t u r e (g ,0 : . . . ) i s the q u o t i e n t o f a F u c h s i a n g r o u p w i t h s i g n a t u r e ( g , k ; m ( 1 ) , m ( k ) ) . T h i s c a n be s e e n by t r i v i a l i z i n g the e l e m e n t s o f f i n i t e o r d e r i n the c a n o n i c a l p r e s e n t a t i o n o f a F u c h s i a n g r o u p . (D_ _ „ m(k) _ 63 e x a m p l e : The m o d u l a r g r o u p i s the group G = PSL^CZ) . I t may be c o n s i d e r e d to be the group o f b i l i n e a r t r a n s f o r m a t i o n s f ( z ) = ( a z + b ) / ( c z + d ) , w h e r e a , b , c , d a r e i n t e g e r s a n d ad - be = 1. I t i s c l e a r l y a d i s c r e t e s u b g r o u p o f P S L ^ R ) , and i t i s known to have a 2 o p r e s e n t a t i o n P S L ^ Z ) = < x , y : x = y = 1 > . T h e g e n e r a t o r s x a n d y o f the group a r e u s u a l l y t a k e n t o be the b i l i n e a r t r a n s f o r m a t i o n s x : z -> - 1 / z a n d y : z -> ( - z - l ) / z , w h o s e f i x e d p o i n t s a r e i and - 1 / 2 + /3 i / 2 r e s p e c t i v e l y . The e l e m e n t (yx)~^ i s the t r a n s l a t i o n z -> z + 1. P S I ^ Z ) i s a F u c h s i a n g r o u p w i t h s i g n a t u r e ( 0, 3 ; 2, 3 , A f u n d a m e n t a l d o m a i n f o r the m o d u l a r g r o u p i s the D i r i c h l e t r e g i o n D p ( G ) = { z = a + b i : - 1 / 2 < a < 1 / 2 , b > 0, | z | > 1 }. T h e p o i n t P may be t a k e n t o be k i f o r a n y k > 1 ( s e e [ 1 4 ] , p . 2 4 2 ) . I t i s the t r i a n g l e o f i n f i n i t e e x t e n t shown b e l o w w i t h v e r t i c e s a t - 1 / 2 +73i /2 , 1/2 + /3 i / 2 , and i n f i n i t y . I t s a n g l e s a re C/3, iT/3, a n d 0; and h e n c e i t s a r e a i s The h y p e r b o l i c t r i a n g l e s shown t e s s e . l a t i n g the upper h a l f p l a n e H a r e a l l images o f the f u n d a m e n t a l d o m a i n u n d e r the a c t i o n o f the m o d u l a r group G = P S L ^ Z ) . S i n c e the a c t i o n i s c o n f o r m a l , a n g l e s a r e p r e s e r v e d and the a r e a o f e a c h image i s a l s o 1T/3. The q u o t i e n t s p a c e H / G i s the once p u n c t u r e d 2 - s p h e r e . 6 4 A F u c h s i a n group w i t h s i g n a t u r e ( g , 0 ; . . . ) , has no p e r i o d s and hence no e l e m e n t s o f f i n i t e o r d e r , and i s c a l l e d a F u c h s i a n s u r f a c e  g r o u p f o r the o b v i o u s r e a s o n t h a t i t i s the f u n d a m e n t a l group fT(S) o f a c l o s e d , o r i e n t a b l e s u r f a c e S o f genus g. e x a m p l e : C o n s i d e r a t r i a n g l e g r o u p . L e t T be a h y p e r b o l i c t r i a n g l e PQR w i t h i n t e r i o r a n g l e s " ^ / p , ^ / q > and "TT/r a t the v e r t i c e s P , Q, a n d R, w h e r e p , q , a n d r a r e i n t e g e r s >^  2. S u c h a t r i a n g l e e x i s t s , o f c o u r s e , o n l y i f 1/p + 1 /q + 1 / r < 1, s i n c e i t s a r e a i s minus the sum o f i t s i n t e r n a l a n g l e s . L e t x, y , and z be r e f l e c t i o n i n the s i d e s o f T o p p o s i t e the v e r t i c e s P, Q, and R. L e t G be t h e g r o u p g e n e r a t e d by x , y , a n d z . 65 Then G has a p r e s e n t a t i o n G = < x, y , z : x 2 = y 2 = z 2 = ( y z ) p = (zx)*> = ( x y ) r = 1 >. G i s the t r i a n g l e group w h i c h we have p r e v i o u s l y d e n o t e d (see p. 33) as T ( p , q , r ) . I t i s t h e f u l l ( p , q , r ) t r i a n g l e g r o u p , a n d i t s a c t i o n on the h y p e r b o l i c p l a n e H does not p r e s e r v e o r i e n t a t i o n . The i l l u s t r a t i o n b e l o w shows T and one o f i t s i m a g e s . R e x a m p l e : L e t us c o n t i n u e the p r e v i o u s e x a m p l e . I f we make the new d e f i n i t i o n s t = y z , u = z x , a n d v = x y ; t h e n t , u , and v a r e e l e m e n t s o f o r d e r s p , q , and r w i t h f i x e d p o i n t s P , Q , and R. L e t T ° ( p , q , r ) denote the o r i e n t a t i o n p r e s e r v i n g subgroup o f T ( p , q , r . ) . T h e n T ° ( p , q , r ) i s g e n e r a t e d by t h e e l e m e n t s t , u , and v ; a n d T ° ( p , q , r ) h a s t h e p r e s e n t a t i o n T ° ( p , q , r ) = < t , u , v : t p = u ^ v r = t u v = 1 >. T ° ( p 5 q j r ) i s then a F u c h s i a n g r o u p . I t s s i g n a t u r e i s ( 0 , 3 : p , q , r ) . A f u n d a m e n t a l d o m a i n f o r T ° ( p , q , r ) c o n s i s t s o f T and one o f i t s images under r e f l e c t i o n i n a s i d e . An e x a m p l e i s shown b e l o w w i t h r = 4. 6 6 R example: A F u c h s i a n s u r f a c e group F w i t h s i g n a t u r e ( g , 0 ; . . . ) , the fundamental group of a s u r f a c e S o f genus g, a c t s i n many d i f -f e r e n t ways on the h y p e r b o l i c plane H i f g > 1. A fundamental domain f o r t h i s group i s a h y p e r b o l i c v e r s i o n o f the 4g - s i d e d polygon we have c a l l e d the fundamental polygon f o r a s u r f a c e of genus g, and the a c t i o n o f the fundamental group e f f e c t s the i d e n t i f i c a t i o n of edges which c r e a t e s the s u r f a c e S. The plane H i s t i l e d w i t h the images of t h i s polygon under the f i x e d - p o i n t f r e e , p r o p e r l y d i s c o n t i n u o u s a c t i o n o f F. H then 'is the u n i v e r s a l c o v e r i n g space f o r the s u r f a c e S, and the q u o t i e n t s u r f a c e H/F i s S. A s c h e m a t i c drawing o f the fundamental domain f o r a F u c h s i a n s u r f a c e group w i t h s i g n a t u r e (2 ,0 ; . . . ) i s shown here i n the d i s k model o f h y p e r b o l i c geometry. I t i s a r e g u l a r 8-gon w i t h a l l a n g l e s e q u a l t o 6 7 F u c h s i a n G r o u p s and Riemann S u r f a c e s A F u c h s i a n g r o u p F , t h e n , t e s s e l a t e s the h y p e r b o l i c p l a n e H w i t h images o f i t s f u n d a m e n t a l d o m a i n . The g r o u p F may a l s o be t h o u g h t o f as a g r o u p o f s y m m e t r i e s o f the t e s s e l a t e d p l a n e H. S i n c e the f u n d a m e n t a l group o f a c l o s e d , o r i e n t a b l e s u r f a c e o f genus g > 1 i s a F u c h s i a n group a c t i n g on H as a p r o p e r l y d i s c o n t i n u o u s g r o u p o f h o m e o m o r p h i s m s , the c l o s e d , o r i e n t a b l e s u r f a c e s a r e the q u o t i e n t s u r f a c e s H / F . In f a c t , t h e s e s u r f a c e s a r e now seen t o be Riemann s u r f a c e s , because they i n h e r i t the l o c a l c o m p l e x s t r u c t u r e o f the upper h a l f p l a n e H. The q u o t i e n t s p a c e H / F i s , o f c o u r s e , the s p a c e o f o r b i t s o f p o i n t s q i n H , w h e r e t h e o r b i t o f a p o i n t q i n H , d e n o t e d F q , i s d e f i n e d as F q = { f ( q ) : f i n F }. T h e p r o j e c t i o n map p : H -> H / F d e f i n e d by p(q) = Fq i s 1-1 on the i n t e r i o r o f the f u n d a m e n t a l d o m a i n D(F) . We r e g a r d the q u o t i e n t space H / F as the s u r f a c e D ( F ) / F o b t a i n e d by i d e n t i f y i n g b o u n d a r y p o i n t s b^ and b 2 i f b 2 = f ( b ^ ) f o r some f i n F . In a d d i t i o n , the p l a n e H i s seen to be the u n i v e r s a l c o v e r i n g s u r f a c e f o r a Riemann s u r f a c e S o f genus g > 1, so the r e s u l t s o f c o v e r i n g space t h e o r y a p p l y . The f u n d a m e n t a l group i s the g r o u p o f c o v e r i n g t r a n s f o r m a t i o n s . L e t ' s now e x t e n d the c l a s s i f i c a t i o n o f R i e m a n n s u r f a c e s . The t h r e e d i s t i n c t s i m p l y c o n n e c t e d Riemann s u r f a c e s a r e C U {oO}, C , and H . They a r e s o m e t i m e s c a l l e d e l l i p t i c , p a r a b o l i c , and h y p e r b o l i c s u r f a c e s . 