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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.,  A THESIS  St.  Olaf  SUBMITTED  CoLlege,  1966  IN P A R T I A L F U L F I L L M E N T  THE REQUIREMENTS  FOR THE DEGREE OF  MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES (Department  We a c c e p t to  THE  this the  of  thesis  required  UNIVERSITY OF April David  Mathematics)  as  conforming  standard  BRITISH COLUMBIA 1989  Norman Medalen,  1989  OF  In  presenting  degree  this  at the  thesis  in  University of  partial  fulfilment  of  of  department  this or  thesis for by  his  or  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  representatives.  an advanced  Library shall make  it  agree that permission for extensive  scholarly purposes may be her  for  It  is  granted  by the  understood  that  head of copying  my 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  Introduction Chapter  One:  Basic  The C a y l e y  The C a y l e y The Genus  Color of  The C a y l e y  Genus  Graph of  of  a Group  a Group  a Graph  Genus  Groups  of of  Maschke,  Chapter  '  p.  2  p.  2  p.  9  p.  15  p.  21  p.  22  1  Graph Theory  1)  jf *  a Group Cayley  Genus  Proulx,  and  0,  1,  Tucker  2:  2)  The Q u a t e r n i o n  Group Q  p.  27  3)  The G r o u p s  (Z  p.  28  4)  H a m i l t o n i a n Groups  p.  29  5)  Theoretical  p.  30  6)  Applications  n  p.  34  7)  The Q u a s i -  H u r w i t z Bound  p.  37  p.  40  p.  40  p.  42  p.  44  p.  47  Two:  Fundamental  2 m  ) : n  Results  The B u r n s i d e  The F u n d a m e n t a l  Covering  G =  to  S  n  Genus  and A  of  a Group  Group  Polygons  Spaces  Fundamental  Groups  and C o v e r i n g  fl  Spaces  Branched  C o v e r i n g Spaces  p.  49  p.  50  Riemann S u r f a c e s  p.  51  Abstract  p.  53  p.  55  p.  55  p.  58  The H y p e r b o l i c P l a n e  p.  58  Fuchsian  Groups  p.  61  Fuchsian  Groups  p.  68  T h e A c t i o n on a R i e m a n n S u r f a c e  p.  70  Existence  p.  72  p.  73  Groups  p.  75  work  p.  75  p.  77  p.  83  Conclusion  p.  90  References  p.  93  The Riemann -  Hurwitz Formula  Riemann S u r f a c e s  Automorphisms  of  a Riemann  Surface  The H u r w i t z Bound The B u r n s i d e  Improved Genus  of  Genus o f  a Group  and R i e m a n n  the  Burnside  Surfaces  Genus  H u r w i t z Bounds  Formulas  for  1)  Burnside's  2)  Machlachlan s  3)  Glover  1  work  and S j e r v e ' s  Hi  work  Introduction  This a  article  f i n i t e group  We c a l l  them  is  that the  i n t e n d to  sible  the  Each  The  genus  group. the  surface  be  of  of  ideas  surface  paper  included,  of  tries  to  and  is  only  to  genus  of  a group,  classifying  finite  groups  not  but  of  least the  isomorphic of  the  of  by s h o w i n g  the  that  homeomorphisms  of  to  for  such  a group  surface  is  a  genus.  C a y l e y genus  determined  conformal  by e m b e d d i n g  possible  a broad mathematical We a t t e m p t  i n the  convey  determined  by  defined  a  to  group.  present  and  is  is  The genus  the  employed  known t h e o r y  but  of  literature.  respon-  d e f i n e d as  a group of  d e f i n i t i o n of group genus.  significant  genus  was  for  a group  genus.  genus  and m e t h o d s  the  mathematician  surface  then  of  to  genus,  in a  is  genus  lesser  Burnside  This each  of  of  mathematical  Burnside  either  a scheme  group  isomorphic  finite  the  The C a y l e y g e n u s  the  this  is  surface the  for  i n the  and  that  two d e f i n i t i o n s  definitions.  criteria.  The B u r n s i d e  group  any  suggest  original  graph  the  prominent  d e f i n i t i o n gives  topological certain  are  of  C a y l e y genus  we do n o t for  a survey  research,  results. a  sense  of  to  show  most  of  of  the  a n d most  Proofs  are  the  nature  1  background  for  the  sporadically of  the  research.  Chapter  Basic  One:  Graph  Theory  A graph  K is  called  the  K,  f  a  T h e C a y l e y Genus o f  and set  a triple  vertices  is  a  {u,v>  of  K,  function  of  two  a Group  (V(K) ,E(K), f ) , is  E(K)  defined  a  finite  o n E(K)  vertices  or  to  = {u,v}  or  f(e)  u and v ,  and  where  an  set  that  V(K) i s a f i n i t e called  maps  ordered  the  edges  of  to  either  an e d g e e  pair  (u,v)  set  of  two  vertices. If  either  incident  with vertices  vertices. If  two  to  v.  The v e r t i c e s  edges  If  either If  two  multiple  The v,  f(e)  or  have  or  loops  d(v)  of  say  also  the  has  f(e)  = (v,v),  allows  a vertex  is  d(v)  is  said  to  regular  of  degree  We w i l l  make  the  directed  edges  Suppose graphs.  K  The c o r r e s p o n d i n g s h o u l d be 2  They are  1  1  to  going  from u  directed.  they  is  are  edges,  called  a  called and a  =  r  of  for  edges  all  incident  vertices  v,  then  r.  definitions  definitions  for  for  graphs  graphs  with  no  with  obvious.  = (V(K ),E(K ),f ) said  e.  adjacent.  edge,  edge e  number  If  edges.  of  edges.  the  following basic  is  adjacent  endpoints are  edge  multiple  twice.  directed  the  edges  every  t h a t edge e  u and v a r e  e is a d i r e c ted  and m u l t i p l e  be  that  called  loop counting  graph  we s a y  same e n d p o i n t s ,  A multigraph  allows  we  i n common,  digraph  the  = (u,v),  v are  that  = {v,v}  edges  degree a  we s a y  graph  edges.  pseudograph  the  = (u,v),  A directed  loop.  u and  h a v e an e n d p o i n t  f(e)  If  with  f(e)  be  2  and K  isomorphic  2  if  = (V(K >,E(K >,f ) are 2  there  2  2  exist  2  bijective  two  functions fjCe)  p : ViK^  = {u,v},  isomorphic  then  graphs,  Theorem vertices  We a l s o morphism,  If  <_ d  morphic  n  is  of degree have  e  n  d  ^  =  regular  (P^ ^  d  v  of degree  f  o  r  a  1  1  1.3:  r and  2. ^ 2 '  t  n  e  n  r.  the f o l l o w i n g n e c e s s a r y soon  observe  t o c^  of K  f o r some  it  of  have  2  condition  that  L e t the v e r t i c e s  and t h e v e r t i c e s  = (V(K ),E(K ),g),  c  c  is  degrees  c  iso-  sufficient.  degrees  c^  n,  f o r graph  not  have  1 < i £  which  c  edges,  £  2  then  d^ < d . . . £  i s not  is  The complement defined  to have  and has V ( K ) = V ( K ) , a n d E ( K ) = { C  i f and o n l y  2  <^  c . n  iso-  union  if U K  C  f(e) £ <u,v} 2  f o r any e i n E(K)  of graphs  and K  2  is  of K is the no l o o p s  or  e  =  : g(e)  }.  the graph  K which has  = V ( K j ) U V ( K ) and E(K) = E C K ^ U E(K >. 2  which  E  h  the graph K = (V(K),E(K), f ) .  The  the  T  2  The V(K)  are  2  to K «  multiple {u,v}  I f K j and K  if  2_ K<>.  ^  If  t h o u g h we w i l l  Let K  ] [  that,  2  2  Let K  d^ i s not e q u a l  graph  E ( K ) such  X  f (q(e)) = {p(u),p(v)>.  1.2:  Corollary ...  2  v in K^.  is regular  2  V ( K ) a n d q : E ( K ) ->  we w r i t e  1.1:  Corollary K  ->  1  - « o f two graphs 2  has V(K) = VCKj) U V(K > 2  set  = {  join K  2  e  of edges E , d e f i n e d :  f(e)  Consider  = <u,v>; graphs  K  i s the graph K = (V(K),E(K), f)  and E(K) = ECKj) U E ( K ) U E , 2  to have  u in V(K ), X  JL  =  (vCKj)  X K  of  edges,  v i n V(K >  }.  2  ,E(K ), f^ 1  The C a r t e s i a n  product  (V(K),E(K),g)  w h i c h has V(K) = VdCj)  2  no m u l t i p l e  and K  t h e two g r a p h s  2  is  3  satisfies  = (V(K ) , E ( K ) , f ) . 2  the graph  X V ( K ) and E(K) = 2  where  2  K =  2  {  e =  (ej,e2) 1)  )  satisfying  the  ej. a n d e2 a r e e d g e s  following  conditions.  i n the j o i n K^+I^ =  (V(K +K ),E(K +K ),f) 1  2)  2  1  2  i f f(e^) = {upUj} and f(e2) = ( v p V 2 ) ,  then  and  are  or u  2  = v  We w i l l graphs 1) are  2  2)  *  ^l^ ^ e  =  and show  u  v  ^  o  r  s  o e m  edge  e  in E(K^).  important well-known  examples.  A bipartite  graph K has V(K) =  and f o r e v e r y  u i n V , and v i n V  P  ^ l> l^  now g i v e t h e d e f i n i t i o n s o f some  non-empty,  we h a v e  a n c  is  the graph  0  edge  Vj U V, 2  w h e r e V± a n d V  e i n E(K), w i t h endpoints  u and v ,  .  which  is a  path  of  length n-1.  P  3)  C  n  , n>3,  i s the graph w h i c h i s a eye l e o f l e n g t h n, a  4  2  closed  4)  K  is  n  path.  the  possible  complete  K  is  n  on n v e r t i c e s ,  the  graph  with  the  4  :  totally  disconnected  graph  on n v e r t i c e s :  E(K. )  empty.  K. :  6)  K  m  is  n  the  complete  bipartite  graph:  K. _ = K + K_ m, n —m —n  '2,3-  7)  Q  n  Q n x  is  all  edges.  K  5)  graph  the  n-cube  = K, XQ . 1 ^n-i.  defined for  n >  recursively  as  1.  5  = K  2  and  n  is  Here vertices below but  observe  with matching  are  the  we w i l l  the  first  2  The a u t o m o r p h i s m of  example:  a n c  bipartite  An a u t o m o r p h i s m  morphisms  degrees  graphs is  that  of  group  K under  of  X Ky the  graph K,  graphs  and s t i l l  *  and  a  two  may h a v e be  K is  denoted  an  are  is  regular  of  of  Shown degree  3,  not.  isormorphism  A(K),  same number  non-isomorphic.  Both  second  the  consists  of  f:  K ->  all  K.  auto-  composition.  As e x a m p l e s  of  the  automorphism  6  group of  a graph,  we  give C ,  here the  n  those  cycle  of  A(K )  2)  A(C ) = D  n  in  this  R  that  that  then  points  the  acts  the  G is  interested  The  The  by  to  in  for  2n.  developments,  is  space  the  points  can if  of  the of  the  we h e r e and  graph at  the  occur  the  introduce  recurring  a  role  at  under  is  well  for  g(u)  edges  in R  3  so  any  of  thought  of  from E u c l i d e a n  homeomorphisms proper, action  Gv = {  g(v)  A(K) a r e  the  we of  or  m i d p o i n t s of  pair  v.  7  of  are  G,  denoted  : g in G points  }.  on K  edges.  vertices  3-  o f K.  some n o n - i d e n t i t y g i n  every =  as  and  u,  G}.  G is  empty.  on K i f that  K is  the  vertices  = p for  if  inherited  necessarily  vertices  as  a g r a p h K, o r  that  actually  subgroup G of g(p)  3  be c o n s t r u c t e d  We n o t i c e  not  in R  vertices  g r o u p A(K) o f  a vertex v of  realized  vertices.  topology  A(K),  Fix(G)  i n G such  can  of K are  set a  may be  representing  graph.  in K :  transitively g  lines  with  orbit  freely  some  graph  automorphisms  points  G acts there  and  letters.  p l a y an i m p o r t a n t  except  on t h e  F i x ( G ) = {p  act  and  a subgroup of  fixed  fixed  said  on n v e r t i c e s ,  group on n  automorphism  Gv a n d d e f i n e d a s  given  later  will  intersect,  topological  If  as  graph  , the d i h e d r a l g r o u p of o r d e r  an a b s t r a c t  of  subgroups,  space,  symmetric  e  - • In a d d i t i o n ,  no e d g e s  a  n  anticipate  that  a set  We s a y  as  n  and t e r m s  respectively.  its  complete  discussion.  as  that  t  n  to  Observe in  the  n  S ,  =  n  ideas  K ,  l e n g t h n.  1)  In o r d e r few  for  v  For in  G a  subgroup  correspondence  1-1  joins  of A(K),  the  with  orbits  Gv a n d G u i n K / G i f  In o t h e r single  words,  vertex,  identified  to  the and  a  example:  edges  single  edge.  the  and  or  that  are  solid  on  dotted  i n V(K),  group  of  solid  through  respectively  in in  orbit  is  a  are  S^,  an a n g l e  of  but  the  2fT/5  shown b e l o w .  identify  the  edge  i d e n t i f i e d to  graph  graph K ^ / Z ^ are  or  and a n  i n the  the  vertices  j o i n s v and u i n K.  Gv a r e  by r o t a t i o n s  quotient  edge  v  joining vertices  The a u t o m o r p h i s m  radians.  dotted  i n an o r b i t  the  acts  Gv,  g r a p h K / G has  and o n l y i f an edge  vertices  s u b g r o u p Z5 a l s o  Edges  the  quotient  quotient  to  a  single  graph.  OO As c a n be s e e n a  pseudograph, If G i s  (K,p), a  from this because  it  the  may h a v e  quotient  covering  the  of  K/G.  In  the  last  |G| = n,  the  covering  covering.  If  sheeted  covering.  We have  projection  is empty,  with  regular  a  5 -  graph  may a c t u a l l y  be  loops.  a s u b g r o u p o f A(K) and F i x ( G )  K together  regular  example,  map p:  sheeted  K ->  example, is  K/G, is p:  also  covering  8  then the  of  K ^ -> called  pair  said  to  be  K^/Z^ is an £  -  K 5 / Z 5 b y K^ i n  a  the  example.  The C a y l e y C o l o r  Let  G r a p h o f a_ G r o u p  G be a g r o u p and l e t  word W i n g i , g , . . .  i  2  belongs of  the set  l  ^2  =  A presentation  related,  generated  is  is said  finite.  proof:  Let {g ,g ,  will  g ,  l s  2  1 ^  =  ••• )  a n <  *  a  n  then  every  finite  d  call  element  {gpg ,... 2  every  this  be a p r e s e n t a t i o n  important  ,...,g  o f G.  H = < g,,...  G=  a presentat ion of  generated,  or  finitely  relations  i s both  finitely  G has a f i n i t e  presentation.  . . . } = G be t h e s e t o f g e n e r a t o r s  2  l  is  r e l a t i o n c a n be  * = 1.  and take  #  t o know how a g r o u p are deleted  is  changed  o r when  relations  Let g  )  i s an e x p r e s s i o n  or d e f i n i n g  group  in a presentation  G = <  f^  = 1, . . . t h e n we w r i t e  2  t o be f i n i t e l y  o f the form g ^ g j g ^  when g e n e r a t o r s added.  Every  be v e r y  ^  each  A  related.  1.4:  relations  ."  >  A f i n i t e presentation  and f i n i t e l y  1  •••  A relation  = 1, W  •••  =  2  where  n  o f G.  w o r d W.  by { g  Theorem  It  f o r G.  i f t h e number o f g e n e r a t o r s  respectively  are  as a w o r d i n g i , g  elements  e  2  2  from the r e l a t i o n s W  D  f^f ...f  gi~*>g ~^>  2  G is generated  < gl>&2' "' '  all  g , •  t h e f o r m W = 1 f o r some  deduced  G.  {g^,  t o be a s e t o f g e n e r a t o r s  If  ••• )  2  f i n i t e product  a  G c a n be e x p r e s s e d  said of  to  s  {g2,g ,  n  :  W  x  = ... = W  C o n s i d e r the  ,gi_!»  8i i,«».g +  m  = 1 >  presentation  n  =  U  l = —  9  =  U  m  =  1  >  s  where  generator  where each replaced  word Uj i s  by  Then H i s  the  the  subgroup  of  been d e l e t e d the  w i t h the  by  Then F  is  normal  subgroup. groups  •• W  n  G = < gp  Then G  the  X H =  where the  the  by K p ( G ) .  elements  elements  g^  only  gjh =  if,  imagined Thus order  :  2  to  and g g  the It  2  2  for  = W  m  that  for  for  for  W  =  1  the  Rj  let  the  . . .  all  i  those the  a  a directed  g r o u p F be  = W  >.  m + p  o f G by a  order  =  ...  1 >, ~  and j  G X H is  g h j g £ i  _  1  h j "  =  1  and g u a r a n t e e s  1  >,  that  o f H. There  is  graph of G w i t h  are  2  y.  1  product  in bijective  unique graph,  =  group G.  generator  of  2  ...  =...=  associated presentation  correspondence  vertices  h.  of  normal  a quotient  = R  for  2 and n g e n e r a t o r s  the  relations: =  m + 1  direct  =...=  and v  2  runs  some  g^  trivialized.  is  G,  is,  =  vertices v^  W  :  Cayley color  If  of  and  g^.  i n G , t h e n a n edge  p r o d u c e edges  Kp(G) i s  ...  with  has  o f G.  G,  Wj  holds  o f G commute  P a graph c a l l e d  the  2  P be a p r e s e n t a t i o n  denoted  with  h ,  presentation  relation  <g^>  w i t h added  = W  ••• :  2  1  been  set  presentations  g > h^,  K.  2  extra  Let  P  1  of  occurrence  where  element  . . .  image  g ,g ,...,h ,h ,..  generators  with  =  standard <  =  x  generating  g^ h a s  presentation  G and H h a v e  H  that  presentation  a homomorphic  Let  the  same  following  F = < gi,...,g  We s a y  g r o u p G/<g^>,  G generated  by t h e  f r o m the  w o r d Wj w i t h e a c h  identity.  quotient  Starting given  g £ has  corresponding  from v^  to  v  Each generator  2  if, h  and  is  color.  and  greater  if  P has  than  10  2,  r  generators  t h e n Kp(G)  of is  to  regular  of degree  2n + r .  A word W i n the g e n e r a t o r s paths, tion  i n K (G),  and a r e l a t i o n  p  of G  graph,  corresponds  o r t o a n edge  In accordance a  Cayley  color  E(K (G))  such  f(e)  1  p  that 2  to a c y c l e ,  many c y c l e s , like  K (G) = p  p : V ( K ( G ) ) -> p  f o r each  g^,  i f , and o n l y  g  if,  p  ([34],  and the c o l o r  p.25)  Theorem group G. graph  gives 1.5:  f(q(e))  =  generator  -> h,  1  graph  2  preserves  adjacency.  directed  White  theorem.  p  group  of  graph  for a  finite  automorphisms  of the  K ( G ) i s G. p  Displayed presentation generators  are  below  is  A^=<r,  the Cayley c o l o r s : r  = s  the permutations  by  (p(g )h,p(g )).  L e t K ( G ) be a C a y l e y c o l o r p  determined p  to each  of the next  Then A(K (G)) = G , the f u l l  is  p  corresponding  a proof  1.  V ( K ( G ) ) and q: E(K (G)) i n G and each  2  color  an a u t o m o r p h i s m o f  (V(K (G)),E(K (G)),f)  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 adjacency  =  -  many  presenta-  i n the C a y l e y  g^g^ *  p  in fact  1 i n the  with our previous d e f i n i t i o n ,  maps,  = (g h,g )  to a path,  o f the form R =  for a relation  graph  two b i j e c t i v e  corresponds  = (rs)  graph  f o r A^ w i t h the  = 1 >,  r = (12)(34)  where  and s =  1 1  the  (123).  A member redundant  if  of  it  generators.  a generating  set  c a n be w r i t t e n  A generating  as  set  is  for  a group G i s  a word  in  the  minimal  if  it  said  to  be  remaining has  no  redundant  generators. 2 example: g r o u p Zg•= fewer  The s e t  < x : x^  If  edges h is  = 1 >,  h is  colored  not  collection the  ,x  }  is  even  a minimal  t h o u g h {x}  generating is  set  for  the  a generating  set  with  elements. A generator  all  {x  3 -  h  redundant  i n Kp(G)  redundant, of  quotient  group  of  and o n l y  leaves  deletion  isomorphic  if,  a  of  disjoint  connected  all  edges  subgraphs,  G generated  if,  by  its  the  deletion  directed  colored each  of  graph.  h leaves  a  representing  generating  set  with h  trivialized. Given  a  presentation graph  has  Cayley color P,  runs  some  generator or  of H w i t h the  Hg^ t o  for  p  corresponding  from coset  Loops  K (G)  and a s u b g r o u p H o f  vertices  edge  graph  G,  to  coset  the  right  Hg  2  if,  a group  G with  Schreier .cosets  of  and o n l y  (right)  coset  H i n G,  and  if,  g^h = g  an  for  2  h o f G i n P.  m u l t i p l e edges  may be  i d e n t i t y element,  so  introduced  a Schreier  by t h e  coset  identification  graph  may be  a  pseudograph. In case H i s is  evidently  Since  the  graph  is  the  a normal  Cayley color  subgroup also  subgroup  just  H acts the  graph  on the  quotient  of  G, the  of  the  graph graph  Schreier  quotient  Kp(G),  the  K (G)/H. p  12  coset  graph  group G / H .  Schreier  coset  It  f o l l o w s from a previous  product  K  pQ)( ) G  x  H has v e r t e x s e t (83.84) o r (2)  i  g  2  K  °^  p  p  (1)  = g  = g^ and g^h = g^  example: presentations  Cayley color  V(K Q)(G)) X V(K ( )(H)),  either  f  p(2)^  definition  Here are are  Z  2  3  2  and g h 2  for  K(Z ),  3  = < y : y  7  = 1 >,  the  graphs  Cartesian  for  groups  and ( g p g ) 2  is  = g^ f o r a g e n e r a t o r  a generator K(Z ),  2  that  h in  G and  joined h in  to  ? > 2  P^.  and K ( Z ) X K ( Z > , 2  3  Z^ = < x : x  3  where  = 1 >,  the  with  the o  standard x  3  = yxy  presentation _ 1  x  _ 1  =  1  of  the  cross  product,  Z  >.  13  2  X Z-j = < y ,  x : y  =  White and  proves  the  following  theorem  in his  text  Graphs,  y^- y  color  Groups,  Surfaces. Theorem  of  1.6:  Let  K  p(i)(G)  a  a group G w i t h p r e s e n t a t i o n  P(2). =  Let  P be  K (i)(G) Here  K  X  P  the  standard  p ( 2 )  different  with  the  this  group  will  have  that  is  P(l),  ^  e  Ca  e  and a g r o u p H w i t h  presentation  different  Cayley color  standard  p(2)^  K  of G X H.  graphs  presentation  T h e n K ( G X H) p  (H).  we n o t i c e  very  n d  graphs.  presentation  isomorphic to  a Cayley color  presentations For  has  Zg,  graph  the  its  of  example,  graph  the  group Z  shown a b o v e .  presentation  isomorphic  a group can  to  as  C^,  X  2  But  a cyclic  the  give  cycle  since  group  of  length  six. Theorem Let  G be a  direct  1.7:  finite  product  where  m(i)  expression  m  of  Z  m(k)>  ^ra(i)  cyclic  is  unique. 1.8:  e  groups,  Let  = < g  h  denotes  a  v  e  t  K  n  e  P  (  and  G be a  presentation w  Then G can G = Z  for G  )  =  finite  each : g  i  C  directed  i  m  the m(l)  m  A b e l i a n Groups be e x p r e s s e d  Q ) X Z ( ) m  2  called  a b e l i a n group  group Z  ^ ^ 1  =  cycle  C  Then,  product  m(2) of  ^ ^ have a  m  1 >.  direct X  the  X  -  length  X  if  m(k)>  m(i).  14  Z  a ^ j,  m  k  This  rank o f G. with G =  Z  m  Q)  presentation P is  w  the  Z- (2)  Z ^ Q J X  C  as  X ... X  | G | = m( 1 ) m ( 2 ) . . . m ( k ) .  T h e number k i s  Let  form  standard X  of  X ... X Z ^ ^ .  2  the  a b e l i a n group.  divides m(i+l),  Theorem X Z ( )  The F u n d a m e n t a l Theorem o f  m  h  e  r  e  X  ...  T h e Genus o f Let discuss  us  a_ G r a p h now d e v e l o p  the  genus  Theorem sphere  of  1.9:  A closed  that  For S\K  are  said  edges  is  the  embedding  embedding F  be  to  be e m b e d d e d  regions  if  homeomorphic Theorem  the  to  every  of  is  region  open u n i t  Let  K be  triangulation vertices,  The  of  edges,  genus  vertices,  closed,  Euler's  Theorem  1.11:  edges, + F=  a surface,  is  a  theorem.  the  g,  finite  S if  it  is  g.  drawn  S,  the  components  of  said  a  embedding. S is  embedding is  to  be  a 2-cell,  that  pseudograph of  genus  faces  with a  g.  Let  2-cell  V,  E,  respectively.  and  Then  It  is  usually  w i t h V, E,  given  and F  for  a  being  triangulation.  graph K, genus(K), surfaces  genus(S) =  2 - 2 g .  of  Every  the  the  and  and  faces  of  surface  genus  orientable  w e l l known.  vertex.  surface  a connected  of  a  to  disc.  a surface  of  S,  in a surface  K in a surface  of  orientable  known as  is  a n d we w r i t e  a common  in a  faces  the  1.10:  number  or  V - E  all  o n l y at  a pseudograph  in a closed  This  2-manifold  needed  g !> 0.  a p s e u d o g r a p h K embedded called  ideas  f o l l o w i n g theorem  orientable  intersect  An e m b e d d i n g o f 2-cell  The  topological  o f S i s d e f i n e d t o be g ,  A pseudograph K is on S so  essential  a graph.  with g handles,  The g e n u s  the  is  the  minimum genus  i n which K can graph  has  a  be  among  embedded.  genus. 2  proof: 2-sphere.  S u p p o s e K has Attach  E edges.  E handles,  Embed e a c h  a n d embed  one  edge  15  vertex i n each  in S , handle.  the Then  genus(K)  < E.  #  Theorem if,  K  5  and o n l y  o  r  K  (Kuratowski)  1.12: if,  it  contains  torus.  The d i a g r a m  Together  genus(Ko  3)  =  minimal  if  surface  S,  proof: argument.  a graph  =  isomorphic  genus(K)  with  =  0,  either  If  1.13:  an e m b e d d i n g o f  theorem,  then  the  White  embedding  ([34],  K lies  on  that  each  handle  has  at  F  a  is  not  that  K in a  a connected  We may a s s u m e  face  is  this  3 in  shows  the  that  surface  of  genus  g is  called  g.  of  is  planar,  1.  genus(K)  Theorem  below  with Kuratowski's  An e m b e d d i n g o f  vertex  no s u b g r a p h  K is  3,3-  example:  a  A graph  the  p.54)  is  least a  a  loss  part one  2-cell.  K is  2-cell  gives  without sphere  graph  m i n i m a l l y embedded embedding.  the  following  of  generality  of  S,  not  edge  of  K meeting  It  must  informal that  each  on a h a n d l e ,  then  16  it.  contain  and  Suppose a  simple  in  closed  curve  C that  within  F,  a point.  has  least  at  Case  to  one  I:  cannot  be c o n t i n u o u s l y  Then the  surface  If  the  closed  curve  capped,  an e m b e d d i n g o f K i n a s u r f a c e  Case on  II:  S \ H , the  a l o n g C , and  C lies  may be c u t  contradicts  the If  when  assumption  the  curve  edges of K that  of  C lies lie  a  the  not  a sphere,  so  it  entirely two  of  holes  lesser  partly  that  genus  on a h a n d l e  on H may be  C on S \ H , g i v i n g an e m b e d d i n g o f  genus  again.  The f o l l o w i n g s k e t c h  on a h a n d l e ,  redrawn  to  K in a surface  illustrates  this  V - E + F = 2 - 2 g  T h i s theorem  result is  are  obtained.  H and  partly  follow of  the  lesser  possibility.  these two cases e x h a u s t the p o s s i b i l i t i e s ,  has been proved.  the  minimal embedding.  curve  Since  S is  entirely  handle.  surface  This  deformed,  the  theorem  a l l o w s us to use the E u l e r  formula  when a graph  i s m i n i m a l l y embedded  17  in a  #  surface.  