68 The f u l l g r o u p o f c o n f o r m a l a u t o m o r p h i s m s o f each o f t h e s e s u r f a c e s i s i n d i c a t e d i n the f o l l o w i n g t h e o r e m . Theorem 2 . 2 2 : (1) A u t ( C U = P S L 2 ( C ) (2) A u t ( C ) = P T 2 ( C ) (3) A u t ( H ) = P S L 2 ( R ) . The n o t a t i o n P T 2 ( C ) s t a n d s f o r the p r o j e c t i v e g r o u p o f 2x2 u p p e r t r i a n g u l a r c o m p l e x m a t r i c e s o f n o n - z e r o d e t e r m i n a n t , o r , i n o t h e r w o r d s , the g r o u p o f b i l i n e a r t r a n s f o r m a t i o n s z -> az + b , where a and b a r e c o m p l e x and a i s n o n z e r o . The compact Riemann s u r f a c e s w h i c h are not s i m p l y c o n n e c t e d a l l a r i s e i n the manner we have j u s t s e e n , as q u o t i e n t spaces R / G , where R i s one o f the t h r e e s i m p l y c o n n e c t e d Riemann s u r f a c e s and G i s the a p p r o p r i a t e g r o u p a c t i n g on R. T h e o r e m 2.23: E v e r y Riemann s u r f a c e S i s c o n f o r m a l l y e q u i v a l e n t t o R / G , w h e r e R i s e i t h e r (1) C U {<*?}, (2) C , o r (3) H ; and G i s a p r o p e r l y d i s c o n t i n u o u s group o f c o m p l e x b i l i n e a r t r a n s f o r m a t i o n s t h a t p r e s e r v e R. A l s o , G i s i s o m o r p h i c to 1T~(S), the f u n d a m e n t a l group o f S. These R i e m a n n s u r f a c e s a r e a l s o c a l l e d e l l i p t i c , p a r a b o l i c , and h y p e r b o l i c r e s p e c t i v e l y . F o r the R i e m a n n s u r f a c e s r e l e v a n t to t h i s p a p e r , e x c e p t f o r the 2 - s p h e r e and t o r u s , the a p p r o p r i a t e Riemann s u r f a c e R i n t h i s t h e o r e m i s R = H , the h y p e r b o l i c p l a n e . In t h a t c a s e , the g r o u p G i s o f c o u r s e a F u c h s i a n g r o u p . 69 The A c t i o n on a_ Riemann S u r f a c e T u c k e r p r o v e s the f o l l o w i n g b a s i c t h e o r e m , w h i c h e x p l a i n s , f i n a l l y , the o r i g i n o f g r o u p a c t i o n s on s u r f a c e s . We have s t a t e d the theorem f o r Riemann s u r f a c e s . We w i l l e x a m i n e the c o n t e n t o f the t h e o r e m f r o m a v a r i e t y o f p e r s p e c t i v e s . Theorem 2.24: L e t F be a d i s c r e t e group o f c o n f o r m a l a u t o m o r p h i s m s o f a Riemann s u r f a c e S. L e t G be a n o r m a l s u b g r o u p o f F . Then S / G i s a Riemann s u r f a c e , and F / G a c t s on S / G as a group o f c o n f o r m a l a u t o m o r p h i s m s . F u r t h e r m o r e S / F = ( S / G ) / ( F / G ) . The a c t i o n o f F / G on S / G a r i s e s as f o l l o w s . F o r a group e l e m e n t f i n F , l e t G f be a n e l e m e n t o f F / G . F o r a p o i n t p i n S , l e t Gp be a p o i n t i n S / G . Gp i s t h e n t h e o r b i t o f p . The e l e m e n t G f t a k e s the o r b i t Gp to the o r b i t G ( f ( p ) ) . More f o r m a l l y , G f : S / G -> S / G i s d e f i n e d by G f ( G p ) = G ( f ( p ) ) . C o n v e r s e l y , a g i v e n g r o u p J i s a group o f a u t o m o r p h i s m s o f a s u r f a c e o f genus g > 1 i f t h e r e i s a F u c h s i a n group F and a e p i m o r p h i s m e : F -> J s u c h t h a t t h e k e r n e l o f e , K = k e r ( e ) , i s t o r s i o n f r e e w i t h 2g g e n e r a t o r s . The group K = k e r ( e ) i s t h e n a F u c h s i a n s u r f a c e group and w i l l have the s t a n d a r d p r e s e n t a t i o n K = < a , , b 1 , . . . , a g , b g : a ^ l a x ~ 1 b L " [ . . . a g b g a g ~ l b g _ 1 = 1 >• O f c o u r s e , F / K = J . So F a n d K a c t on t h e u p p e r h a l f p l a n e H a n d J = F / K a c t s on H / K . We may s u m m a r i z e the s i t u a t i o n by s a y i n g t h a t t h e r e i s a s h o r t  e x a c t sequence o f groups 1 > k e r ( e ) — i — > F - - e - - > J > 1. T h i s means t h a t i i s a m o n o m o r p h i s m , e i s an e p i m o r p h i s m , and 70 ker (e ) = i m a g e ( i ) . In f a c t , we may say t h a t whenever t h i s a l g e b r a i c c o n d i t i o n i s met , t h a t i s , whenever t h e r e i s a s h o r t e x a c t sequence o f groups 1 > K — i—> F — e—> J — > 1 w i t h F a F u c h s i a n g r o u p , t h e n J a c t s on H / K as a group o f c o n f o r m a l a u t o m o r p h i s m s . A d i r e c t c o n s e q u e n c e o f the Gauss - Bonnet f o r m u l a f o r the a r e a o f a h y p e r b o l i c t r i a n g l e i s t h a t the h y p e r b o l i c a r e a o f the f u n d a m e n t a l d o m a i n o f a F u c h s i a n group F c a n be computed f r o m the p r e s e n t a t i o n o f F. Theorem 2.25: L e t the F u c h s i a n group F have s i g n a t u r e ( j , k ; m ( l ) . . . m ( k ) ) . Then the h y p e r b o l i c a r e a o f a n y f u n d a m e n t a l d o m a i n f o r F i s 2TT< ( 2 j - 2) + ( l - l / m ( l ) ) + . . . + ( l - l / m ( k ) ) >. T h i s t h e o r e m w i l l a l l o w us to g i v e a c o n c r e t e d e s c r i p t i o n o f the a c t i o n o f a group o f c o n f o r m a l a u t o m o r p h i s m s on a Riemann s u r f a c e . Suppose F i s a F u c h s i a n group w,ith s i g n a t u r e ( j , k ; m( 1 ) , m ( k ) ) , and l e t e : F -> G be an e p i m o r p h i s m f r o m F t o a f i n i t e g r o u p G. I f the n o r m a l s u b g r o u p K = k e r ( e ) i s a F u c h s i a n s u r f a c e g r o u p w i t h s i g n a t u r e , s a y , ( g , 0 ; . . . ) , t h e n G = F / K a c t s o n the q u o t i e n t s u r f a c e H / K , where H i s the h y p e r b o l i c p l a n e . I f D(F) i s a f u n d a m e n t a l d o m a i n f o r F , i t s a r e a i s g i v e n by 2T{ ( 2 j - 2 ) + ( 1 - l / m ( l ) ) + . . . + ( 1 - l / m ( k ) ) >. I f D(K) i s a f u n d a m e n t a l d o m a i n f o r K, t h e n i t s h y p e r b o l i c a r e a , a c c o r d i n g to the l a s t t h e o r e m , i s j u s t 21T"{ (2g - 2) >. Now s i n c e H / K g i v e s a b r a n c h e d c o v e r i n g o f H / F , w i t h ( H / K ) / G = H / F by T u c k e r ' s l a s t t h e o r e m , we may i n v o k e the R i e m a n n - H u r w i t z 7 1 f o r m u l a to ge t 2 - 2g = |G|( 2 - 2 j - ( 1 - l / m ( l ) ) - . . . - ( 1 - l / m ( k ) ) ) . T h i s g i v e s |G| = ( a r e a o f D ( K ) ) / ( a r e a o f D ( F ) ) . Because F i s a c o n f o r m a l , o r a n g l e - p r e s e r v i n g a c t i o n , a l l images o f i t s f u n d a m e n t a l d o m a i n D(F) have t h e same a n g l e s , and hence the same a r e a by the Gauss - Bonnet f o r m u l a . T h e r e f o r e | G | i s j u s t the number o f images o f D(F) m a k i n g up a c o m p l e t e t e s s e l a t i o n o f the f u n d a m e n t a l domain D(K). And the a c t i o n o f G on the Riemann s u r f a c e H / K , seen most c o n c r e t e l y , i s the t i l i n g o f the s u r f a c e , i n a f i n i t e number o f s t e p s , by the images o f D(F) . E x i s t e n c e o f the B u r n s i d e Genus L e t us now show t h a t e v e r y f i n i t e group G does i n f a c t a c t on a s u r f a c e o f some f i n i t e g e n u s , s h o w i n g t h a t e v e r y f i n i t e g r o u p does have a B u r n s i d e genus . Suppose G has a p r e s e n t a t i o n o f the form < g j , . . . , g n : . . . y, w h e r e n > l . L e t F be t h e f u n d a m e n t a l g r o u p o f a s u r f a c e S^ o f genus n , so F = < a 1 , b 1 , . . . , a n , b n : a ^ ^ 1 b 1 1 . . . a n b n a n 1 b n 1 =1 >. - 1 D e f i n e a h o m o m o r p h i s m h : F -> G by h ( a k ) = g ^ a n d h ( b ^ ) = g ^ G i s then i s o m o r p h i c to F / K , where K i s the k e r n e l of the homomorphism h. As we know, S^ i s c o v e r e d by' the upper h a l f p l a n e H, and F i s i s o m o r p h i c to the g r o u p o f c o v e r i n g t r a n s f o r m a t i o n s . In o t h e r w o r d s , F a c t s on H , and t h e q u o t i e n t s p a c e H / F i s t h e s u r f a c e S^. In a d d i t i o n , H / K i s a l s o a s u r f a c e , c a l l i t S2 , and F / K a c t s on H / K = S2. T h i s e s t a b l i s h e s the d e s i r e d r e s u l t . 7 2 Improved H u r w i t z Bounds L e t ' s r e c a l l t h a t i f G i s a group o f c o n f o r m a l a u t o m o r p h i s m s o f a s u r f a c e S o f genus g> l , t h e n the H u r w i t z bound on G g u a r a n t e e s t h a t | G | _< 84 (g - 1 ) . T . W. T u c k e r , i n a p a p e r e n t i t l e d " F i n i t e G r o u p s A c t i n g on S u r f a c e s and the Genus o f a G r o u p " ( [ 3 0 ] ) , has i m p r o v e d the H u r w i t z bound by s h o w i n g t h a t groups w i t h o r d e r " c l o s e " t o the b o u n d , soon t o be d e f i n e d p r e c i s e l y , must be q u o t i e n t s of a c e r t a i n few t r i a n g l e g r o u p s . H e r e ' s h i s r e s u l t . T h e o r e m 2 . 2 6 : L e t G a c t on a s u r f a c e S o f g e n u s g > 1 as a group o f c o n f o r m a l a u t o m o r p h i s m s . Then |G| < 84(g - 1). In a d d i t i o n , i f | G | >^  12(g - 1), t h e n G i s a q u o t i e n t o f some t r i a n g l e group T ° ( p , q , r ) , the o r i e n t a t i o n - p r e s e r v i n g s u b g r o u p of T ( p , q , r ) . F u r t h e r m o r e , i f |G| >^  24(g - 1), t h e n the t r i p l e ( p , q , r ) must be ( 2 , 4 , 5 ) o r ( 2 , 3 , r ) , 7 < r < 11 . p r o o f : Suppose the genus o f S / G i s q. A c c o r d i n g to the R i e m a n n - H u r w i t z f o r m u l a , we have 2 - 2g = |G|( 2 - 2q - ( 1-1 / m( 1 ) ) - ( 1 - 1 / m( 2) ) - . . . - ( 1 - 1 / m( k ))) , where m( i ) i s the o r d e r o f the i t n b r a n c h p o i n t o f the c o v e r i n g p r o j e c t i o n p: S -> S / G . C o n s i d e r the f o l l o w i n g c a s e s . C a s e I : I f 2 - 2q < 0, t h e n 2 - 2g < - | G | . C a s e I I : I f 2 - 2q = 0, t h e n 2 - 2 g < - | G | / 2 , b e c a u s e i f t h e r e a r e no b r a n c h p o i n t s t h e n 2 - 2g = 0, c o n t r a r y t o o u r a s s u m p t i o n . C a s e I I I : I f 2 - 2q = 2 and i f the number o f b r a n c h p o i n t s k s a t i s f i e s k > 3, t h e n 2 - 2g < - | G | / 6 . T h i s i s e a s i l y v e r i f i e d . 73 T h e r e f o r e , t a k i n g the c o n t r a p o s i t i v e , we have so f a r shown t h a t i f 2 - 2g > - | G | / 6 , o r | G | > 12(g - 1 ) , t h e n S / G i s a s p h e r e c o n t a i n i n g e x a c t l y t h r e e b r a n c h p o i n t s , because two b r a n c h p o i n t s g i v e 2 - 2g > 0. Hence G must be the q u o t i e n t o f a t r i a n g l e group T ° ( p , q , r ) . I f i n a d d i t i o n , |G| > 24(g - 1), an e x a m i n a t i o n o f the c a n d i d a t e s f o r the t r i p l e ( p , q , r ) i n the Riemann - H u r w i t z r e l a t i o n shows t h a t the o n l y ones a r e t h o s e i n the t h e o r e m . # The r e s u l t c a n be r e s t a t e d when g i s the B u r n s i d e genus o f the g r o u p G , t h a t i s , when G a c t s on s u r f a c e S o f g e n u s g , b u t on no s u r f a c e o f l e s s e r genus. T h e o r e m 2 . 2 7 : L e t g e n u s B ( G ) = g > 1. T h e n | G | < 8 4 ( g - 1 ) . I n a d d i t i o n , | G | > 24(g - 1) i f and o n l y i f G i s a q u o t i e n t o f T ° ( 2 , 4 , 5 ) o r T ° ( 2 , 3 , r ) , 7 < r < 11. p r o o f : The " i f " p a r t o f the t h e o r e m comes f r o m f u r t h e r work o f T u c k e r ' s t h a t d u p l i c a t e s known r e s u l t s abq>ut F u c h s i a n g r o u p s . I f G i s a q u o t i e n t o f T ° ( p , q , r ) , t h e n G has a p a r t i a l p r e s e n t a t i o n G = < t , u , v : t p = u ^ = v r = t u v = 1 = . . . >. By R i e m a n n -H u r w i t z we get 2 - 2g = |G|( 2 - ( 1 - 1 / p ) - ( 1 - 1 / q ) - ( 1 - 1 / r ) ) . E x a m i n a t i o n o f the t r i p l e s ( p , q , r ) i n the t h e o r e m r e v e a l s t h a t i n a l l c a s e s | G | > 24(g - 1) . # Theorem 2.28: I f G i s a f i n i t e q u o t i e n t o f T ° ( 2 , 3 , 7 ) , t h e n g e n u s B ( G ) = | G | / 8 4 + 1. p r o o f : G a c t s on a s u r f a c e S o f genus g , and the genus o f S / G i s 0. T h e r e f o r e , 2 - 2g = |G|( 2 - ( 1 - 1 / 2 ) ) - ( 1 - 1 / 3 ) ) - ( 1 - 1 / 7 ) ) ) . 74 The r e s u l t f o l l o w s . # T h e r e f o r e , a H u r w i t z g r o u p , one w h i c h a t t a i n s the 84(g - 1) u p p e r b o u n d , i s a homomorphic image o f T ° ( 2 , 3 , 7 ) . Conder [4] has shown t h a t A n i s a q u o t i e n t o f T ° ( 2 , 3 , 7 ) f o r n >^  168. H e n c e , T u c k e r ( [30 ] ) g e t s the next r e s u l t . T h e o r e m 2 .29 : g e n u s B ( A n ) = n ! / 1 6 8 + 1, f o r n > 168. A c o n c l u s i o n r e a c h e d as one s t e p i n the p r o o f o f the i m p r o v e d H u r w i t z bounds seems to me to r e q u i r e s p e c i a l e m p h a s i s . I t ' s a r e s u l t t h a t has i m p o r t a n c e i n i t s own r i g h t , so I w i l l s t a t e i t as a t h e o r e m . Theorem 2.30: L e t the f i n i t e group G a c t on a Riemann s u r f a c e S o f g e n u s g >1, and l e t | G | > 12 (g - 1) . T h e n S / G i s a s p h e r e , a n d p: S -> S / G i s a b r a n c h e d c o v e r i n g w i t h e x a c t l y t h r e e b r a n c h p o i n t s . Genus F o r m u l a s f o r Groups D e t e r m i n i n g the B u r n s i d e o r a c t i o n genus o f a group i s v e r y d i f f i c u l t , and I t h i n k i t i s f a i r t o say t h a t the p r o b l e m i s f a r f r o m b e i n g c o m p l e t e l y s o l v e d f o r a l l f i n i t e g r o u p s . We w i l l now d e s c r i b e the t h r e e m a j o r r e s u l t s i n the f i e l d . 