The from  following  what  lower  bounds  we k n o w . T h e y a r e  Corollary  1.14:  If  very  K is  genus(K) Equality  holds  proof: so  that  Assume K i s  the of  size  its  n>3,  i n the  with  equality  follows  and o n l y  of  crude,  but  a connected > E/6  if,  -  a  a g r a p h K now of  graph,  V/2  K has  of are  +  some  follow  interest.  w i t h V ^> 3,  then  1.  triangular  m i n i m a l l y embedded  face  boundary.  of  1.15:  K has  be  Let  if  embedding.  in a surface  of  genus  g  holds  proof:  Corollary quadrilateral  If  K is  and o n l y as  1.16:  K is  embedding,  then  example: K^Q ^Q,  corollary  as  a complete that  edges  number + 4F  3  faces  faces  + ...  4  are  of  traversed  ,  of  size  and so  triangular.  in a n,  2E >_ 3 F ,  The  result  a connected  graph,  regions,  > E/4  -  K has  3,  and  the  then  V/2 + a  with V ^  1.  quadrilateral  embedding.  a connected  = E/4  -  bipartite  V/2 +  graph  having  a  1.  above.  From c o r o l l a r y  yields  the  of  above. If  Same  all  if,  genus(K) proof:  be  n  no t r i a n g u l a r  if,  Same  F  number  #  genus(K) Equality  the  T h e n 2E = 3 F  holding only  immediately.  embedding  a  embedding.  Corollary  graph  genus  V - E + F = 2 - 2 g .  Let circuit  if,  on t h e  the  1.14  above,  we  bipartite  graph,  has  graph  K^Q,  the  find  that  genus  complete  18  >_ 8.  graph  the The on  ten  same  vertices, graphs  has  will  In ([28]),  be  a  1980  T.  W.  Theorem vertices,  3  +  4F  reports  the  theorem  and F of  faces  1,  -  2g)  C l e a r l y 2E = 3 F +  ...  +  (n-l)F _ n  Theorem embedded  in  a  >  the  6 embeddable On  we o m i t ,  the to  Theorem  Let  surface  Therefore, d  his  other show 1.19:  these  Let  F  Genus"  concerning  technical  of  2-cell n  that  be  2  degree  tool  graph that  d with V  embedding  the  ¥ ^ = ¥  < (n-l)F + 4F^  3  number = 0.  If  from  this  in  of  an  faces  n is  of  any  of  a  of  in a given  g.  + ¥  2  + ...  inequality  is  The shown  graph  of  T h e n V < 6(2g  connected  surface  Tucker  l  +  ( D F ^ .  + F _ >) n  using  1  <  2E.  the  result  follows.  in  following  the  #  3.  regular  genus  (¥  + ...  2  Therefore,  2g).  inequality  K be  + (n-2)F  1  + ....  + n( ¥ -  1  this  number  hand,  of  work. graph  a  d > 6 and n =  1.18:  a Given  important  dV = 2E and F = E - V + ( 2 -  by l e t t i n g  genus  then  d)V + n(2  One a p p l i c a t i o n o f result  an  with  g.  Now F and E c a n be e l i m i n a t e d relations  is  understanding than  the  Groups of  a regular  genus  for  following results  K be  n -  4  " T h e Number o f  Let  the  value  sequel.  1.17:  proof: 3F  the  i n much o f  surface  -  in  first  greater  (nd/2  The e x a c t  repeatedly  n, w i t h  integer  paper,  E edges,  orientable size  obtained  The  uses  > 4.  Tucker  embeddings. Tucker  genus  gives  is  regular  degree -  d >  6  2).  graphs  of  degree  finite.  an e x p l i c i t  construction,  which  that: The n u m b e r  3 or 4 of a g i v e n genus  is  of  connected  regular  infinite.  19  graphs  of  degree  Tucker of  also  construction Theorem  reports  that  t o show  that:  1.20:  graphs paper  T. White  and r e p o r t s entitled  Bipartite  and P r o u l x  T h e number o f c o n n e c t e d  5 o r 6 o f a g i v e n genus Arthur  Gross  is  considerable  much o f h i s own w o r k  " T h e Genus o f R e p e a t e d  source  for several  Ringel  and J .  This  graphs  of  w o r k o n t h e genus  in this  field  Cartesian  paper  in a  Products  of White's  well-known results  W. T . Y o u n g s ,  regular  found a method  degree  infinite.  h a s done  Graphs" ([32]).  have  is  also  of  1970  of a  secondary  due t o t h e m a t h e m a t i c i a n s G .  sometimes  working together  ([25],  [26],  [27]). For earliest genus  of  example, genus  formulas  1.21:  {x}  = n  the f i r s t  Youngs  showed  construction  f o r a graph,  genus(Q )  ([25])  w i t h one o f t h e  when he e s t a b l i s h e d  if  n -  o f the f o l l o w i n g  the next  one i n 1 9 6 8 ,  genus(K  Theorem  1.23:  genus(K )  m  n  T h e g r a p h K^Q  o f 4.  Our e a r l i e r  8 and 4 r e s p e c t i v e l y own w o r k  n  )  i n 1955 t h e  Ringel  i n 1965 a n d R i n g e l a n d by a c t u a l  t o be m i n i m a l .  then  i n t h e same  4 ) , n > 2.  i n both cases  = < (n -  f o r these  -  A c c o r d i n g to White,  = < (m -  coarse  (n  results  o f a n e m b e d d i n g shown  example:  n - 3  1 < x £ n.  1.22:  White's  = 1 + 2  n  Theorem  a genus  Ringel  n  proved  of  credits  the n-cube Q :  Theorem  Let  White  2)(n 3)(n -  2 ) / 4 }, 4)/12  has a genus  estimates  >,  o f 16,  had g i v e n  m , n > 2. n > 3. a n d K^Q h a s  lower  bounds  two v a l u e s . paper g e n e r a l i z e s  20  the r e s u l t  for  complete  bipartite  embedding,  to  yield  Theorem genus(K  9 m  X K III  recursively  of  the  1.25:  Because  every  n -  t  special  h  e  c  v  -  2)(r  as  + s)  follows. n  Q(2m)  cycle  able  to  e  .> 2.  case n  l  as  o  graph J n  Let  J  Recall  n  is  + rs),  x  K  s  s  ) =  n  Let f  r  1 +  extend  his  Q(s)^  o  r  2  2 n _ 2  —  n  a bipartite  minimum  and  is  (  m  )  an  < 2m,  s <  2m.  =  K  -  n  graph, to  and  g  '  2  m (mn  methods  g  2).  though  not  Cartesian  products  of  is  of  not  vertices.  = m for  as  Let  the  n  genus(J ) n  a  J  follows. =  R  product  J  n  -  Let X C  1  2 m  = ( )  where  n  m( 1) m( 2 ) . . . m ( n ) .  i = l,2,...,n  we w i l l  denote  In  the  the  =  1 + 2  n - 2  (,n -  2)M  ( n )  ,  n >  2.  Group  genus(Kp(G))  the  g r o u p G.  genus(Kp(G))  denotes Then  over  the  the all  genus  of  C a y l e y genus presentations  the of  Cayley a  P of  group the  color G  is  group G,  genus^,(G). clear  a group,  definition  ( D  recursively  .  that  denoted  It  m  n  M ^ ^ denote  1.26:  Kp(G) f o r  the  2  when m ( i )  T h e C a y l e y Genus o f  genus  of  theorems.  = Q(s) _i  n  even  the  c  Theorem  graph  cube  genus(  W h i t e was  m(i)  graph J  construction  cycles.  2m(l)'  each  explicit  *  define Q(s)  We d e f i n e C  two  ) = 1 + m((m  Theorem  even  next  through  L j O  Generalize  complete,  the  again  1.24:  9  y  Ill  graphs,  who f i r s t  and we do n o t  originated  formulated suggest  this  by t h i s  with Cayley. A l l that  is  21  d e f i n i t i o n of terminology known i s  that  the that the  the  earliest [21]  significant  i n 1896.  1)  He c l a s s i f i e d ,  Groups of  Theorem  = 0 if,  and G  n  D  is  and A  is  n  In  a  alternating  1978  paper  G r o u p s , " V. K.  Proulx  Maschke groups  82 of  Maschke's  years  determination  Tucker,  who h a s  intricate  have  has  order that  of  greater  1.28:  Let  order than  3 < k + 5m/3  2, 3.  +  Maschke,  is  n  the  order  2n,  and  group  G = G^ X G2, where Z  is  n  of  the  genus  Tucker  G has  Gj = Z^ or  group of S  groups  Proulx,  A finite  entitled  This  of  the  her of  determined  C a y l e y genus  Theorem generators  1.  advanced  studied  finite  Z  integers  2  mod  symmetric  "Classiftcation  the  research  by a n n o u n c i n g  manipulations  Proulx G to  more  follows,  all  done by H . M a s c h k e  group,  group.  continuted  C a y l e y genus  considerably  of  ([24])  earlier  was  [21])  if,  group  C a y l e y genus  2:  A ^ , or A5.  dihedral  the  1,  and o n l y  n  the  0,  as  (Maschke,  = Z , D , S^,  2  n  genus  1.27:  genus^(G)  n,  work on t h e  is  the  work  groups  over  generators  one and  a necessary  the  begun  presentation  involving,  Toroidal by  types  same s p i r i t  C a y l e y genus  in d i f f i c u l t y , work,  program  i n the  of  of  0,  of  all  as  though  according  h u n d r e d pages  to  of  relations.  condition  for  a finite  group  1. a presentation m generators Then  2n <  for  G to  of be  of  the  order  finite 3,  toroidal  4.  22  group G have  and n g e n e r a t o r s it  is  necessary  k of  0.  two,  Corollary  1.29:  three,  four  Theorem  or  if,  satisfied.  t o r o i d a l g r o u p has  (Proulx,  1.30:  [ 2 4 ] ) Let  a presentation  and o n l y  if,  at  The cases  are  not  least  of  S  2  = R SRS  2)  G = <  R, S:  S  2  = R  3)  G = <  R,  S:  s  4)  G = < R, S:  s  5)  G = < RJ  S:  R  6  = S  2  = (RS) ,  6)  G = < R, S:  R  4  = S  2  = (RS)  7)  G = < R>  S:  R  4  = S  2  8)  G = <  R, S:  R  3  =  s  2  =  (RSRSR  9)  G = < R, S:  R  3  =  s  2  =  (RS) (R  =  (RSR  =  (RSR  10)  G = < R, S: R  11)  G = <  12)  G = z  R) S:  m 13)  some  G = < for  2  s  =  2  R  2  P  R  G = < R>  16)  G = < R, S:  S:  R  with  R  2  p,q >  P  p > 2<  ,= 1;  1, . . . >  s* 2  . . .  1  _ 1  =  S)  S) S)  2  3  = 1, .. . .  2  1, . ..  =  2  S),  _ 1  > >  = 1, .  3  1, ... >  =  =1,  ... >  > 3 _ 1  S  1, q >  p,q >  =  _  S)  _ 1  - 1  3  o f m,n  4  =  4  even o r d e r ,  R S 2  = 1 2;  ... >  = 1  2  1, . . . >  (RS)  2  = (RS)  = 2  1, . . . > = ( R  _  1  S )  >  23  2  = 1  for  with G is  cases  even  ... >  2  = RSR  s  1, R h a s  =  - (RS) (R  = s *  =  4  following  = 1, R a n d RS h a v e  2  2  ,5  2P  the  Then  1 >  3  2  integers  15)  integers  R S R ~' S  integers  R, S:  some  =  2  s  group G,  = 1 >  ( R S !I  = 3  SRS  =  g-c .d  n ,  G = < R, S: for  14)  Z  2  4  R  =  3  2  finite  mutually exclusive.  R, S:  3  the  one  G = <  x  presentation  G = < R, S: . . . >.  1)  m  a  generators.  g e n u S ( - , ( G ) > 0, h a v e toroidal  Every  is  17)  G = < R,  S:  R  3  = S  6  =  (RS)  2  =  1,  ...  >  18)  G =  < R,  S:  R  3  = S  3  =  (RS)  3  =  1,  . . .  >  19)  G = < R,  S:  R  3  = S  3  = RSRS  =  1  Theorem genus (G)  >  c  is  0,  toroidal  conditions  1)  1.31: have  if, is  (Proulx,  if,  at  R  _ 1  Let  a presentation  and o n l y  S  - 1  the  finite  G = < R,  least  one  >.  S,  of  S,  other  T: R  = S  2  reduced  2)  G = < R,  S,  T:  3)  G = < R,  S,  T:  for  k > 0,  T:  the  some  R R  = (RS)  2  relations = S  2  = S  2  = T  2  ...  = T  2  = T  2  are =  2  =  3  of  (RST)  =  2  Then G  = RSRTST =  2  2  = S  2  = T  2  =  (TRTS)  5)  G =  < R,  S,  T:  R  2  = S  2  = T  2  =  (RT) (ST)  6)  G = < R,  S,  T: R  (ST)  1,  ...  G = < R,  S,  T: R  (ST)  1,  ...  S,  T: R  3  =  G = < R, where  r  >  1,  >  > = (TS)  2 k  =  2  2  (RS) 2  =  2  =1,  (RS)  =  2  = S  2  = T  2  = (RS)  2  =  (RT)  4  =  2 k  =  ...  >  1  2  =  S  2  = T  2  = (RS)  2  = (RT)  6  =  r  = S  2  = T  2  =  2  = (RT)  2  = (ST)  1 >  >  >  G = < R,  S,  for  integers  some  ...  2  2 and p i s T:  =  3  >  R  =  ...  (RS)  T:  4  = (ST)  3  length, 1,  S,  9)  >.  with  following  even  G = < R,  8)  ...  (RT)  4)  7)  group G,  satisfied.  G = < R, all  [24])  - 1  R  2 p  (RS)  relatively  = S  2  p,q >  = T 1;  2  prime  = (RS)  ...  2  to  r,  = RTR  >  24  ... - 1  T  P  = 1  >  = (ST)  2 q  = 1  10)  G = < R, where  Theorem genus (G) is  is  = (ST)  2  even or  dihedral  subgroup  cases  The o r d e r  the is  think  x  it  historical the  genus, becomes  attempt as  2  is  96  =y =z is  Maschke,  completely  an  S,  the  to  S,  =  q  2  ...  U: ...  = S  1 where  direct  with  >.  = T  2  = 1  P  >  group G,  T,  U: R  *T = ( S T )  r,  finite  T,  (SU)  are  fair  find  single  that  for  class some  Then G = U  2  p =  product  Tucker  [29] is  to  all  Proulx,  has  shown  exactly  one  presentation  = (xy)  = (yz)  2  say of  =  2  2 and q  of  two  that  groups  defined,  groups.  these cases  the  normality  has  shown  we o m i t  the  Proulx and  that  index  all  such  them.  following:  group  = (xz)  the  for  is  G with  genus -,(G)  =  (  2.  is  3  research  =  8  logical  y(xz) y(xz) 4  beginning,  on a d e f i n i t i o n o f  however,  done.  for  even  25  This  =  4  1>.  and  often  group  genus  a given small value  and T u c k e r h a v e  impractical,  of  [31,] so  group of  concerning  unnecessary,  and a 2  a  infinite  However,  beginning, to  = RTR  prime  the  =  above  restrictions  There  2  = (RT)P  2  S  G = < R,  G = < R,  cases  Tucker  G is :  if,  be n o t e d  1.33:  of  G = <x,y,z I  it  restrictions  In a d d i t i o n ,  Let  L  X D^.  subgroups.  Theorem  relatively  The group G i s  i n some  additional  certain  = RSR  2  [24])  = (UR)  >.  should also  included of  2  G =  that  i n other It  and o n l y  groups,  = T  2  a presentation  = (TU)  2  = S  r  (Proulx,  have  if,  R  2 and p i s  p,q > 2  Notice while  >  T:  1.32:  toroidal  (RS)  r  > 0,  c  S,  of  the  approach  moderate  soon  values  of  the  genus,  genus the  of  and  each  the  that  are  ideas  theorem  will  of  contains  no  tends  are  1.34: 3.  If  corresponding  G be  P be a  a  relation  But  now y ( l )  its  inverse  has  of  assume  the  is  = y(3),  order  the  to  have  3,  giving  the  +1  this  research,  with  the  formulas  looking  their  the  one  type.  few g e n u s  at  proof.  group  for  classes  elementary The  whose  presentation  triangle,  or  -1.  or e l s e  P for  g is  a group  results  following  order for  of  genus  then  is  G.  1>  not  a  Then K ( G ) p  |G|, a  a group,  always  it  a  each the  g£  g^ 2  is  it  is  = g^ = g.  T h e r e f o r e g or  role  because exists.  26  in  important  genuS(-,(G),  does  minimal.  a  are  contradiction.  therefore  P is  is  H e n c e g^ = g  trivial.  generators,  presentation  where  any two o f  a group G gives  a group  there  play a c r i t i c a l  a minimum genus of  If  and 3 d i v i d e s of  =  redundant.  no r e d u n d a n t  C a y l e y genus  that  a  C a y l e y genus  a presentation  presentation about  = y(2)  y(i)  presentations  assumed  of  gj^^g2^^g3^"^  t h e n one o f them i s  be  i s o l a t i o n of  finite  Kp(G) c o n t a i n s  distinct,  if  Tucker's  the  minimal  and each  that  Recent  for  tool.  generator  definition  search  groups.  be  in  form of  triangles.  proof:  Since  the  We b e g i n by  involved  Let  Let  to  now some o f  be a u s e f u l  Theorem multiple  2,  of  work a n d  known.  that  takes  a class  Proulx's  us c o n s i d e r  of groups and  of  C a y l e y genus  Let  quickly  member o f  exception  group of  research  In  the to  then  such  #  note P may  a  arguments  no h a r m  to  Q,  a  2)  The Q u a t e r n i o n Group Q  Let  us  that  genus^CQ) =  minimal. It readily of  has  a  K (Q) p  Therefore, is  clear  order  2.  is  1.15  One o f  the  subgraph,  of  have order  graphs  = y is  at  at  three  least  least  two  two  of Q  is theorem.  generators.  of  order  that  the  It  then 2,  can  neither  because Q  Cayley  graph  T h e r e f o r e 2E >_ 4V = 3 2 .  that  presentations  P  last  generators  means  4.  that  by t h e  generators  This  we h a v e  q u a t e r n i o n group  We may a s s u m e  precisely  2.  the  P be a p r e s e n t a t i o n  no t r i a n g l e s ,  have  P has  9  y : x  presentation  if  standard  9  Q = < x,  P must  genus o f Let  p  of degree  for  8.  genus(K (Q)).  P cannot  regular  order  p  that  unique element  Cayley  K ( Q ) has  be s h o w n t h a t  corollary  a  the  n o n - a b e l i a n group of  such  is  now d e t e r m i n e  genus(Q)  of  the  >  E/4  -  V/2  +  By  1 >  q u a t e r n i o n group Q i s  9  = (xy)  >.  shown b e l o w .  A Cayley It  Kuratowski's theorem  is  color  ^.  verifies  graph  Since the  it  last  for  this  contains inequality:  ^  1  The d i a g r a m on t h e the  same  next  presentation,  page  shows  a Cayley  embedded i n the  color  torus.  27  a  s  Q is  non-planar.  with  1.  g r a p h o f Q,  Theorem genus  1.35:  Let  1,  genus (Q)  of Q i s Since  list.  In  Q is fact,  a a  toroidal  for  since  x and y b o t h have  Then  the  Cayley  group  it  of  Q by t h e order  must  type  be  14  included  may be  in  Proulx's  obtained  from  s u b s t i t u t i o n R = x and S = y  4,  as  can  be s e e n  f r o m the  the  ^,  Cayley  above.  3)  The Groups G =  The n e x t non-trivial  (Z  theorems,  genus  Theorem  formulas  1.36:  X G  Cayley  n  _p  If  G  n  and i f  color  graph  n  is G K  n  )  n  to for  (White, c  proof:  2 m  due  genus (G )  2  quaternion•group.  1.  presentation  presentation  Z  the  =  c  given  graph  Q be  A. T .  White,  certain  [34])  Let  = 1 + 2  n _ 3  (n  infinite  G -  defined recursively has  p(  G  n  the )  is  standard Q  n >  t  h  e  give  Q  the  classes  =  (Z ) .  4),  n >  as  G^ = Z  2  of  groups.  2. 2  P,  and G then  Therefore,  28  decidedly  Then  n  presentation n-cube.  first  n  = the  using  Rirtgel's  result  we  genus(G )  have  on  other  n generators,  has  1.15  genus (G ) White  has  graph  J ^ ^  copies  of  with G  n  last  > E/4  n  the  =  Theorem  1  already  finite  n  i n the  by  X  Z  -  n - 3  this  Let  ra  1.37:  groups,  are  if  -  i n Theorem  1.21,  4). G  graph  must  n  p(  K  3 does  the  2  result  2 m  .  us  graph Then  (White,  every  1.38:  and o n l y i f ,  quaternions, a  2  the  not  graph  G  n  have  at  least  |G| =  2 ,  )«  divide  theoretic  n  ( m )  + 1 = 1 + 2  n _ 1  as  follows.  the  by G ^  J ^ ^.  That  n  n  m  White  gets  n _ 3  ^ the  m  is,  the  n  result  m  -  4).  that  product  abelian  G ^  next  (n  Recall  Cartesian  denote  # the  of  n  group  ^ =  and  theorem.  [34])  )  =  1 +  =  for  few  2  n _ 2  (n  known r e s u l t s  H a m i l t o n i a n groups.  -  2)m  n  abelian a  in  of  2  , where  2  A  2  is  another  family  A n o n - a b e l i a n group G i s  by t h e  odd o r d e r ,  A -  concerning  normal  and M o s e r )  G = Q X Aj X A  all  G is  characterized  (Coxeter  A^ i s 1  a  subgroup of  completely  Theorem  which  (n  P for  b e e n d e f i n e d as  C2 -  l o o k now a t  Hamiltonian  if,  n - 3  reported  H a m i l t o n i a n Groups  Let's  groups  2  and b e c a u s e  V/2 + 1 > n 2  c  of  +  Therefore  genus (G  4)  n-cube  section,  -  ^  1  nV = n 2  Cayley color  G ^ ^ "  =  minimal,  cycle  minimal  ^  2E ^  the  presentation  generalized  has  m  n  h a n d , any  P is  the  of  n  no t r i a n g l e s .  from c  genus  genus(Q )  and s o  We may a s s u m e n  <  n  On t h e  Kp(G )  the  i n G.  next  G is Q is  theorem.  a H a m i l t o n i a n group the  and A  then  Hamiltonian  2  group is  (Z ) . • 2  29  n  of  a group  for  White his  gives  at Western  about  credit  Michigan University,  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  1.39:  genus (Q X (Z ) )  Theorem  1.40:  genus (Q X Z  genus  X  m  2  results  Ph.D. t h e s i s . n  X (Z ) )  Q X Z  m  = ( m ) ( n ) ( 2 ) + 1, m o d d .  n  2  n  X (Z ) , 2  m o d d , have  8  Cayley  order.  (Z ) ,  X  few  of  = n ( 2 ) + 1.  n  G i s a H a m i l t o n i a n group,  m(k)  Z  The g r o u p s  to t h e i r  Obviously,if X  2  c  1.41:  asymptotic  m(2)  c  a student  f o r p r o v i n g the next  Theorem  Corollary  Z  t o P. H i m e l w r i g h t , a p p a r e n t l y  where  n  each  then G = Q X Z  m  (i) X  m(i) i s odd and m ( i - l )  d i v i d e s m(i). Theorem Z  m(2)  2 (k  *"  X  + n -  n  m(k)  X  ^ 2 ^ '  Let  us now c o n s i d e r  of a group.  groups  genus (G)  n  c  construction  theoretical  o f erabeddings  kinds  of  ([12])  to  on the C a y l e y c a n be o b t a i n e d by  of Cayley c o l o r  graphs f o r  presentation.  ([11]).  uses  so-called  voltage  graphs  The p a r t i c u l a r  method  we w i l l  the o r d i n a r y v o l t a g e  entitled  results  b o u n d s on g e n u s ^ ( G )  method o f c o n s t r u c t i o n  i s known as  is asymptotic  Q) X  i f 1 < k < n-1.  some  the permutation voltage  paper  e  m  Results  t o G r o s s and A l p e r t  here  n  Important  with c e r t a i n  The  t  the H a m i l t o n i a n group G = Q X Z  l)(m(l)m(2)...m(k))  Theoret i c a l  explicit  to  Z  If G is  5)  genus  due  X  1.42:  graph c o n s t r u c t i o n ,  graph c o n t r u c t i o n ,  and i t  and i s use  as o p p o s e d  appears  in a  " 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 " b y J . L . G r o s s a n d T . W. T u c k e r .  30  Let E ( K ) ->  K be a d i r e c t e d G f r o m the  assignment graph. as  follows.  = E(K) X G. (k,g).  (k,g)  finite  of K to G is  K,  and  the  graph,  the  d e r i v e d graph  A vertex  is  i f edge  of  and G a  set  The v e r t e x set  Then  edge  edge  in G for  A second  graph  pair  group.  said  (K,f)  to  A mapping  be a v o l t a g e  an o r d i n a r y will  k of K runs  (v,g)  constructed  edge  f  as  voltage  now be  V ( K ) = V(K) X G and the  then denoted  and an  set  edge  E(K ) f  as  from v e r t e x u to v e r t e x v,  run from v e r t e x  (u,g)  f:  to v e r t e x ( v , g b ) ,  let  where  b =  f(k). In rules  this  situation  p((v,g))  define  the  = v for vertices  projection  (v,g)  p:  ->  and p ( ( k , g ) )  K by  the  = k for  edges  (k,g). Theorem ordinary n,  is  1.43:  voltage  The p r o j e c t i o n graph (K,f)  a regular  n -  p:  K  f  ->  with voltages  sheeted  covering,  K associated i n group  and G a c t s  with  G having  on  by  an |G| =  covering  transformations.  We w i l l  also  Theorem  1.44:  group G i f , some  the  and o n l y  if,  assignment  proof:  If  G has  ki,...,k . g  K = f:  Cayley  The next  for E(B)  Let  f(k^)  ->  = g-  is  some  with Cayley  a Cayley graph  bouquet  B of  graphs. for  circles  a  and  G. gp...,g , s  take a bouquet B of  be a v o l t a g e  assignment.  s  Then K  #  graph c l e a r l y  theorem  connection  graph K i s  generators  the d e r i v e d g r a p h . Every  following  The c o n n e c t e d  voltage  circles = B^,  want  the  arises  result  in this  we r e a l l y  way. want  31  from  the  theory  of  voltage  having be  graphs.  a certain  calculated  It  realized  as  in  a bouquet  G for  Theorem  g^g2...g  is  surface  d  of  so  cell.  There  for  G is  are  of  ( 388^> v  so  on.  back  at  circuit. circuit, produce n/d  9  any  to  (1/dj)  After  s  for  that  The C a y l e y g r a p h  with g^  |G| = n ,  is  d^  and  K for  can  is  assignment  generated the  G can  the  edge  by  order be  of  embedded  in  repeat  n/d^ 2-cells  (v,g).  at  the  a n d s o on  s  loops  (l/d  and e a c h  to  1  )).  i n the  2-  l o o p bounds a  2-  The C a y l e y g r a p h K  for  g  g +  embedded  embedding.  gp...,g  the the  been a  follows.  voltage s  loops.  (v,h)  with  The  the  s  g r a p h K = B^ this  goes  the d^  not  to  order edges  vertex  g^  of to  will  generator  generators.  g^,  we  and are  this last  clearly g  2  Finally,  32  to  (v,gg^g^),  i n c l u d e d i n the  Generator  construction,  the  from  and  traversed,  2-cell  In  runs  from here  procedure.  for  of  -  s  g r a p h B^  a vertex  by t h i s  (l/d )  An edge  continues have  -  a n d ns e d g e s . ,  Attach  again  this  e m b e d d i n g as say  ...  another  derived  n vertices  (v,g).  Start  a bouquet  generators  d^ e d g e s  vertex  2-cells,  genus  with a voltage  graph  -  intersects  the  vertex,  another  and  of  Cayley  faces  the  We c o n s t r u c t at  1 -  s+1  g r a p h K = B^ has  begin  -  no l o o p  identical f  of  a group G  genus  that  assignment  order  the  Suppose B i s  sphere  G.  associated  a group,  the  Then  1 + (n/2)(s proof:  G be  where  j.  g +  of  for  circles.  Let  g  Cayley graph in a surface  presentation  of  1.45:  gj,...,g  a  the  the  embeds  d e r i v e d graph  elements s  that  presentation  from  the  shows  will  produce  let  h =  with  g^g2—.gg.  the  Using  generators,  to  generator to  one g^  map t h e  loop  of  p:  the  using  loops  m a p p i n g to  interior the  same  we o b t a i n  The p r o j e c t i o n 2-cells  the  of  one  construction  n/d  ^  additional  B  takes  of  B,  all  n/d^ circuits  loop,  and  so  example  the  2-cell z ->  boundary of  on.  z .  the  Then  d  m o r p h i s m on s m a l l e n o u g h n e i g h b o r h o o d s o f  points  since  also  B is  surface. (n/d  g +  faces,  follows.  fact,  if  S is  is  a branched  d  and  has  is  orders  further which  the  surface 2 S ,  2-sphere  covering  G = < x,y,z  and  with  the  extended  a  of  the  homeo-  i n the  2-cells,  and  be e m b e d d e d i n a s  The genus  +  formula  s+1  branch  embedded and B  • projection  natural  :  ...  points  > be a g r o u p .  a (p,q,r) generating  of  yz,  we w i l l  xy,  and xz  G is  describe  G = < x,y  :  the  q respectively, relations,  are so  p,q,  having  from S to  S  2  orders  The g e n e r a t i n g  generating  G is  the  xy  group  i f x r  full  7  set  9  = y  =z  =1  and  respectively.  Without  triangle  T(p,q,r),  group  later.  . . . > be a g r o u p .  and e i t h e r  set and  called  in detail  } is called a ( p , q ; r ) °  further  be  interior  i n w h i c h K = B^ i s  called  relations,  Let <x,y  p r o d u c e d by  p is  9  the  the  s + l *  Let  <x,y,z>  of  (n/d^) + ... + (n/d )  ns e d g e s .  used  #  embedded i n the  l » * " >  B^ m u s t  so c o n s t r u c t e d  n vertices,  is  d  in a surface,  The s u r f a c e  p  In  embedded  each  Now p c a n  onto  we  2-cells.  ->  any d - s i d e d  canonical  g +  with h that  or  x  Then set  the  set  i f x and y have o r d e r s  ^y h a s  T°(p,q,r),  generating  order the  33  r.  Without  orientation  p  preserving Let (p»P;q)  C  subgroup  of  the  full  triangle  The s e t {  generating  o f x, y , a n d y x y x ~ *  q respectively.  y.  Tucker  ([31])  generating  set  Tucker  set i f the orders Notice  uses  behaves  ([31])  that  this  ways  the  image  of T°(p,q,r)  set  is said  generating an index  a  1.46:  o f x and "a  (p,p;q)  generating  observations  c  set."  and d e f i n i t i o n s . If  2 subgroup o f G, the g e n e r a t i n g A group G w i t h a  of T(p,p,q).  (Tucker,  Theorem generating  1.47:  set  If  (p,p;q)  the image set  of  is said  (Tucker,  (p,q,r).  [31])  f o r the f i n i t e  (Tucker)  -  c  T°(p,p,q)  t o be  c  c  < 1 + (|G|/4)(1 -  A p p l i c a t i o n s to S theory  will  n  allow  and A  {  x,y,z >  1/r). (p,q;r)°  Then 1/p -  1/q -  L e t the g e n e r a t i n g  g r o u p G be p r o p e r l y ( p , p ; q ) . genus (G)  1 / q -  L e t < x , y } be a  -  set  Then  1 / p -  group G.  < 1 + (|G|/2)(1  c  1.48:  L e t the g e n e r a t i n g  < 1 + (|G|/4)(1  genus (G) Theorem  [31])  g r o u p G be p r o p e r l y c  This  a (p,p,q)  a r e p , 2,  c  genus (G)  6)  he s a y s ,  a  (p,p;q) .  finite  finite  like  2 subgroup o f G, the g e n e r a t i n g  Theorem for  i s an i m a g e  is called  s e t i s an image o f T ( p , q , r ) .  t o be p r o p e r 1 y ( p , q , r ) . set  because,  the f o l l o w i n g  i s an i n d e x  x,y }  ^ i s the commutator  notation,  i n many  makes  yxyx  A group G w i t h a (p,q,r) g e n e r a t i n g  properly  T(p,q,r).  G = < x , y : . . . > be a g r o u p .  and  is  group  1/r).  set  {  x,y }  for a  Then 2/p -  1/q).  R  us t o c o n s i d e r  a few r e s u l t s  34  concerning  the  C a y l e y genus  alternating  of  groups  who u s e d  the  slightly  different  Let  genus (A )  (2)  genus (A )  <  proof :  If  c  c  n  (l,n-l)(2,n) . n is  Tucker, exact  Since  Groups  revealed  odd,  n >3 ,  T h e n x and  x  = y  n  n>4,  let  2  -  2  It  by  White,  by a  = (xy)  n _  3n + 1 ) ,  is  *  well  = 1,  will  White  soon of  S  see.  have  to  n>2.  known  the  result  1 < g <  f o r G w i t h an  +  1);  n odd,  n even,  A  Q  and  x  n  and x  sharpened has  for  the  2  2  and  =  (xy)  =  n  1.  y =  = (xy)  n _ 1  =  but its  | G| /168  considerably  actually  n >^  1980  and  irredundant  = y  n  n>4.  and y =  = y  1 1 - 2  n>3  1.  #  G e n u s , " by T u c k e r ,  If  4n + 2);  worked out  f o l l o w i n g theorem  1.51:  A  Tucker  n  alluded  -  x = ( 1, 2 , . . . , n - 2 )  been  of A , in both cases  i n the  3n  1.47.  a n d has  n  2  (1,2,...,n-l)(2,n)  x and y g e n e r a t e  already  Theorem  let  x =  of  a Given  -  2  y generate  results we  graph  cited  #  from Theorem  genus  of  1.47.  follows  We h a v e  color  n is  C a y l e y genus  Cayley  1 + ((n-2)!/4)(n  and y = ( 1 , 2 ) .  1 + ((n-2)!/8)(n  Then  as  are  and  w h i c h he o b t a i n e d  1 + ((n-1)(n-3)!/8)(n  even,  (l,2)(3,n).  These  results  1.45,  <  n  x = (l,2,...,n)  <  n  result  symmetric  1.50:  (1)  If  These  genus (S )  from Theorem  c  the  n  method.  n  Theorem  and A ,  n  Theorem  x and y g e n e r a t e S .  follows  S  on n l e t t e r s .  1.49:  proof:  The  groups  embedding of  Theorem  that  the  a  determined  lower  by the  bound f o r  the  168.  paper its  [28],  full  " T h e Number  significance  of  is  implications. + 1,  and i f K ( G ) i s p  generating  35  set  that  a Cayley  embeds  in  a  surface  G,  is  of  0 or  genus  genus (G)  > 1 then genus (G)  Corollary  1.53:  If  genus (G)  > 1 then  > 1 is  of  the  group  1.54:  c  >  c  c  The number o f  groups  | G | /168  | G | < 168(g of  -  +  1.  1).  a given Cayley  finite. arrives  difficult  at  this  theorem  manipulations  case  by c a s e  of  through e x c e e d i n g l y  generators  derivation.  and  We w i l l  intricate  relations  not  in a  attempt  to  reproduce  here. most  corollary  bounded some is  C a y l e y genus  If  The of  the  1.52:  laborious it  genus^CG),  Corollary  Tucker and  then  1.  Corollary genus  g,  above  useful 1.53, by  i n w h i c h the 168(g  Cayley color  a quasi -  important  f o r m u l a t i o n of  -  1),  d e f i n i t i o n of  the  second  half  of  this  there  is  another  bound  of  greater  1.51. 1 + than  1.46,  that  bound  on  Therefore,  a its  |G|.  o f H u r w i t z i n the  second  theorem  group  |G|/168  group  on t h e  But  far  bounds c o i n c i d e ,  for  the  is  probably  group G i s  genus  of  We w i l l  that  is  nineteenth  to  that be  i n which  168(g  -  an a n a l o g u e  century  which w i l l  seen  a surface  say  The theorem  reaching i n the  1) of  an  concerning  be d e a l t  a  (p,q,r)  (p,q,r)  1 +  following =  of  s h o w n , as  proper of  consequence  first  C a y l e y genus  T u c k e r has  with  the  result  with  in  paper.  C a y l e y genus  we h a v e  the  genus,  The f o r m u l a t i o n  one.  of  g r a p h Kp(G) e m b e d s .  the  But  order  g being  H u r w i t z bound on  theorem  Tucker's  result  corollary  for  Tucker's  places  a  a group  whose g e n u s  we h a v e  seen  generating  (|G|/4)(1  of  -  1/p  the  (2,3,7).  36  set -  case  1/q  lower is  i n Theorem has  an  -  1/r).  when t h e s e  upper  two  Theorem generating  M.  1.55:  set.  L e t the  Then  finite  genus (G)  i s an image  n  that 2  the image  in S  genus  the  gives  1.  entitled  that  f o r a l l n > 168.  then  Theorem  1.56 ( T u c k e r  We w i l l  discuss  alternating  order  "Generators  the symmetric  Tucker ([29])  is  t h e most  A  n  observes  , a subgroup  spectacular  group  of  index  of Cayley  that  specifically quotients those  Let degree  I:  n  =  n!/168  +  1,  n>168.  A  R  i n the c o n c l u s i o n s e c t i o n  of this  of  paper.  H u r w i t z Bound  of Irredundant  -  H u r w i t z upper bound o n t h e  entitled  " A R e f i n e d H u r w i t z Theorem  Cayley Graphs" [31].  groups  whose  order  greater  than  24(g -  i s near 1),  t h e 168(g -  must  kinds of t r i a n g l e  have  H e r e he i s 1) u p p e r  presentations  groups.  We w i l l  able  bound, as  now c o n s i d e r  in detail.  K ( G ) has d e g r e e p  4.  K p ( G ) be a n i r r e d u n d a n t  4 and genus^(G) g.  c  i n a long paper  of certain  results  Case  genus (S )  has i m p r o v e d h i s q u a s i  of a group  show  -  [29]):  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  groups  The Q u a s i  Imbeddings  genus  Groups" ([4])  (2,3,7)  formulas.  Tucker  to  1.55  +  a proper  i n a paper  o f the subgroup T ° ( 2 , 3 , 7 )  Theorem  n >  7)  for  of T(2,3,7)  G have  |G|/168  i n 1980  f o r A l t e r n a t i n g and S y m m e t r i c S  =  c  D. E . C o n d e r showed  group  Cayley graph  = g > 1 embedded  L e t X be t h e g e n e r a t i n g  f o r group G h a v i n g  i n an o r i e n t a b l e  s e t o f the p r e s e n t a t i o n  37  surface P.  of  Theorem 24(g  -  |G|(r  1), -  1.57:  X is either  6)/6r,  Theorem 24(g  -  Suppose  or X i s ( 4 , 5 ; 2 ) °  1.58:  1),  already  Now  that  half  II:  in a l l cases  let  2g -  with  2 =  2g -  if  | G |  >  2 =  | G | / 20.  four  involutions  when  the C a y l e y  | G | is actually  of the q u a s i  -  less  Hurwitz  then  |G| £  than  upper  or  bound  graph  has degree  equal  to  that  T u c k e r has  3 and g e n u s ^ C G )  o f genus  1)  then  g.  1.59:  Suppose  one o f  X is  b)  X is (5,5;2)  the  (5,2;4)°  if,  or  following (4,2;5)°  properly  2 =  d)  X is  and t h e r e  is  Theorem (7,2;3)°  length  a n d 2g -  (5,5;2)  a n d 2g -  of  i n an set  orientable  of  the  If  presentation.  | G |>  holds:,  2 >_ | G | / 4 0 ,  X is ( r , 2 ; 3 ) ° (3,2;r)°  f o r a group G  X has one i n v o l u t i o n .  c)  than  graph  = g > 1 embedded  and 2 g -  c  X is  generators  Cayley  L e t X be t h e g e n e r a t i n g  a)  only  3.  K p ( G ) be an i r r e d u n d a n t  Theorem -  1),  here,  K p ( G ) has d e g r e e  degree  surface  24(g  with  Then  proved.  Case  having  7 < r < 11,  I f X has two o r  4 and | G | > 24(g -  (3,r;2)°,  .  1).  Notice  84(g  X h a s no i n v o l u t i o n s  less  2 =  with equality  reduced  than  i f , and  c  |G|(ra  |G|/20  6)/6r relator  24 b u t none  of  involving length  14. 1.60:  a n d 2g -  If 2 =  | G | > 48(g -  1) t h e n  X is (3,2;7)°  |G| / 4 2 .  38  or  both  less  Theorem If (3,3,r)  with  r =  7 < r  Equality properly  |G|  (2,4,5)  > 24(g -  bound  <  168(g  -  or  "near"  unnecessary because n = 4,  1),  X i s ( 2 , 4 , 5 ) a n d 2g -  3 < p < 5,  or X i s  w i t h 2 g -  2=  theorem  |G|(r -  2 =  |G|/40,  or X  6)/12r.  i f , and o n l y  i f , X is  (3,2,r). cases,  as  when  less  the Cayley graph  than  or equal  to  has d e g r e e  the q u a s i  -  3 and  Hurwitz  expected.  to d e t e r m i n e  the upper  bound,  to consider  for these, gives  involutions.  1) t h e n  |G| is  In a t t e m p t i n g groups  three  X is (2,p,q),  in this  these  1),  of  1) t h e n  11,  holds  a l l of  X consists  4,5.  |G| > 48(g -  (3,2,r),  In  Suppose  |G| > 24(g -  If is  1.61:  these with  | G | _< 2 4 ( g -  presentations  |G| > 24(g  C a y l e y graphs  the fundamental  partial  -  of degree  inequality  1),  it  for  is  greater  than  4,  o f Theorem  1.17  ,  1).  39  with  Chapter  Two:  The B u r n s i d e  The F u n d a m e n t a l We now n e e d topological of  A path f:  [0,1]  a path at  ->  with  Group  to review  space.  the space.  This  We make  the i d e a o f the f u n d a m e n t a l  group  the  following  in a topological Y of  i s an i m p o r t a n t  the c l o s e d  algebraic  preliminary  interval  f ( 0 ) = f ( l ) = a i n Y.  into  Such a path  invariant  definitions.  space Y i s a continuous unit  group o f a  Y.  map  A closed  is said  path  t o be  is  based  a. The  paths  is  inverse  of a path  the path  The  identity  all  0 < s < Two  written Y such  paths  that  at  path  -  1)  s).  If  f and g are  the produc t o f f and g  0.5  the c o n s t a n t  and g ,  with  if  there  exists  two p a t h s  f as  = f(s)  0.5  < s < 1  path  i ,  f(0)=g(0)  d e f i n e d by i ( s )  and f ( l ) = g ( l ) ,  a continuous  and h ( s , l )  are homotopic, This  a point.  paths based  = f(l -  0 < s <  f  h(s,0)  We d e f i n e  h.  is  ^(s)  = a  for  1.  the o t h e r .  based the  path  f  then  (f(2s) I  fg =  f ~ g,  If into  f is  i n Y w i t h f ( l ) = g(0),  lg(2s  ~  Genus o f a G r o u p .  = g(s)  denote  [0,1]  for a l l s in  one c a n be c o n t i n u o u s l y  i s an e q u i v a l e n c e  We w i l l  map h :  are  relation  the e q u i v a l e n c e  homotopic, X [0,1] [0,1]. deformed  among c l o s e d class  ->  paths  containing  [f]. a binary  operation  a t a i n Y by l e t t i n g  The e q u i v a l e n c e  classes,  among t h e e q u i v a l e n c e [ f ] [g]  with  this  = [h]  i f , and o n l y i f , f g  operation, 40  classes  form a  group.  of  The  identity The  is  group  base point  in  and c o n s i d e r  groups  2.1:  is  _ i  [f  ].  the fundamental  group  o f the space Y,  with  ITCYja).  a few o f the b a s i c  some o f t h e m o s t If  |T(Y,a)  important  the t o p o l o g i c a l  and f T ( Y , b )  are  facts  about  fundamental  examples.  space Y i s pathwise  isomorphic  connected,  f o r a n y two p o i n t s  a,b  Y. proof:  L e t f be a c l o s e d  f r o m b t o a. between  Then  "fT(Y,a)  Because in  is called  now r e c o r d  Theorem the  and [ f ]  a, and i s denoted  We w i l l groups  [i]  t h e map [ f ]  path ->  b a s e d a t a , a n d l e t g be a  [gfg  defines  path  an i s o m o r p h i s m  and fT(Y,b).  