1) Burns i d e ' s work The f i r s t work i s the d e t e r m i n a t i o n o f a l l g r o u p s o f a c t i o n g e n u s 0, 1, a n d 2. I f i n d i t r e p o r t e d by B u r n s i d e i n h i s t e x t , T h e o r y o f G r o u p s of F i n i t e O r d e r f i r s t p u b l i s h e d i n 1896. H o w e v e r , 75 the a u t h o r ' s f o o t n o t e s s u g g e s t t h a t some o r a l l o f the work s h o u l d be c r e d i t e d to D y c k , who p u b l i s h e d i t i n 1880 and 1882. A d d i t i o n a l f o o t n o t e s h i n t t h a t H u r w i t z had s o m e t h i n g t o say about t h i s d e f i n i -t i o n o f g r o u p genus at l e a s t as e a r l y as 1893. I w i l l g i v e B u r n s i d e ' s v e r s i o n o f the r e s u l t s . The groups o f a c t i o n genus 0 t u r n out to be j u s t Z n , c y c l i c g r o u p s o f o r d e r n ; D n , d i h e d r a l g r o u p s o f o r d e r 2 n ; A ^ ; S ^ ; a n d A ^ . T h e s e groups a r e a l l i n c l u d e d among those o f C a y l e y genus 0 . The g r o u p s G o f B u r n s i d e genus 1 f a l l i n t o f o u r d i s t i n c t c l a s s e s o f p r e s e n t a t i o n t h a t i n v o l v e a number o f p a r a m e t e r s i n a d d i t i o n to the g e n e r a t o r s and r e l a t i o n s . Here t h e y a r e . I . A 2 = B 2 = C 2 = ( A B C ) 2 = 1 ( A B ) d ( B C ) e = ( A B ) f ( B C ) 8 = 1 | G | = 2 (dg - e f ) . I I . A 3 = B 3 = ( A B ) 3 = 1 ( A B 2 ) d ( B A B ) e = 1 | G | = 3 ( d 2 - de + e 2 ) . I I I . A 2 = B 4 = ( A B ) 4 = 1 ( A B 2 ) d ( B A B ) e = 1 | G | = 4 ( d 2 + e 2 ) . I V . A 6 = B 3 = ( A B ) 2 = 1 ( A 2 B 2 ) d ( A B 2 A ) e = 1 | G | = 6 ( d 2 + de + e 2 ) . B u r n s i d e o b s e r v e s t h a t f o r a few s p e c i a l v a l u e s o f the p a r a m e t e r s i n t h e s e p r e s e n t a t i o n s the group G may have genus 0. F o r e x a m p l e , i n c l a s s I, i f dg - e f i s a p r i m e , t h e n G i s a d i h e d r a l g r o u p . T h e r e a r e e x a c t l y f o u r g r o u p s G, up to i s o m o r p h i s m o f c o u r s e , o f B u r n s i d e genus 2, g i v e n as f o l l o w s , I . A 4 = 1, B 2 = A 2 , B - 1 A B = A - 1 I I . A 8 = B 2 = 1, B _ 1 A B = A 3 I I I . A 4 = B 3 , A _ 1 B A = B 2 I V . A 8 = B 2 = ( A B ) 3 = ( A 4 B ) 2 = 1. 2) M a c l a c h l a n ' s work The s e c o n d i s the work o f C. M a c l a c h l a n , r e p o r t e d i n a 1965 paper ( [18] ) e n t i t l e d " A b e l i a n Groups o f A u t o m o r p h i s m s o f Compact R i e m a n n S u r f a c e s . " In t h i s paper the B u r n s i d e genus o f a l l a b e l i a n g r o u p s i s d e t e r m i n e d . The methods t h a t M a c l a c h l a n e m p l o y s t o o b t a i n t h e s e i m p o r t a n t r e s u l t s a r e based on F u c h s i a n g r o u p s a c t i n g on the u p p e r h a l f p l a n e H. As we now know, i f R i s a compact Riemann s u r f a c e o f genus g > 2, t h e n i t s u n i v e r s a l c o v e r i n g space i s t h e h y p e r b o l i c p l a n e H , and R i s r e p r e s e n t a b l e as H / K , where K i s a 77 t o r s i o n f r e e F u c h s i a n s u r f a c e g r o u p . In f a c t , K i s the f u n d a m e n t a l g r o u p o f R. M a c l a c h l a n c a l l s a homomorphism f : F -> G f r o m a F u c h s i a n group F t o an a b e l i a n group G a s u r f a c e - k e r n e l homomorphism i f K = k e r ( f ) i s a F u c h s i a n s u r f a c e g r o u p . He t h e n c o n s i d e r s a l l F u c h s i a n g r o u p s F s u c h t h a t t h e r e e x i s t s a s u r f a c e - k e r n e l homomorphism o f F o n t o G. L e t F be a F u c h s i a n g r o u p w i t h s i g n a t u r e ( g , k ; m( 1 ) , . . . , m( k ) ) . C o n s i d e r a g a i n the r e l a t i o n 2(g - 1 ) / | G | = 2 (q - 1) + ( l - l / m ( l ) ) + . . . + ( l - l / m ( k ) ) . To m i n i m i z e g f o r a f i x e d g r o u p G, i t i s n e c e s s a r y to m i n i m i z e the r i g h t hand s i d e o f t h i s e q u a t i o n o v e r the s e t o f F u c h s i a n g r o u p s F t h a t c a n be mapped onto G by a s u r f a c e - k e r n e l h o m o m o r p h i s m . T h i s i s e q u i v a l e n t to f i n d i n g the F u c h s i a n group F w i t h q u o t i e n t G whose f u n d a m e n t a l d o m a i n has l e a s t a r e a . I w i l l g i v e some i n d i c a t i o n o f M a c l a c h l a n ' s methods i n d e t e r m i n i n g the genus o f a l l a b e l i a n g r o u p s and w i l l show h i s d e t a i l e d work i n one c a s e . T o g e t h e r w i t h h i s f i n a l r e s u l t s , t h i s w i l l c o n v e y the q u a l i t a t i v e n a t u r e o f h i s w o r k . R e c a l l t h a t a f i n i t e a b e l i a n group G i s d e t e r m i n e d by i t s i n v a r i a n t s n ( l ) , n ( 2 ) , . . . , n ( r ) ; w h e r e n ( i ) d i v i d e s n ( i + l ) a n d G = Z n ( l ) X Z n ( 2 ) X *** X Z n ( r ) " T n L S i s t h e c a n o n i c a l r e p r e s e n t a t i o n o f G. The g e n e r a t o r s o f the c y c l i c groups Z n Q ) w i l l be c a l l e d c a n o n i c a l g e n e r a t o r s f o r G. The number r , the r a n k o f the g r o u p , i s a l s o the m i n i m a l number o f g e n e r a t o r s i n any p r e s e n t a t i o n o f G. M a c l a c h l a n c o n s i d e r s i n t h r e e c a s e s a l l p o s s i b l e F u c h s i a n g r o u p s F s u c h t h a t t h e r e e x i s t s a s u r f a c e k e r n e l homomorphism 78 h : F -> G . C a s e I c o n s i s t s o f t h o s e g r o u p s F w i t h s i g n a t u r e ( 0 , k ; m( 1 ) , . . . , m ( k ) ) . T h e o r b i t g e n u s o f t h e s e g r o u p s i s 0 and t h e r e a r e k g e n e r a t o r s w i t h the f i n i t e o r d e r s g i v e n . Case I I c o n s i s t s o f g r o u p s F w i t h s i g n a t u r e ( g , 0 ; . . . ) . These a re F u c h s i a n s u r f a c e g r o u p s . Case I I I c o n s i s t s o f " m i x e d " F u c h s i a n g r o u p s h a v i n g a s i g n a t u r e ( g , k ; m( 1 ) , . . . , m ( k ) ) . We w i l l e x a m i n e Case I more c l o s e l y now, but to do so we w i l l need two p r e l i m i n a r y t h e o r e m s M a c l a c h l a n has worked o u t . F i r s t , c o n s i d e r a F u c h s i a n g r o u p F w i t h s i g n a t u r e ( g , k ; m ( l ) , . . . , m ( k ) ) . I f t h e r e i s an e p i m o r p h i s m h : F -> G, w i t h a t o r s i o n - f r e e k e r n e l , t h e n G m u s t h a v e g e n e r a t o r s a^ , b ^ , . . . ,a_g >b_gJ a n d Z . i > " * » y _ k " H e r e , o f c o u r s e , h ( a ^ ) = a _ £ , h ( b ^ ) = b ^ , and h ( y j ) = £ j . In a d d i t i o n , the d e f i n i n g r e l a t i o n f o r the F u c h s i a n group F must a l s o be s a t i s f i e d by the c o r r e s p o n d i n g g e n e r a t o r s i n G. F u r t h e r m o r e , the o r d e r o f y_^  must be e x a c t l y m ( i ) , because i f i t s o r d e r were d ( i ) < m ( i ) , t h e n y ^ d ^ ^ w o u l d , b e l o n g to the k e r n e l o f h , c o n t r a d i c t i n g the f a c t t h a t the k e r n e l i s t o r s i o n - f r e e . I f the group G i s a b e l i a n , the c o m m u t a t o r s a r e a l l t r i v i a l , and the F u c h s i a n d e f i n i n g r e l a t i o n r e d u c e s to j u s t X l X 2 " ' ^ k = T h e r e f o r e X i ^ = i . i y . 