of  this  our notation  theorem  we w i l l  and s i m p l y w r i t e  usually  suppress  the fundamental  the base  group  point  o f Y as  fT(Y). The  next  Theorem then  IT(X)  2.2:  is  Theorem  topological orientable  at  X, a n d Y a r e  homeomorphic,  fT(Y). group  the d i r e c t  of a product  product  of  space  Y X W,  fundamental  a few e x a m p l e s w i t h most  of  fundamental  attention  given  groups  to those  of of the  closed  surfaces. 2.4:  and hence  punctured  to  spaces  importance.  fT(W).  spaces,  Theorem cyclic  to  The f u n d a m e n t a l  isomorphic  look  are of primary  topological  isomorphic  IT(Y) X  Let's  If  2.3:  TT (Y X W), i s groups,  two r e s u l t s  plane  The f u n d a m e n t a l isormorphic  h a s t h e same  group  of the c i r c l e  to Z , the i n t e g e r s fundamental  group.  41  under  is  infinite  addition.  The  Theorem orientable In  2.5:  The  surface  of  fact,  for  be c o n t i n u o u s l y trivial  genus  any  is  the  to  a point,  of  the  sphere,  trivial  compact  group.  space every  and so  the  every  closed such  path  space  can  has  a  group.  2.6:  The  orientable  surface  of  generators  and  This  0,  group  simply connected  deformed  fundamental  Theorem  fundamental  hence  fundamental genus  is  1,  is  group the  isomorphic  f o l l o w s because  the  free  to  torus  of  torus,  abelian  the  is  the  direct  the  the  group  compact  on  product  Cartesian  two  Z X Z.  product  of  two  circles. Theorem connected and  the  2.7:  A closed,  sum o f  g tori,  following  has  1  g  The f u n d a m e n t a l form,  vacuously  torus,  the  confirms Z X Z.  so  the  group  Fundamental  identified, marked  with  :  g  groups  i n the  fT(S)  =  group  is  For a c l o s e d  A torus  fundamental  S of  group  genus  with  2g  g,  the  generators  a b  *b ~ x  {  of  the  case  of  sphere the  1  - - -  and  a  g  b  g  the  sphere.  In  a  g ~  l  b  torus the  g  ~  1  =  >•  1  do h a v e  case  of  this  the  presentation  that  fundamental  a  surface  presentation:  -TT(S) = < a , b , . . . , a , b 1  orientable  is as the  < a,  b :  free  non -  - 1  abelian  orientable  is  aba  surface  b  _ 1  with  =  1 >  two  of genus  generators, g > 1,  that  the  abelian.  Polygons often  i n the  seen  as  diagram  same v e c t o r .  a  square  below.  with opposite Edges  to  be  edges  identified  are  The two e d g e s a and b c o r r e s p o n d  42  to  is  the  two  circuit  generators  of  around  square  the  The s u r f a c e s e e n as which  genus  correspond  of to  edges the  and one  are  four  group  corresponds  to  two,  the  circuit  for  that  group.  In  the  same  way,  pictured  as  a 4g  -  be  There  much more  be  around  the  of  the  the  about  this  torus,  relator  sum o f  and  aba  two  Again  the  fundamental  orientable the  the  tori,  edges way o f  43  of  is showing  distinct group  represents  surface  one  ^b  same m a r k i n g s c h e m e  octagon  polygon with said  the  identified.  a closed,  sided to  to  of  connected  with  generators  relator  is  fundamental  t h e o c t a g o n shown b e l o w ,  pairs  surface,  of  the  the  genus  identified  of  edges this  defining  g in  understanding  is pairs. a  surface  of  genus  generators able  to  the  or  But  that  Covering  Spaces  The  theory  a great  group.  In  to  full  ->  c  point  such  that  p.  all  that If  a  single  facts  on g r o u p  0,  not  called a  a group  to  when we  of  the  are  motions  of  the  identification  with  will  £  give  the  together  sometimes only  of  call the  S  the  c  case  of  that  closed,  covering for  space of  S is  covering  a  will  be  orientable planes.  theory  these  surfaces.  a pair  map,  c  map such  that  neighborhood  space  a connected  in  (S ,p)  mapped h o m e o m o r p h i c a l l y covering  of  space  with a continuous  is  genus  and h y p e r b o l i c  open c o n n e c t e d  p ^(N)  the  spaces,  given.  results  map o r  an  covering  spaces  the  complex  present  projection  each component  now be  only with  A covering S  covering  will  the  to  branched  d e f i n i t i o n of  about  genus  in S there e x i s t s  We c o n s i d e r  correspond  effect  and o f  second  attempt  but  the  as  that  concerned and  a surface  We w i l l  space.  salient  g ^  edges  be c l a r i f i e d  group  spaces, the  we a r e  will  of  S,  each  by  genus  will  motions  do w i t h  work  the  wait.  S be a s u r f a c e .  consisting S  the  group  plane,  must  to  paper  we  in which  fundamental  covering  generality,  Let  p:  the  of  Therefore, its  of  this  surfaces  of  deal  Some  relevant  the  hyperbolic  edges.  has  The way  fundamental  interpret  complex of  of  g.  and S t h e  covering  for  N of  a  onto N base  space  in  follows. V is  a  point  connected  component  for  i n N.  any  a  of  p ^(N),  Therefore,  then  |p * ( a ) |  44  p ^(a)f|V =  n is  the  consists same  for  any  a i n N and  the  number  an  n-sheeted Let  of  of  covering be  c  is  c  all  b i n S.,.  for  a 11  a  in  next  to  of  S.  f:  ->  S  S  c  f  as  are  c  maps  A(S ),  say  A covering such that the  set  called  that  S  is  c  transformation  p(b)  =  p ^(a)  p(f(b))  to  -  itself  automorphisms.  form a group  where  c  introduce  is  often  space  on a s e t .  space  the  the  the  under  map p i s  convenient  understood.  and v e r y  and p r o b a b l y  the  All  composition  general  A group of homeomorphisms  an e x a m p l e ,  G ac ts  following  convenience, a n d (2)  g(g  A group b i n S,  Then G i s a  _ 1  on a s e t  = g  G be said  (g(s))  a g i n G such  act  of  pair  gp  g  2  = s  is  one  of  idea  a  motivating  the  gh(s)  a m a p p i n g m: G X S -> notation  = g(h(s))  for  all  on S i f  for  the  b.  homeomorphisms  of  a  that  N  for  all  pair  of  topological  discontinuously gj(N)f\g2( )  m(g,s)  is  i n G.  45  S  =  g(s)  g,  h in  i n G.  each  =  properly  that  g  g(a)  neighborhood N such  distinct  (1)  transitively  a group to  _ 1  there  We use  and r e q u i r e :  (s))  is  S if  properties.  G acts  there  Let  has  We w i l l  equals  definition.  with  G;  p ^(N).  space  words,  a covering  acting  A group  for  a covering  denoted  We now want  topological  of  and  S.  In o t h e r  of  be  a group  of  transformations  automorphisms  of  components  a,  S.  Covering  will  open n e i g h b o r h o o d N o f  a homeomorphism  for  that  any  connected  (S ,p)  (S ,p)  for  on X i f empty  points  s p a c e X.  each  for  a,  point  each  x  Theorem A(S )  of  c  S .  If  covering  Hence,  c  2.8:  if  S  is  c  a covering  transformations  g  is  space  acts  a nontrivial  of  S,  properly  then  the  group  discontinuously  automorphism,  it  has  on  no  fixed  us  to  obtain  a group of homeomorphisms  of  a  points. The c r i t e r i o n converse  of  the  of  proper  last  first  that  topological  space  X,  by a  quotient p(a)  set  is  c  covering sometimes This concerns are  a  orbit  Ga = {  space open  called  if,  regular  the  important us  in  this  2.9:  If  automorphism  group  homeomorphic  to  orb i t  a  defined  A(S ), C  topology  only  if,  space  a,  for  w i t h the  of  S  such  next is  each  spaces, or  is  if  on  spaces  equivalent,  the  p *(A)  transitively  For  c  The s e t  by  the  "look  in  set  p  given  rule  that  X.  group  A(S )  of  c  ^(a),  S.  the  the  the  open  in is  X/G is  form  one  that  elements alike."  really  of  the  Their  theorem.  a regular quotient  covering  space  homeomorphism  space  S /A(S ) c  c  of  S,  with  is  pairing a point  i n S and  c  the of  a  denoted  of o r b i t s  X ->  the  a  set  in S «  Now f o r that groups  S,  map p :  covering  (S ,p)  the  projection  of  the  in X is  the  over  paper.  a  : g i n G }.  acts  class  from  a point  covering  interchangeable,  Theorem  allow  g(a)  and  fiber  comes  of  X / G has  transformations  significance  its  G is  X / G , and  The  A in X/G is  (S ,p)  fiber  space  = Ga.  if  the  Ga and d e f i n e d as  the  will  theorem.  Recall  as  discontinuity  converse  of  Theorem  properly discontinuous  2.9  above.  This  homeomorphisms  46  result have  the  shows  "right"  properties.  Theorem and  locally  2.10: pathwise  discontinuous covering p:  X ->  connected,  group  space  and suppose  of homeomorphisms  s p a c e o f X / G , where  p is  G r o u p s and C o v e r i n g  Suppose  (S ,p) c  that  o f X.  i s an i n d u c e d  map p * :  mapping can i n f a c t  This  surface c  [pof].  o f the c o v e r  of S with projection T^CSjpCb))  point  in S is  Thus, surface  a  c  with  actually  be shown t o be a  =  group  p(b) = a . a  i s s i m p l y connected covering  Then the homomorphism p ^ : 'fP(S ,b) c  t h e two f u n d a m e n t a l space  of the fundamental and f T ( S , b ) c  is  group o f the c o v e r i n g  group  trivial,  o f the base. we s a y t h a t  When S  c  is  space. d e f i n i t i o n of a regular  groups (S ,b) c  c  of sheets  covering  c a n now be g i v e n . is  called  p*(7T(S ,b)) i s a n o r m a l s u b g r o u p o f The n u m b e r  ->  monomorphisra.  equivalent  A covering  s p a c e o f S a n d l e t b be  c  we may t h i n k o f t h e f u n d a m e n t a l  A second, of  between  a n d t h e b a s e d e f i n e d by p ^ X t f ] )  L e t ( S , p ) be a c o v e r i n g  as a s u b g r o u p  universa1  p.  '  Theorem 2.11:  c  properly  projection  homomorphism.  S  G is a  connected  Then (X,p) i s a  the n a t u r a l  TT(S ,b) ->  groups  '(7~(S,a)  is  Spaces  is a covering  fundamental  a  that  X / G ; a n d , i n a d d i t i o n , A(X,p) = G.  Fundamental  There  L e t X be a t o p o l o g i c a l  regular  i f the group  fT(S,a).  o f the c o v e r i n g  is related  47  to the  i n terms  fundamental  groups  Theorem  p(b) = a .  ence  with  this  for  the s e t p~^(a)  2.13:  A(S ,b)  is  If (S ,b)  i s then  equal  is a regular  c  isomorphic  to  the  2.14:  covering  Even  If (S ,b)  correspondThe  c  to the order o f  covering  the f a c t o r  group  space  group  A(S ,p)  is related  of  a n d S.  Recall  S  c  normalizer  to 4r(S,a),  is equal  i f a covering c  is a universal  c  i s isomorphic  c  p: S  surface  o f S,  7r(S,a)/p*(TT(S,,b))  ->  c  that  covering  a n d t h e number  to the order  of  space  o f S,  o f sheets  of  ^T(S,a).  S i s not r e g u l a r ,  i n a simple  the automorphism  way t o t h e f u n d a m e n t a l  groups  i f H i s a subgroup o f G, then the  of H i n G, N[H],  i s the largest  subgroup  of G i n which H  normal. Theorem quotient  2.15:  The a u t o m o r p h i s m  group  N [ p ^ ( J ? ( S , b ) ) ] / p ^ C 1f"(S , b ) ) ,  N[p^( lf"(S b))] /  is  c)  example: t:  y  of S  any a i n S and any b i n p ^(a).  then A ( S , b )  the  space  fr(S a)/p^( Tr(S ,b). )  result.  group.  Theorem  is  covering  is in bijective  o f the group  o f the c o v e r i n g  Theorem c  i n the f o l l o w i n g  be a r e g u l a r  c  the elements  quotient  then  Let (S ,b)  Then  of sheets  and ^ ( S j a )  c  2.12:  with  number  fT(S ,b)  z ->  /  the n o r m a l i z e r  with  we s e e t h a t  i s isomorphic  where  the  to  group  i n t h e g r o u p -fPCSja). plane.  on C and i s i s o m o r p h i c  C / ( Z X Z) i s t h e t o r u s , below  c  The t r a n s l a t i o n s  a n d u : z -> z + i g e n e r a t e a g r o u p  discontinuously  diagram  A(S ,b)  c  L e t C be t h e c o m p l e x  z + 1  group  to Z X Z.  fundamental  group  the i d e n t i f i c a t i o n  that  acts  The q u o t i e n t  Z X Z.  space  In the  of opposite  48  properly  edges i n  the of  fundamental its  square  fundamental  regular  covering  for  the  group.  of  the  torus  is  Furthermore  torus,  and  effected p:  in  C ->  fact  by t h e  generators  C / ( Z X Z)  a universal  is  a  covering.  * I 1  2-<l  * *  1  "]  4 1  I  | t  (  r  |  <  Branched Let a is  group not  Covering us of  suppose  a group  still  covering.  In  be  G acts  S will  discontinuous  S/G w i l l  branched  that  homeomorphisms.  properly  points,  Spaces  but  be has  a surface  this  topological  a surface a  and  situation  on-a  in  manageable p:  the  S ->  this set  space paper.  