2 * " i . i - l £ i + l ' " . v 4 c » a n d s o must d i v i d e the l e a s t common m u l t i p l e o f { m( 1) , m( 2 ) , . . . , m( i - 1 ) , m( i + l ) , . . . , m ( k ) } . T h i s c o n d i t i o n on the p e r i o d s o f the g r o u p F , n e c e s s a r y f o r t h e r e to be an e p i m o r p h i s m h : F -> G w i t h t o r s i o n - f r e e k e r n e l , w i l l be c a l l e d the L . C . M . c o n d i t i o n . M a c l a c h l a n p r o v e s the f o l l o w i n g t h e o r e m . T h e o r e m 2.31: L e t F be a F u c h s i a n g r o u p . The c o m m u t a t o r s u b g r o u p [ F , F ] o f F i s a F u c h s i a n s u r f a c e g r o u p i f and o n l y i f the 79 p e r i o d s m(i) o f F s a t i s f y the L . C . M . c o n d i t i o n . The next r a t h e r t e c h n i c a l r e s u l t w i l l be needed to s k e t c h the methods o f t h i s p a p e r . Theorem 2.32: In an ( a b e l i a n group G l e t x ^ , . . . , x q be e l e m e n t s s u c h t h a t X l x 2 . . . x q = 1 a n d x x p ( 1 ) = x 2 p ( 2 ) = . . . = x q p ( q ) = 1. T h e n t h e r e a r e e l e m n t s y , , . . . , y t s a t i s f y i n g y ^ y 2 . . . y t = 1 a n d y , w ^ = y 2 w ^ 2 ^ = . . . = y t W ^ t ^ = 1 t h a t g e n e r a t e t h e s a m e s u b g r o u p as x , , . . . , x q ; and i n a d d i t i o n , w ( i ) d i v i d e s w ( i + l ) , w h i l e ( 1 - l / w ( D ) + . . . + ( 1 - l / w ( t ) ) < ( 1 - l / p ( D ) + . . . + ( 1 - l / p ( q ) ) . Now f o r M a c l a c h l a n ' s main a r g u m e n t . Case I: L e t G be a g i v e n a b e l i a n group w i t h i n v a r i a n t s n( l ) , . . . , n ( r ) w h e r e n ( i ) d i v i d e s n ( i + l ) . C o n s i d e r t h e c l a s s FQ o f F u c h s i a n g r o u p s w i t h o r b i t genus 0, t h a t i s w i t h s i g n a t u r e ( 0 , k ; m ( l ) , . . . , m ( k ) ) , whose q u o t i e n t by a t o r s i o n - f r e e n o r m a l subgroup i s G. D e f i n e the s u b c l a s s F^ o f FQ t o be t h o s e F i n FQ such t h a t F / [ F , F ] = G. That i s , a b e l i a n i z a t i o n o f , g r o u p s i n F^ g i v e s the g r o u p G . Take a group F i n F j , h a v i n g g e n e r a t o r s x ^ , . . . , x k and s i g n a t u r e ( 0 , k ; m ( l ) , . . . , m ( k ) ) , w i t h h : F -> G the a b e l i a n i z i n g homomorphism. G i s t h e n d e f i n e d by t h e g e n e r a t o r s x^ = h ( x ^ ) , 1 < i £ k , w h e r e m ( i ) t h e o r d e r o f x^ i s m ( i ) , a n d t h e r e l a t i o n s x^ ^ = 1 , 1 < i < k , and ^ . . . x ^ = 1. The v a l u e o f the q u a n t i t y we w i s h to m i n i m i z e i s t h e r e f o r e - 2 + (1 - l / m ( l ) ) + . . . + (1 - l / m ( k ) ) . B u t by t h e t e c h n i c a l theorem o f M a c l a c h l a n ' s a b o v e , G has g e n e r a t o r s y j , . . . y t w i t h t h e r e l a t i o n s y , u ( 1 ) = y 2 u ( 2 ) = . . . = y t u ( t ) = y l y 2 . . . y t = 1, 80 a n d a l s o , u ( i ) d i v i d e s u ( i + l ) , a n d ( 1 - l / u ( l ) ) + . . . + ( 1 - l / u ( t ) ) < ( 1 - l / m ( l ) ) + . . . + ( 1 - l / m ( k ) ) . We may t h e r e f o r e u s e t h e g e n e r a t o r s y ^ , . . . , y t i n s e e k i n g the minimum - a r e a f u n d a m e n t a l d o m a i n f o r F. S i n c e u ( i ) d i v i d e s u ( i + l ) , the l e a s t common m u l t i p l e o f u ( l ) , u ( 2 ) , . . . , u ( t - l ) i s u(t - l ) , a n d s o t h e L . C . M . c o n d i t i o n i m p l i e s t h a t u ( t ) = u ( t - l ) . B e c a u s e o f t h e r e l a t i o n y ^ y 2 . . . y t = 1, G i s a c t u a l l y g e n e r a t e d by y ^ , y 2 > » - > y t : - i w i t h t h e r e l a t i o n s y ^ u ^ ^ = 1, 1 < i _< t - 1 . T h e r e f o r e y^,y2> — > y t - i a r e t h e c a n o n i c a l g e n e r a t o r s o f G , a n d s o t - 1 = r and u ( i ) = n ( i ) . H e n c e , t h e m i n i m u m we s e e k i s - 2 + ( 1 - l /u(D) + . . . + ( 1 - l / u ( t ) ) = - 2 + ( 1 - l / n ( l ) ) + . . . + ( 1 - l / n ( r ) ) + ( 1 - l / n ( r ) ) . The m e a n i n g o f t h i s r e s u l t i s t h a t the minimum c a n be computed f r o m the c a n o n i c a l r e p r e s e n t a t i o n o f the a b e l i a n group G as the d i r e c t sum o f c y c l i c g r o u p s . M a c l a c h l a n next shows t h a t the minimum i s the same f o r g r o u p s F i n F Q \ F ^ . The f i n a l s t e p i s t o f i n d the minimum - a r e a f u n d a m e n t a l d o m a i n f o r F u c h s i a n groups F f o r c a s e s I I and I I I , where F a g a i n has q u o t i e n t G, and t h e n to t a k e the minimum o v e r a l l t h r e e c a s e s . The n a t u r e o f t h i s r e m a i n i n g work i s s i m i l a r to the work I have p r e s e n t e d h e r e . L e t ' s l o o k a t the f i n a l r e s u l t s . C y c l i c g r o u p s w i l l not e n t e r i n t o the d i s c u s s i o n , s i n c e e v e r y c y c l i c g r o u p o b v i o u s l y has B u r n s i d e g e n u s e q u a l t o 0. 81 Here we p r e s e n t the genus o f the f o u r n o n - c y c l i c groups o f o r d e r l e s s t h a n 10: g e n u s B ( Z 2 X Z 2 ) = 2 g e n u s B ( Z 2 X Z^) = 1 g e n u s B ( Z 2 X Z 2 X Z 2 ) = 3 g e n u s B ( Z 3 X Z3) = 4. Now f o r the g e n e r a l r e s u l t . L e t G be an f i n i t e a b e l i a n g r o u p w i t h | G | > 10 , a n d l e t G = Z n ( 1 ) X Z n ( 2 ) X . . . X Z n ( r ) , w h e r e r > 2 a n d n ( i ) d i v i d e s n ( i + l ) f o r a l l i . C a s e 1: ( r = 2) g e n u s B ( G ) = 1 + ( | G | / 2 ) ( 1 - l / n ( l ) - 2 / n ( 2 ) ) . In the n e x t c a s e , p i s a p a r a m e t e r and the minimum i n d i c a t e d i s t a k e n o v e r p s u c h t h a t 0 < 2p < r . Case 2: ( r e v e n , r > 2) g e n u s B ( G ) = 1 + ( | G | / 2 ) m i n { 2(p - 1) + (1 - l / n ( l ) ) + (1 - l / n ( 2 ) ) + . . . + (1 - l / n ( r - 2p)) + (1 - l / n ( r - 2p) ) }. I n the l a s t c a s e , p i s a p a r a m e t e r and the minimum i n d i c a t e d i s t a k e n o v e r p s u c h t h a t 0 < 2p < r . C a s e 3: ( r o d d ) g e n u s B ( G ) = 1 + ( | G | / 2 ) m i n { 2 (p - 1) + ( l - l / n ( l ) ) + (1 - l / n ( 2 ) ) + . . . . + (1 - l / n ( r - 2p)) + (1 - l / n ( r - 2 p ) ) > . 