of  S/G w i l l  following  S  as If G  fixed be  a  terminology  applies. The g(s) is  stabilizer  = s },  not If  and  trivial G acts  of  a point  the  set  of  }.  The a c t i o n  on a s u r f a c e  fixed  s  in S is points  the  is  o f G on S i s  S i n such  a  way  group G  Fix(G) free that  49  g  =  = { {  s  i f Fix(G) Fix(G)  g in G  is  :  in S : G  is  g  empty.  non-empty  but If  discrete, p:  S ->  branched  and  |G I  is  S  S/G is  the  covering  finite  natural  of  S/G,  for  all  s,  projection,  and  p(Fix(g))  then  then  S/G is  (S,p)  consists  a  is  of  surface.  called  the  a  branch  points. Under point,  then  component the  case  then  the  many -  it  of of  there  an is  to -  each  which of  s  one.  in  fiber  Suppose (S,p)  is  points  s.  and  _ 1  (b)  as  is  N by p ,  If  p(s)  It  and = b,  the  is  a point the it  b is  is  | G | s  of  the  then e v i d e n t  that  in a  the  branch the  point  one  is  1G I,  in  locally  to -  g  order  as  a branch  mapping p i s  stabilizer,  the  connected  same  point  b,  cardinality  |G|/m . b  Hurwitz Formula that  a  finite  group  covering  of  from  order  the  of a  genus  G by  the  with each  branch  to  The edges  and  the  Riemann of  are  orders  Hurwitz  a vertex.  surface  S/G,  Then  S / G has  then  on t h e  surface  S / G , the  point  faces  G acts  m^,...,!!^.  of  triangulation  faces, S.  branch  onto  called  b^,...,bk with orders  Suppose  a  homeomorphically  of  m^.  not each  if  is  in S/G is that  neighborhood,  denote p  a point  N such  covering.  The o r d e r  a branched  determined the  mapped  In f a c t ,  p~^(b)  we w i l l  the  is  ordinary  of  if  an o p e n n e i g h b o r h o o d  no s u c h  The R i e m a n n -  and  has  p~^(N)  neighborhood for  same c o n d i t i o n s ,  having  the of  S and  genus  the  that  branch of S can  branch  be  points,  formula.  V vertices, We l i f t  replicated  50  the  E edges,  and  F  triangulation  |G| t i m e s ,  once  for  each  sheet,  as  point  b^ of  vertex  Then  -  2g)  overcounts classical  lifting  the  S / G have  equals  At each  t r i a n g u l a t i o n to  genus  genus  (2  each  Hurwitz  |G|(2 -  j  and E u l e r  g and E u l e r  |G| times  Riemann -  theory  of  Riemann s u r f a c e s analysis.  -  2j)  branch  S replicates  the  characteristic  characteristic  minus  the  branch point.  2  sum o f  This  -  2g.  the  gives  the  formula:  2j  -  ((l-l/ra ) + ... + ( l - l / m ) ) ) . 1  For  branched coverings  for certain  example,  w  a,  b,  sphere  S  domain  for  sheeted This  is  1 -  replaced  9  = (z  k  is  -  the  a)(z  -  arises -  i n the  construction  valued functions  in  of  complex  equation:  b)(z  -  c)  complex numbers.  Since  2,  except  b , c,  oroO ; the  complex  by a  two -  surface  to enlarge  the  z and p r o d u c e surface  multiple  consider  and c d i s t i n c t  b e t w e e n z and w i s  -  branch p o i n t s .  times.  S have  2g =  at  Surfaces  The  with  not  ( | G | - | G | / m ^ ) at  2 -  Riemann  m^,  surface  and l e t  (2  vertices  |G1/m^  the  2j,  the  order  only  Let 2 -  are  the  the  f o r z = a, sheeted  desired  1 - 1  Riemann s u r f a c e  e q u a t i o n i n w and z  is  said  to  the  correspondence  correspondence. for  the  have  This  equation.  a branch p o i n t at  51  two  z  =  ZQ  if,  (1)  small  circle  around a,  w has  c,  around ZQ once, In our  WQ.  b,  a s i n g l e v a l u e WQ a t  and  Then,  on t h e  (1) 6j  ->  2  b,  Let  does  change  number  of  (3) -> S  l  z -  b =  r2cis(92)  z -  c =  r cis(0 ).  Y  /  r  3  we  r  -  ->  1  curve  branch points  are  =  3  i 2 3 r  r  r  >  and c  a closed  r  c  i  c i s  ^  s  ^  6  l  e  l  +  e  2  +  9  2  02> are  +  branch  or  9  3^ ^ 2  3 ^  2  the  0  ^  +  about  © 3 ; and h e n c e  ^3  0 | + 2("7? ),  sheets,  e  +  ^  a (or  b, or c).  w changes  Then  sheets.  points. about a and b (or 2  ->  0  circle  £  + 2(7? ),  cuts  branch  0  a and c, 3  ->  lines  ©3.  or b Hence w  an e v e n  times. Let +  z describe  2(fT),  ©  2  a  have:  V l 2 3  =  for  z describes  abbreviation cis(9)  z describe a c i r c l e  Then 0  not  the  a = r jc i s ( 0 ^ )  02  f  a,  candidates  z -  =  describe  z describe a small circle  and c ) .  x  w  ©j + 2(' T),  (2)  ©  l  not  as  let:  sheets,  w  the  Using  we now  two  Let  Therefore  example,  infinity.  cos(0)+isin(0),  w does  Z Q , a n d (2)  ->  a circle 0  2  +  about a,  2(-7T), 0  3  ->  b , and c. 0  3  +  52  2(1?)  Then .  Hence w  changes  sheets.  One c a n lines.  is  This  is  general  Riemann s u r f a c e (z,w)  a branched theory. on R i s  formula. is  a  :  the  w  of  the of  point.  covering the  by a d d i n g  Riemann s u r f a c e  branch for  our  Z ,  this It  genus  of  A calculation  of  of  a)(z  S  S  each  -  we h a v e  branch  with g  interchanges is  known i n  c)  }.  is  2,  the  = z, it  the  in  and  pair  the  the  group  sheets.  compact,  topology  orientable,  that  such  a  handles.  R c a n be d e t e r m i n e d reveals  -  defined  point  Riemann s u r f a c e well  b)(z  d e f i n e d by p(z,w) as  action  is  be a s p h e r e  -  R ->  whose  2  thus  = (z  2  p:  The o r d e r  boundary.  must  Thus  branch  diagram  R is  covering  By c o n s t r u c t i o n and w i t h o u t  a  sheets  schematic  with projection  G acting  surface  the  is  shown b e l o w .  R = { Together  infinity  construct  A standard  equation  (R,p)  Thus  that  the  from the  genus  of  Riemann -  R is  1,  and  Hurwitz so R  torus.  Abstract  Riemann  Surfaces  The R i e m a n n s u r f a c e s  that  arise  i n complex a n a l y s i s  53  motivate  the  d e f i n i t i o n of  such a surface makes  is  an a b s t r a c t  is  that  locally  it  R together *  i  f  °i  :  with  _  >  a  *  C  s  u  1)  U 0  = R  2)  each  f^  i  if  0—  f ^ f j  -  of  pair  charts  atlas  is  )  1  =  z;  covered and  (S  1  )  We w i l l simply  h  a  S  a complex  open  t  0^  }  of  T h e key  coordinate  subsets  a connected  of  the  Hausdorff  feature system  complex  o f R and  that  plane.  topological  open s u b s e t s  of  space  maps  :  a homeomorphism  is  j(°ij)  f  these  not  of  _  >  f  empty i(°ij)  subsets  f^)  is  0^  onto  then  the  for  local  2  = C U {<*>}  -{0},f2)»  open s e t s  are need  connected  analytic the  a  an o p e n s u b s e t  of  complex  the  of  space  =  S -{0}  the  R.  entire  Clearly,  coordinate  two c h a r t s l /  z  *  T  n  defined  This  it  with  (C.fj), is  the  its  atlas  where  fj(z)  compositions  on  is  obviously f^(f2  2.16:  Let  Then R i s c o n f o r m a l l y  R be  a  called  s i m p l y connected  equivalent  to  exactly  one  three:  54  )  C-{0}.  classification is  collection  systems.  sphere  e  and t h e  f o l l o w i n g important  Riemann s u r f a c e s .  plane.  a Riemann s u r f a c e  the  functions  and  map  the  theorem  Riemann of  the  for  Uniformization  Theorem. Theorem  the  composition  analytic  n  a chart  £2^^  C and  s  complex  with  where  i  the  called  an a t l a s  R with  of  A s i m p l e example  by t h e  f j ^ r  :  called  sphere  and  t  (0^,  t h a t endows  the  h  = O ^ D 0j  example: is  c  surface.  plane C  between Each  is  to  family {  is  complex 3)  carries  identical  A Riemann s u r f a c e  Riemann  surface.  following  (1)  the sphere  (2)  the complex  (3)  the u n i t  Topologically, connected surfaces  finite  Automorphisms It  analytic h:  R ->  Riemann and  genus.  as a c o m p l e x  R.  i s synonymous  with  S is analytic  to  is called  as  closed,  orientable  say about  with  an a t l a s  i f each  and a n a l y t i c . well.  an  this.  maps  h f • ^ is  An a n a l y t i c  means  A  1  1  function  automorphism angle  -  a  i s sometimes  o f the  preserving,  C o n f o r m a l mappings  In a d d i t i o n ,  between  atlas  {(Oj'.fj )}.  map f j  a conformal  Conformal, of course,  preserving  are not s i m p l y  or conformal  f u n c t i o n on i t s d o m a i n .  automorphism o f a Riemann s u r f a c e the  more  analytic  surface  w i t h 1-1  that  our f a m i l i a r have  }.  Surface  to d e f i n e  R w i t h an i n v e r s e  orientation  Riemann s u r f a c e s  We w i l l  Riemann  map h : R ->  surface  | z| < 1  L e t R be a R i e m a n n s u r f a c e  and S a  continuous  or  with handles,  o f a_ R i e m a n n  Riemann s u r f a c e s .  C,  disk D = < z in C :  i s now p o s s i b l e  {(O^fp}  plane  the compact  are spheres of  C U { oO } ,  are  conformal called  a symmetry  of  surface.  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 surface 1878  R clearly  Schwarz  R is greater  forms  proved than  that  automorphisms  a group such  under  a group  o f a compact  composition must  of  be f i n i t e  1, and i n 1893 H u r w i t z e s t a b l i s h e d  55  Riemann  functions. i f the genus  In of  t h e upper bound  84(g  -  1)  best  possible,  many  surfaces.  called  on  the  order  as  groups  any  such  G with  A group  a Hurwitz  Theorem  of  2.17:  Suppose  the  g  -  Then  | G | < 84(g  proof: Hurwitz 2g  -  R/G is  |G|(2p -  m(l), Now  g  (2p  is  2  the +  this  for  the  We w i l l  always  Case  I:  Case  II:  of  some  1)  can  84(g  -  group  be 1)  found upper  G acts  as  the for  bound  a group  Riemann s u r f a c e  genus  Case  so  the  p,  R of  and by t h e  t  a  to  If  the  is  of  genus  Riemann  then  t = 3,  triples  l/m(l) of  the  +  the  -  convenience.  quantity  points.  factor  (1-  l/m(t))).  Clearly p = 0 will  We now c o n s i d e r t,  l/m(t))),  f o r |G|  the  +...+  (1-  branch  value  for  l/m(2))  +...+  the  the  number o f  m(l)  <. m(2)  A >^ 1 / 2 ,  branch  <. . . . <  since  A =  the  and  Hence that  the  least  value  each  l/m(l)  l/m(2)  +  +  l/m(3)  (2,2,m) are  l/m(2)  m(i)  <  >^ 2.  is value  1/6.  value +  points.  m(t).  positive  is A =  minimum p o s i t i v e  for  form  0.  be  smallest  (m( 1 ) , m ( 2 ) , m ( 3 ) , m ( 4 ) ) = ( 2 , 2 , 2 , 2 )  a maximum  condition:  of  value  ordering:  g i v e n by ( 2 , 2 , 2 , 3 ) ,  corresponds  orders  f u n c t i o n of  = 4,  because  (1-  o f A.  the  > 4, t  +  A for  assume  III:  the  l/m(2))  maximum p o s s i b l e  l/m(D)  o f , A as  If  + (1-  minimum p o s i t i v e  (1-  If  inadmissible,  All  the  compact  are  minimum v a l u e  value  the  finite  l/m(l))  m(t)  quantity  positive  A is  2 + (1-  fixed,  to  -  Call  for  -  bound i s  1).  a surface  m(2),...,  corresponds  best  84(g  upper  f o r m u l a we h a v e  2 =  where  This  group.  on the  1.  |G| =  G which attains  conformal automorphisms >  group.  for A  l/m(3)  subject  to  1.  therefore  56  inadmissible,  as  the  sum o f t h e r e c i p r o c a l s Triples  of  these,  of  these,  seen  of  A =  A l l (2,n,m)  o f the form (3,3,m) a r e a c c e p t a b l e  (3,3,4),  to g i v e  makes  a larger  triples  value  1/20.  Triples  of  best  i f m>5,  and the  best  triples  with  i f m>4,  and the  triples  with  n>5  are  for A.  A =  value  and t h e  1/12.  A l l (3,n,m)  n>4  best are  for A.  the form (p,m,n)  w i t h p>4 a r e c l e a r l y  inferior  (3,3,4). the l e a s t  value  f o r A, with  three  branch  points,  is  1/42. Case  IV:  If  Combining for  makes  i f m>7,  1/42.  a larger  Therefore, A =  A =  to give  All to  makes  o f the form (2,4,m) a r e a c c e p t a b l e  (2,4,5),  these,  seen  1.  o f the form (2,3,m) a r e a d m i s s i b l e  (2,3,7),  Triples  exceeds  these  A i s A = 1/42  84(g  -  1)  The genus  as  < 2,  A < 0.  results,  1.  t h e maximum v a l u e  we g e t  representing  be a g r o u p o f c o n f o r m a l  for  a sphere  case  that  |G|.  is  ruled out.  the least  triple.  1) b o u n d d o e s  For either  order,  This  f o r the (2,3,7)  H u r w i t z 84(g -  0 or  arbitrary  t  This  or a torus,  automorphisms  value  value of A gives  #  not a p p l y  rotations  positive  about  for a surface a cyclic  group o f  the obvious a x i s ,  f o r some  57  analytic  of  will  structure.  T h e B u r n s i d e Genus o f One c o n s e q u e n c e Riemann s u r f a c e conformally  g as  the  the  is  least  the  genusg(G). the  the  is  of  number o f  finite,  natural  up t o  that  We w i l l  sometimes  the  also  for  groups  a  Given  G acts  which  a  on a  This  act  finite  to  group G,  surface  of  genus  least value  of  g  g r o u p G , and w i l l refer  compact  isomorphism.  