82 3) G l o v e r and S j e r v e ' s work The t h i r d m a j o r a c h i e v e m e n t i n c a l c u l a t i n g t h e B u r n s i d e genus o f g r o u p s , a f t e r the v e r y e a r l y work l a t e i n the l a s t c e n t u r y , and a f t e r M a c l a c h l a n ' s s o l u t i o n o f the p r o b l e m f o r a b e l i a n g r o u p s , i s the d e t e r m i n a t i o n o f the genus o f the g r o u p s P S L 2 ( q ) by G l o v e r and S j e r v e . T h e g r o u p P S L 2 ( q ) , w h e r e q = p n , a n d p i s a p r i m e , i s t h e p r o j e c t i v e s p e c i a l l i n e a r g r o u p o f 2 X 2 m a t r i c e s o v e r the f i n i t e f i e l d G F ( q ) . In a 1985 paper ( [7 ] ) e n t i t l e d " R e p r e s e n t i n g PSL^Cp) on a R i e m a n n s u r f a c e o f l e a s t g e n u s , " G l o v e r and S j e r v e f i r s t c a l c u l a t e the genus o f P S L 2 ( p ) » We now d i s c u s s t h a t p a p e r . R e c a l l t h a t P S L 2 ( Z ) i s the m o d u l a r g r o u p , o r the p r o j e c t i v e s p e c i a l l i n e a r group o v e r the i n t e g e r s . I t has a p r e s e n t a t i o n o f t h e f o r m P S L 2 ( z ) = < S , T : S 2 = ( S T ) 3 = 1>, w h e r e S = and T = / ^ j . I t i s known t h a t r e d u c i n g c o e f f i c i e n t s mod p g i v e s an e p i m o r p h i s m h : P S L 2 ( Z ) -> P S L 2 ( p ) , a n d , t h e r e f o r e , the f o l l o w i n g p r e s e n t a t i o n : P S L 2 ( p ) = < A , B , C : A 2 = B 3 = C p = ABC = 1, . . . >, w h e r e A = h ( S ) , B = h ( S T ) , a n d C = h ( T _ 1 ) . From the g i v e n p r e s e n t a t i o n f o r P S L 2 ( p ) , i t i s c l e a r t h a t t h i s g r o u p i s a l s o the e p i m o r p h i c image o f the F u c h s i a n t r i a n g l e g r o u p T ° ( 2 , 3 , p ) . T h e r e f o r e , as u s u a l , we have a s h o r t e x a c t s e q u e n c e : 1 -> k e r ( f ) — i — > T ° ( 2 , 3 , p ) — f —> P S L 2 ( p ) -> 1 and P S L 2 ( p ) a c t s on the s u r f a c e H / k e r ( f ) as a group o f c o n f o r m a l a u t o m o r p h i s m s . The genus o f t h e s u r f a c e H / k e r ( f ) i s known to be 1 + ( p ( p 2 - D / 4 X 1 / 6 - 1 / p ) , w h e r e p ( p 2 - l ) / 2 = | P S L 2 ( p ) | . 83 T h e r e f o r e the B u r n s i d e genus o f PSLjCp) i s l e s s t h a n o r e q u a l to t h i s u p p e r b o u n d . I d e a l l y , G l o v e r and S j e r v e w o u l d l i k e to f i n d a l l t r i p l e s ( r , s , t ) f o r w h i c h t h e r e i s an e p i m o r p h i s m f : T ° ( r , s , t ) -> P S L 2 ( p ) , d e t e r m i n e the genus o f H / k e r ( f ) f r o m the Riemann - H u r w i t z f o r m u l a , and t h e n m i n i m i z e o v e r a l l s u c h t r i p l e s . T h i s p r o c e d u r e w i l l y i e l d t h e genus o f the group P S L 2 ( p ) o n l y i f the c l a s s o f groups T ° ( r , s , t ) i s s u f f i c i e n t l y l a r g e f o r the p u r p o s e , i f i t i s u n n e c e s s a r y to c o n s i d e r any o t h e r g r o u p s F whose q u o t i e n t i s P S L 2 ( p ) . T h i s p r o b l e m d i d n ' t a r i s e f o r M a c l a c h l a n i n d e t e r m i n i n g the genus o f a b e l i a n g r o u p s because o f the e s s e n t i a l l y u n i q u e p r e s e n t a t i o n o f s u c h g r o u p s . In the t h e o r e m to f o l l o w , G l o v e r and S j e r v e p r o v e t h a t the o r i e n t a t i o n p r e s e r v i n g t r i a n g l e g r o u p s are i n d e e d the o n l y ones needed to c a l c u l a t e the genus o f P S L 2 ( p ) . F i r s t o b s e r v e t h a t i f S = H / k e r ( f ) i s a s u r f a c e a r i s i n g when P S L 2 ( p ) i s r e a l i z e d as a q u o t i e n t o f some t r i a n g l e group T ° ( r , s , t ) , t h e n the s u r f a c e S / P S L 2 ( p ) has genus 0 and the p r o j e c t i o n map p: S -> S / P S L 2 ( p ) i s a b r a n c h e d c o v e r i n g w i t h e x a c t l y 3 b r a n c h p o i n t s . The c o n v e r s e , g i v e n i n the f o l l o w i n g t h e o r e m , i s what i s r e q u i r e d . Theorem 2.33: I f S i s a R i e m a n n s u r f a c e o f l e a s t genus f o r P S L 2 ( p ) , t h e n the s u r f a c e S / P S L 2 ( p ) has genus 0 and the p r o j e c t i o n map p: S -> S / P S L 2 ( p ) i s a b r a n c h e d c o v e r i n g w i t h e x a c t l y 3 b r a n c h p o i n t s . The p r o o f o f t h i s i m p o r t a n t theorem i s i n t e r e s t i n g and i n s t r u c t i v e , and we w i l l p r e s e n t i t h e r e . 84 p r o o f : L e t G be the group PSL2(p) . Suppose g i s the genus o f the s u r f a c e S i n the t h e o r e m and h i s the genus o f the s u r f a c e S / G . I f the p r o j e c t i o n p: S -> S / G has b r a n c h p o i n t s y^, . . . , }^ w i t h o r d e r s m ( l ) , . . . , m ( k ) ; t h e n the R i e m a n n - H u r w i t z f o r m u l a g i v e s 2 - 2g = |G|(2 - 2h - (1 - l / m ( D ) - . . . - (1 - l / m ( k ) ) ) , o r g = 1 + ( | G | / 2 ) ( 2 h - 2 + (1 - l / m ( l ) ) + . . . + (1 - l / m ( k ) ) ) . S i n c e we have a l r e a d y s e e n t h a t our group a c t s on a s u r f a c e o f genus 1 + ( | G | / 2 ) ( l / 6 - 1 / p ) , we get the i n e q u a l i t y 2h - 2 + (1 - l / m ( l ) ) + . . . + (1 - l / m ( k ) ) < 1/6 - 1 / p . T h i s i n e q u a l i t y c l e a r l y f o r c e s e i t h e r h = 0 o r h = 1. The a s s u m p t i o n that h = 1 q u i c k l y l e a d s to a c o n t r a d i c t i o n . The i n e q u a l i t y i s not s a t i s f i e d i f even one m(i) >^  2. Hence t h e r e a r e no b r a n c h p o i n t s . Then PSL2(p) a c t s w i t h o u t f i x e d p o i n t s on S, and S/PSL2(p) has genus 1. T h e r e f o r e , PSL2(p) i s r e q u i r e d to be a q u o t i e n t o f Z X Z , the f u n d a m e n t a l group o f the t o r u s . But the p r o j e c t i v e s p e c i a l l i n e a r g r o u p s a r e non - a b e l i a n and Z X Z i s a b e l i a n . T h e r e f o r e h = 0 and S / P S L 2 ( p ) = S 2 , the 2 - s p h e r e o r s u r f a c e o f genus 0. Our i n e q u a l i t y i s now: - 2 + (1 - l / m ( l ) ) + . . . + (1 - l / m ( k ) ) < 1 /6 - 1 / p . The number o f b r a n c h p o i n t s h e r e must be k < 4. I f k = 0, then the p r o j e c t i o n p: S -> S / G i s an u n b r a n c h e d c o v e r i n g , and the a u t o m o r p h i s m g r o u p o f t h e c o v e r S i s P S L j C p ) . But t h i s i s c l e a r l y i m p o s s i b l e s i n c e S i s a s p h e r e and has a t r i v i a l f u n d a m e n t a l g r o u p , w h i l e we know f r o m c o v e r i n g s p a c e t h e o r y t h a t the 85 a u t o m o r p h i s m group o f the c o v e r i s i s o m o r p h i c to a q u o t i e n t o f a s u b g r o u p o f the f u n d a m e n t a l g r o u p o f the base s p a c e . Suppose k = 1. Denote the b r a n c h p o i n t y , . Then the — 1 9 p r o j e c t i o n p: S \ {p ( y p ) -> S \ {y^} i s a r e g u l a r u n b r a n c h e d c o v e r i n g , and the a u t o m o r p h i s m g r o u p o f the c o v e r i s a g a i n P S L 2 ( p ) . T h i s a l s o c a n n o t be , because S \ { y i > , the s p h e r e minus a p o i n t , has a t r i v i a l f u n d a m e n t a l g r o u p . T r y k = 2, a n d l e t t h e t w o b r a n c h p o i n t s be y ^ a n d y 2 < T h e n p : S \ p iy\)Y2^ S \ {yjj }^} *- s a g a i n a r e g u l a r u n b r a n c h e d c o v e r i n g . Now a s p h e r e w i t h two p u n c t u r e s has a f u n d a m e n t a l g r o u p w h i c h i s i n f i n i t e c y c l i c , and so the f u n d a m e n t a l group o f the c o v e r , an i n f i n i t e group w h i c h i s a s u b g r o u p o f Z , must a l s o be Z. S i n c e P S L 2 ( p ) a c t s on the c o v e r , whose f u n d a m e n t a l group i s Z , we get the s h o r t e x a c t s e q u e n c e 1 -> Z -> Z -> P S L 2 ( p ) -> 1. B u t t h i s i s a c o n t r a d i c t i o n b e c a u s e P S L 2 ( p ) i s not an e p i m o r p h i c image o f Z . I f t h e r e are f o u r b r a n c h p o i n t s , k =>4, t h e n our b a s i c i n e q u a l i t y i s 2 - l / m ( l ) - l / m ( 2 ) - l / m ( 3 ) - l / m ( 4 ) < 1/6 - 1 / p , w h i c h c o u l d be s a t i s f i e d by m ( i ) = 2, 1 £ i <^  4 ; b u t t h i s g i v e s g = 1 and i t i s known t h a t the a c t i o n genus o f P S L 2 ( p ) >^  2. T h e r e f o r e we must have 2 - l / m ( l ) - l / m ( 2 ) - l / m ( 3 ) - l / m ( 4 ) > 2 - 1/2 - 1 / 2 -1/2 - 1 / 3 = T / 6 , w h i c h c o n t r a d i c t s our b a s i c i n e q u a l i t y . Hence t h e r e a r e e x a c t l y t h r e e b r a n c h p o i n t s . # K n o w i n g t h a t the c l a s s o f t r i a n g l e g r o u p s T ° ( r , s , t ) i s a s u f f i c i e n t c l a s s f r o m w h i c h t o f o r m e p i m o r p h i c images o f P S L 2 ( p ) , G l o v e r and S j e r v e d e t e r m i n e a l l t r i p l e s ( r , s , t ) > 2 so t h a t t h e r e 86 e x i s t s an e x t e n s i o n 1 -> k e r ( f ) — i — > T ° ( r , s , t ) — f — > P S L 2 ( p ) -> 1 and t h e n c a l c u l a t e the genus g o f the group as g = m i n <1 + ( p ( p 2 - l ) / 4 ) (1 - 1 / r - 1 / s - l / t ) > , w i t h the m i n i m u m t a k e n o v e r a l l s u c h t r i p l e s ( r , s , t ) . T h e r e i s a g r e a t d e a l o f d i f f i c u l t g r o u p t h e o r y i n v o l v e d i n d e t e r m i n i n g e x a c t l y w h i c h t r i p l e s ( r , s , t ) a l l o w an e p i m o r p h i s m f : T ° ( r , s , t ) -> P S L 2 ( p ) , as a f u n c t i o n o f p , so we w i l l o m i t the work. We w i l l now p r e s e n t the genus o f the g r o u p s P S L 2 ( p ) , p ^ 5. F i r s t the f o l l o w i n g d e f i n i t i o n i s n e e d e d . I f p > 13 , l e t d = m i n < k : k > 7 and e i t h e r k d i v i d e s ( p - l ) / 2 o r k d i v i d e s (p+ l ) /2 }. Theorem 2 . 3 4 : g e n u s B ( P S L 2 ( p ) ) i s : a) 1 + ( p ( p 2 - l ) / 4 ) ( l / 6 - 1 /p) i f p = 5 , 7 , 1 1 b) 1 + p ( p 2 - l ) / 4 0 i f p > 13 , p = +1 ( m o d 5 ) , p n o t +1 ( m o d 8 ) , and d > 15 c ) 1 + p ( p 2 - l ) / 4 8 i f p > 13 , p n o t +1 ( m o d 5 ) , p = +1 ( m o d 8 ) , and d > 12 d) 1 + p ( p 2 - l ) / 8 0 i f p > 13 , p = +1 ( m o d 5 ) , p = +1 ( m o d 8 ) , and d > 9 e) 1 + ( P ( p 2 - l ) / 4 ) ( l / 6 - 1 / d ) i n a l l o t h e r c a s e s . In the 1987 paper ( [ 8 ] ) "The genus o f P S L 2 ( q ) , " G l o v e r and S j e r v e e x t e n d t h e i r p r e v i o u s r e s u l t s by d e t e r m i n i n g the B u r n s i d e genus o f P S L 2 ( q ) , where q = p n , f o r a l l p r i m e s p and a l l e x -p o n e n t s n . 87 R e c a l l t h a t T ° ( r , s , t ) i s an o r i e n t a t i o n - p r e s e r v i n g group o f s y m m e t r i e s o f the s p h e r i c a l p l a n e P i f 1 / r + 1/s + 1 / t > 1, o f the e u c l i d e a n p l a n e P i f 1 / r + 1/s + 1 / t = 1, and o f the h y p e r b o l i c p l a n e P i f 1 / r + 1 / s + 1 / t < 1. As b e f o r e , t h e t a s k i s t o f i n d t h e t r i p l e s ( r , s , t ) o f p o s i t i v e i n t e g e r s such t h a t t h e r e e x i s t s an e p i m o r p h i s m f : T ° ( r , s , t ) -> P S L 2 ( q ) f r o m the t r i a n g l e group T ° ( r , s , t ) , w i t h k e r ( f ) t o r s i o n f r e e . P S L 2 ( q ) then a c t s on the q u o t i e n t s u r f a c e P / k e r ( f ) and the R i e m a n n -H u r w i t z f o r m u l a g i v e s the genus o f the s u r f a c e : g e n u s ( P / k e r ( f ) ) = 1 + ( | P S L 2 ( q ) I / 2 ) ( 1 - 1 / r + 1/s + 1 / t ) . M i n i m i z i n g o v e r the r e l e v a n t t r i p l e s ( r , s , t ) g i v e s g e n u S g ( P S L 2 ( q ) ) . To s t a t e the r e s u l t s , we need the f o l l o w i n g d e f i n i t i o n s . (1) I f p=2 l e t d=d R d e n o t e the l e a s t i n t e g e r k s a t i s f y i n g the c o n d i t i o n s (a) k>7 (b) k d i v i d e s one o f the i n t e g e r s 2 n - l , 2 n + l (c) k does not d i v i d e any o f the i n t e g e r s * 2 m - l , 2 m + l where m d i v i d e s n and K m < n . (2) I f p i s an odd p r i m e , l e t d=d n (p) d e n o t e the l e a s t i n t e g e r k s a t i s f y i n g the c o n d i t i o n s (a) k_>7 (b) k d i v i d e s one o f the i n t e g e r s ( p n - l ) / 2 , ( p n + l ) / 2 (c) k does not d i v i d e any o f the i n t e g e r s ( p r a - l ) / 2 , ( p m + l ) / 2 where m d i v i d e s n and l<m<n (d) k does not d i v i d e any o f the i n t e g e r s p m - l , p m + l where 2m d i v i d e s n and l<2m<n. T h i s d e f i n i t i o n o f the i n t e g e r d does not a p p l y when q = 2 , 3 , 4 , 5 , 7 , 9 , 1 1 . T h e c a s e s q = 5 , 7 , 1 1 a r e c o v e r e d by T h e o r e m 2 .34 . F o r the o t h e r s we h a v e : 88 Theorem 2 . 3 5 : (a) I f q = 2 , 3 , 4 then genusg( P S L 2 ( q ) ) = 0. (b) I f q = 9 then genusg( P S L 2 ( q ) ) = 10. G l o v e r and S j e r v e p r e s e n t t h e i r main r e s u l t s i n the f o l l o w i n g t h e o r e m s by i n d i c a t i n g w h i c h ( r , s , t ) t r i p l e s d e t e r m i n e the B u r n s i d e genus o f P S L 2 ( q ) from the R i e m a n n - H u r w i t z f o r m u l a . Theorem 2.36: The B u r n s i d e genus o f P S L 2 ( q ) i s d e t e r m i n e d by a ( 2 , 3 , d ) t r i p l e i n the f o l l o w i n g c a s e s : (1) p=2 and n>3. (2) p>2 and n = 4 , 5 , 7 , 8 , . . . . Theorem 2.37: Suppose p>3. The B u r n s i d e genus of P S L 2 ( p ) i s d e t e r m i n e d by an ( r , s , t ) t r i p l e , where ( r , s , t ) i s g i v e n as f o l l o w s : (1) ( 2 , 4 , 5 ) i f p = +2 (mod 5) and d>9. (2) ( 3 , 3 , 4 ) i f p = +1 (mod 5 ) , p = +3 (mod 8) and d>12. (3) ( 2 , 3 , d ) i n a l l o t h e r c a s e s . Theorem 2.38: Suppose p>2. The B u r n s i d e genus o f P S L 2 ( p ) i s d e t e r m i n e d by an ( r , s , t ) t r i p l e , where ( r , s , t ) i s g i v e n as f o l l o w s : (1) ( 2 , 5 , 7 ) i f p = +1 (mod 5 ) , p = +2,+3 (mod 7) and d>105. (2) ( 2 , 3 , d ) i n a l l o t h e r c a s e s . Theorem 2.39: Suppose p>2. The B u r n s i d e genus o f P S L 2 ( p 6 ) i s d e t e r m i n e d by an ( r , s , t ) t r i p l e , where ( r , s , t ) i s g i v e n as f o l l o w s : (1) ( 2 , 5 , 7 ) i f p = +2 (mod 5 ) , p = +2, +3 (mod 7) and d M 0 5 . (2) ( 2 , 3 , d ) i n a l l o t h e r c a s e s . 89 C o n e l u s i o n At t h i s p o i n t we s h o u l d d e m o n s t r a t e t h a t the B u r n s i d e genus and the C a y l e y genus o f a group need not be the same. C o n s i d e r the group G = X w i t h the p r e s e n t a t i o n G = < x, y : x - y = x y x y = 1 >. A C a y l e y g r a p h f o r G i s shown b e l o w , c l e a r l y embedded i n a s u r f a c e o f genus 0. T h e r e f o r e , genus^C Z 2 X Z ^ ) = 0. *1 H o w e v e r , the B u r n s i d e genus o f t h i s group i s not 0, b e c a u s e i t i s n o t i s o m o r p h i c t o Z n , D n , A ^ , S ^ , o r A ^ . A. T . W h i t e , i n h i s p a p e r "On the Genus o f a G r o u p " ( [ 3 3 ] ) , i n f o r m s us t h a t the a c t i o n genus o f Z 2 X Z ^ i s 1 and t h a t B u r n s i d e ' s work shows i t . The two d e f i n i t i o n s o f group genus a r e f u n d a m e n t a l l y d i f f e r e n t , and t h e r e are few e s t a b l i s h e d r e l a t i o n s b e t w e e n them. I t i s known, h o w e v e r , t h a t Theorem 2 . A O : g e n u s c ( G ) < g e n u s B ( G ) . T h i s f o l l o w s f r o m the next t h e o r e m , due to T u c k e r , w h i c h shows t h a t i f a group G a c t s on a s u r f a c e t h e n a C a y l e y g r a p h f o r G embeds i n the s u r f a c e . 90 Theorem 2.41: Suppose G a c t s on the o r i e n t a b l e s u r f a c e S p r e s e r v i n g o r i e n t a t i o n . L e t g be the genus o f S / G and l e t p: S -> S / G h a v e k b r a n c h p o i n t s o f o r d e r s m( 1 ) , . . . , m ( k ) . T h e n a C a y l e y g r a p h Kp(G) f o r G embeds i n S, where the p r e s e n t a t i o n P i s t h a t o f a F u c h s i a n g r o u p w i t h s i g n a t u r e ( g , k ; m ( l ) , . . . , m ( k ) ) , w i t h a d d i t i o n a l r e l a t i o n s . I n a d d i t i o n , T u c k e r has made the e m p i r i c a l o b s e r v a t i o n t h a t the C a y l e y genus o f t e n seems to be a b o u t h a l f the B u r n s i d e genus . L e t ' s c o n s i d e r t h a t . Theorem 1.55 shows t h a t i f a f i n i t e group G i s a p r o p e r q u o t i e n t o f T ( 2 , 3 , 7 ) , t h a t i s , i f the image o f the T ° ( 2 , 3 , 7 ) s u b g r o u p i s an i n d e x 2 s u b g r o u p o f G , t h e n genus^CG) = 1 + |G 1/168. Theorem 2.28 showed t h a t i f G i s a f i n i t e q u o t i e n t o f T ° ( 2 , 3 , 7 ) , t h e n g e n u s B ( G ) = 1 + | G | / 8 4 . I f a f i n i t e group G were s i m u l t a n e o u s l y a q u o t i e n t o f T ° ( 2 , 3 , 7 ) and a p r o p e r q u o t i e n t o f T ( 2 , 3 , 7 ) , t h e n we w o u l d have (genus^CG) - 1) = ( l / 2 ) ( g e n u S g ( G ) - 1). In f a c t , h o w e v e r , the two c o n d i t i o n s a r e m u t u a l l y e x c l u s i v e , w h i c h c a n be s e e n as f o l l o w s . L e t f: T ° ( 2 , 3 , 7 ) -> G be an e p i m o r p h i s m . S i n c e any s u b g r o u p o f i n d e x 2 i s a l w a y s a q u o t i e n t g r o u p as w e l l , t h e r e i s a l s o an e p i m o r p h i s m g: T ( 2 , 3 , 7 ) -> T ° ( 2 , 3 , 7 ) . Then the c o m p o s i t i o n o f t h e s e two i s an e p i m o r p h i s m f o g : T ( 2 , 3 , 7 ) -> G , b u t t h e i m a g e o f T ° ( 2 , 3 , 7 ) i n G i s t h e w h o l e o f G , not an i n d e x 2 s u b g r o u p . T h e r e f o r e , a l l we c a n d e t e r m i n e i s the next t h e o r e m . Theorem 2.42: I f G i s a f i n i t e q u o t i e n t o f T ° ( 2 , 3 , 7 ) t h e n ( g e n u s c ( G ) - 1) > ( 1 / 2 ) ( g e n u s g ( G ) - 1) . 91 T h i s i s the p r o b l e m t h a t p r e v e n t s T u c k e r ( [ 2 9 ] ) f r o m d e t e r m i n -i n g the C a y l e y genus o f A n e x a c t l y : Conder has shown t h a t A f l i s a q u o t i e n t o f T ( 2 , 3 , 7 ) , but i t i s c e r t a i n l y not a p r o p e r one. Theorem 2 .43 ( T u c k e r ( [ 2 9 ] ) g e n u s c ( A n ) > n ! / 3 3 6 + 1, n>168. T h e r e i s a n o t h e r d e f i n i t i o n o f group genus i n the l i t e r a t u r e t h a t i s r e l a t e d to the B u r n s i d e g e n u s . I f a g r o u p G a c t s on an o r i e n t a b l e s u r f a c e S o f genus g , p o s s i b l y w i t h o u t p r e s e r v i n g o r i e n -t a t i o n , but does not so a c t on any s u r f a c e o f l e s s e r g e n u s , t h e n the s y m m e t r i c genus o f G i s d e f i n e d to be g. T h i s i s T u c k e r ' s t e r m i n o l o g y , and the o n l y work done on the s y m m e t r i c genus o f a group i s T u c k e r ' s d e t e r m i n a t i o n t h a t the s y m m e t r i c genus o f S n i s e x a c t l y n ! / 1 6 8 + 1 f o r n >^  168. He uses C o n d e r ' s r e s u l t t h a t S n i s a q u o t i e n t o f T ( 2 , 3 , 7 ) f o r n > 168. Because A n i s the o n l y i n d e x 2 subgroup o f S n , i t must r e p r e s e n t the o r i e n t a t i o n p r e s e r v i n g p a r t o f the a c t i o n o f S n on a s u r f a c e o f genus n ! / 1 6 8 + 1. T h e r e f o r e , ^ n i s a q u o t i e n t o f T ° ( 2 , 3 , 7 ) f o r n > 168. H e n c e , g e n u s B ( A n ) = n ! / 1 6 8 + 1 f o r n > 168. But t h i s i s Theorem 2.29 a g a i n . 92 REFERENCES [I ] S. R. A l p e r t and J . L . G r o s s , Components o f b r a n c h e d c o v e r i n g s o f c u r r e n t g r a p h s , J . C o m i n a t o r i a l T h . B, 20 (1976) , 2 8 3 - 3 0 3 . [2] A. F. B e a r d o n , The g e o m e t r y o f d i s c r e t e g r o u p s , S p r i n g e r -V e r l a g , New Y o r k , 1983. [3] W. B u r n s i d e , T h e o r y o f g r o u p s o f f i n i t e o r d e r , C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , E n g l a n d , 1911. [4] M. D. E . 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T . W h i t e , On the genus o f f i n i t e a b e l i a n g r o u p s , E u r o p . J . C o m b i n a t o r i c s 1 (1980), 243-251. [16] H. L e v i n s o n and B. M a s k i t , S p e c i a l e m b e d d i n g s o f C a y l e y d i a g r a m s , J . C o m b i n a t o r i a l T h . B 18 (1975) 12-17. [17] A. M. M a c b e a t h , The c l a s s i f i c a t i o n o f n o n e u c l i d e a n c r y s t a l l o g r a p h i c g r o u p s , C a n a d . J . M a t h . 6 (1967), 1192-1205. [18] C. M a c l a c h l a n , A b e l i a n g r o u p s o f a u t o m o r p h i s m s o f compact Riemann s u r f a c e s , P r o c . L o n d o n M a t h . Soc . 15 (1965), 699-712. [19] W. Magnus, N o n e u c l i d e a n t e s s e l a t i o n s and t h e i r g r o u p s , A c a d e m i c P r e s s , New Y o r k , 1974. [20] W. Magnus, A. K a r r a s s , and D. S o l i t a r , C o m b i n a t o r i a l g r o u p  t h e o r y , I n t e r s c i e n c e , New Y o r k , 1966. [21] H. 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