question.  g such  of  that  this  as  be  the  will  denoted act ion  genus  group.  less  words,  i f G acts of  genus  as  a group o f c o n f o r m a l  g and d o e s  genus,  t h e n g e n u S g ( G ) = g.  is  being  It  not  responsible very  for  early  characterization textbook, the  2,  conformal automorphisms?  on a R i e m a n n s u r f a c e  some  g ^  surface  value  H u r w i t z bound i s  B u r n s i d e genus  In o t h e r  of  the  following  a group of  be c a l l e d  of  on  the  of  o f genus  We a s k what  a_ G r o u p  this  of  by t h i s  d e f i n i t i o n of  work w i t h all  Theory of  subject  suggested  this  groups  Groups o f  available  in  not  so  act  on any  nomenclature  the  genus  of  that  genus  Finite  English  0,  1,  Order,  (see  pp.  and  is  2  the  surface  Burnside  a group.  n o t i o n of group genus, of  automorphisms  He d i d  is do  and h i s  in his  earliest  1896 w o r k on  76-77).  The H y p e r b o l i c P l a n e At  this  automorphisms exploits Let plane,  point of  a deep  it  is  a problem to  a Riemann s u r f a c e connection  the  how t h e  denote  hyperbolic plane.  conformal  be r e a l i z e d .  with hyperbolic  H = { a + b i : b > 0 }  known a s  can  see  the H is  The  solution  geometry. upper a  half  model of  58  of  the  non  complex -  Euclidean straight half if  -  hyperbolic lines  circles  they  because  connected,  proper  equivalent. concerned, z in C :  geometry,  perpendicular two  models  to  of  the  the  A hyperbolic a + b + c < ^ c,  with  up t o  upper  element  of  H, w i t h  to  circle  |z| = 1 .  hyperbolic  plane  with  < IT  ,  halfplane length  line  L  there  two  simply  C are  to  most  the  disk  or  conformally  unit  disk  D =  hyperbolic half  -  occasionally  circles use  the  interchangeably. a,  three  exists a,  any  a model of the  fails  L.  plane  We w i l l  angles  for  there  of  or  parallel  postulate  on a g i v e n  but  axis  are  w h i c h we a r e  also  usual,  real  Lines  equivalent  diameters  with angles  arc  P not  as  the  parallel  complex  either  in hyperbolic  The  the  The d i s k D i s  In a d d i t i o n ,  congruence,  to  axis.  parallel  conformally  triangle  a + b + c  congruent  an  .  of  halfplane  | z | < 1 }. lines  real  a point lines  subdomains  therefore  with  measured  Riemann Mapping Theorem,  The upper is  the  The E u c l i d e a n  through  the  are  perpendicular  to  i n f i n i t e l y many d i s t i n c t to  Angles  rays  intersect.  According  {  either  perpendicular  do n o t  dramatically, are  are  geometry.  b,  b,  and c  positive  has  numbers  a hyperbolic  and  c.  an a n g l e a,  triangle,  Similar  triangles  sum  b,  and  unique are  geometry. H has  a non -  equal  to  Euclidean metric  |dz|/y,  where  which  z = x +  iy,  yields and  9  element  of  distances Under derived.  area near  the  this The  given  by  real  metric,  result  is  dxdy/y axis the  and  .  The m e t r i c  reduces  magnifies  them  far  f o l l o w i n g remarkable  usually  attributed  to  from theorem  Gauss  59  and  Euclidean it. can  be  Bonnet.  an  Theorem and c .  2.18:  The h y p e r b o l i c  Consider where  r,  implies  s,  and t  that  1/r  triangles,  This  has  S  real  ->  2  S  the  1/s  + 1/t  H to  1,  b -  >2.  The G a u s s  < 1.  a,  b,  c.  with angles  T T / r , f T / s , '[T11  -  Bonnet  It  also  follows  minimal  area  has  transformation  is  f ( z ) = (az + b ) / ( c z  it  itself.  - a -  IT  with angles  theorem  that  (r,s,t)  among a l l =  (2,3,7).  consequences.  a bilinear  be =  triangle  is easily In f a c t ,  an a n a l y t i c  + d).  shown t h a t  these  If  a,b,c,d  f takes  bilinear  function  the  are  upper  transformations  form  a u t o m o r p h i s m g r o u p o f H. The  over  matrix  the r e a l  a,b,c,d special  real  a n d ad -  linear  PSL2(R).  group  SL (R),  the s p e c i a l  ?  numbers R c o n s i s t s  group  transformations to  +  triangle  integers  of the form  and a d -  halfplane  are  reaching  that 2  of T is  t h e one w i t h  far -  Recall  area  a hyperbolic  such  f:  L e t T be a h y p e r b o l i c  be = over  1.  of a l l 2 X 2  The group  the  reals  is  f o r m i n g the s y m m e t r i e s  The c o r r e s p o n d e n c e  linear  taking  group  matrices  PSL2(R),  I  the  c d j with  projective  SL2(R)/{+I,-I).  The b i l i n e a r  of H are  isomorphic  in fact  the b i l i n e a r  transformation  / a. M f(z)  = (az + b ) / ( c z  + d) t o t h e e l e m e n t  isomorphism.  Therefore  automorphisms  of H.  Theorem is  the group  2.19:  we r e g a r d  The f u l l  group  PSL2(R)  o  ^  of PSL (R)  as  the group o f  2  i s the conformal  of conformal automorphisms  of H  PSL (R). 2  I n any g r o u p G , t w o e l e m e n t s in G i f there  (  i s an e l e m e n t  f and g a r e s a i d  h i n G such that  t o be  f = h g h ^.  60  conjugate Conjugacy  ,  is  an e q u i v a l e n c e  conjugacy  classes.  classified For  by  of  the  The e l e m e n t s  of  conjugacy  We w i l l  PSLjCR)  by  their  a great  role  for  geometric  give  the  conjugacy  i n what  a group  therefore  may  realization,  share  are  called  be  PSL^CR),  classification  we i n c l u d e of  the  the  geometric of  The c l a s s i f i c a t i o n  but  understanding  like  significant  standard  class.  follows,  classes  it  for  elements  will  not  in  play  completeness  and  action  of  PSI^CfO on H t h a t  class  of  a non-identity  conveys. It  can  be  shown t h a t of  of  class  equivalence  class.  with a geometric  a conjugacy  properties.  it  and  their  groups  elements  the  relation,  f,  tr(f)  =  the  PSL (R) 9  |a + d | .  conjugacy is  almost  An e l e m e n t  determined  element  if  has  tr a  (f)  point  and  for  translation  real  fixed  axis.  =  single  that  a  tr (f) z  4,  and  fixed  point.  conjugate  hyperbolic  point  to  z ->  a  z  if  i n H and  A parabolic  tr is  +  k,  or has  (f)  has  The  the  point  two  >  4.  fixed  a single  fixed (a  to a  -  points  fixed  point  d)/2c  rotation  is on  e i t h e r oO the  i n H.  Fuchs i a n Groups  between  define  Poincare  was  the  first  PSI^CR) and h y p e r b o l i c g e o m e t r y ,  systematic  study  these  4,  An e l l i p t i c  conjugate  element  translation.  A hyperbolic element  Apparently  <  9  about  is  trace  f is c a l l e d e l l i p t i c i f  9  parabolic  by t h e  of  the  so  called  to  notice as  the  a result  F u c h s i a n groups  groups.  61  connection of  his  i n 1880.  We now  The g r o u p identifying group  this  the  PSL (R) and  has  a  topology  matrix  is  2  (a,b,c,d)  SL~(R)  then  which i t  J with the  quotient  (-a,-b,-c,-d).  The  the  inherits  point  space  (a,b,c,d).  which  following  from R The  identifies  definition  by  the  is  points  based  on  topology. A Fuchsian  group  is  a  topologically  discrete  subgroup  of  PSL (R)2  There  are  characterizing PSL (R) 2  is  equivalent  definitions  its  on H .  discontinuous  and e a c h  disc  finite.  In o t h e r  discrete  subset  subgroup  of  idea  of  a  F  if,  words,  of  H.  PSL (R). 2  fundamental subset  the  The  domain D of  each is  a  a  group  subgroup  each  point  in D :  point then  Fuchsian  such  a  for  < f(p)  of  of  for  H is  F  p in H  f in F >  p i n H is a  of  is  a  discontinuous  groups  depends  on  the  group.  fundamental  domain for  a  Fuchsian  if a non-empty,  (2)  H = {  U f(D)  (3)  for  each  distinct f(D) We n e e d  element  if,  set  orbit  study  D has  Let  the  a Fuchsian  example,  A F u c h s i a n group  (1)  domain.  For  and o n l y  D c o n t a i n i n g p,  A closed group  action  of  and to  To t h i s p be of  :  f  point  connected  in F  },  and  p in H, i f  elements  f,g  interior,  of  p is F,  i n f(D)  then  p is  and  i n g(D)  on t h e  for  two  boundary of  g(D).  know t h a t end,  a point  a F u c h s i a n group does  consider in H that  a Fuchsian group  F.  the is  have  a  fundamental  following. not  Then  fixed  the  by any  Dirichlet  62  non region  trivial for  F  both  centered f  in F  at  p i s D (F) = < q i n H : d(q,p)  fixed  D (F) P  is  a  theorem  2.20:  If  F is a Fuchsian  by a n y n o n - t r i v i a l fundamental  We w i l l  need  element  domain  2.21:  and t h e p o i n t  of F, then  p in H is  the D i r i c h l e t  region  f o r F. important  classification  groups.  Every  Fuchsian  group  F has a  presentation  the form F  -  < x^,...,x ,  ai,...,a  k  ,  b^,...,b  x ...x a b a " b " ... 1  In  k  this  1  1  1  1  element,  exponents  m ( i ) , groups  non-isomorphic  then  Fuchsian  century which  denoted  redundant, signature,  ture  m(i) = ° 3 .  1  m  ( "D' _~  =  •••  -  _ j„ m(k) _ x  c  1 >• I f x^ i s a  f o r permutations  presentations  the s i g n a t u r e authors  by t h e  (g,0:...)  the q u o t i e n t  Felix  symbol  o f the group.  use K l e i n ' s  the t o r s i o n  This  genus.  sometimes  that  (g,k;m( 1 ) , m ( k ) ) .  the m(i) are c a l l e d  a group  b u t we w i l l  order  i  Except  and g t h e o r b i t  We o b s e r v e  finite  x  k , g >^ 0 a n d a l l m ( i ) >_2.  with different  such  modern  is  agbg^g'^g"  1  terminology,  group,  he c a l l e d  :  of this  o f the  form are  groups.  In customary the  1  presentation  parabolic  of  group  to know t h e f o l l o w i n g  for Fuchsian  Theorem of  for a l l  >.  Theorem not  < d(q,f(p))  p  of  Klein  i n the 19th  (g,k;m(l),...,m(k)), Since  the k i s  use (g;m(l),...,m(k))  f r e e F u c h s i a n group a  of  for  the  notation.  Fuchsian  c a n be s e e n  i n the c a n o n i c a l  the p e r i o d s  group  with  with  signa-  signature  by t r i v i a l i z i n g t h e  presentation  of a Fuchsian  63  elements group.  example:  The modular group  be c o n s i d e r e d (az is  t o be t h e g r o u p  of bilinear  + b ) / ( c z + d ) , where a , b , c , d clearly  a discrete  i s the group G = PSL^CZ).  are integers  subgroup o f P S L ^ R ) , 2  presentation  PSL^Z) = <x,  y o f the group  are usually  x  and y :  : z -> - 1 / z  and - 1 / 2 + / 3 i / 2 z ->  z + 1.  taken  PSI^Z)  and ad -  and i t  = y  =1>.  Thegenerators  l ) / z , whose  point is  D (G) = { p  The element  (yx)~^  the t r i a n g l e  -1/2  +73i/2,  z = a + b i : - 1 / 2 < a < 1/2,  of i n f i n i t e extent  The h y p e r b o l i c H a r e a l l images modular are  space  H/G i s  are i  translation ( 0, 3 ; 2, 3 ,  i s the D i r i c h l e t | z | > 1 }.  The  shown  o f the f u n d a m e n t a l  and the area t h e once  with vertices  Its angles  a r e C/3,  It  at iT/3,  is  triangles  group G = P S L ^ Z ) .  preserved  the  b > 0,  shown b e l o w  and i n f i n i t y .  i t s area  a  xand  t o be k i f o r a n y k > 1 ( s e e [ 1 4 ] , p. 2 4 2 ) .  1/2 + / 3 i / 2 ,  and 0; and h e n c e  is  i s a F u c h s i a n group w i t h s i g n a t u r e  P may be t a k e n  It  transformations  fixed points  A fundamental domain f o r the modular group region  be = 1.  i s known t o h a v e  t o be t h e b i l i n e a r  z -> ( - z -  f(z) =  o  y : x  respectively.  transformations  I t may  Since  domain under  the a c t i o n  o f each  punctured  tesse.lating  image  the upper  the action  is conformal,  is also  2-sphere.  64  halfplane  1T/3.  o f the angles  The q u o t i e n t  A Fuchsian  group  hence  no e l e m e n t s  group  for  of  the  a closed,  vertices triangle area  P,  is  minus  reflection Let  G be  i n the  of the  course,  only  of  its  the  Let  internal  by x,  y,  the  a Fuchsian  T be  ^/q>  and r are +  periods  and  surface  fundamental group  "^/p,  1/p  no  fT(S)  g.  group.  if  has  called  genus  T opposite  the group g e n e r a t e d  is  is  S of  p, q ,  sum o f  sides  it  angles  where  (g,0;...),  and  triangle  interior  Q , a n d R,  that  surface  Consider a  exists,  order,  reason  orientable  PQR w i t h  signature  finite  obvious  example: triangle  of  with  a hyperbolic  and  integers  1/q  +  1/r  angles. vertices  and z.  65  at  "TT/r >^ 2.  the  Such a  <  1,  since  Let  x,  y,  P,  Q,  and  its  and z R.  be  T h e n G has G = < x, G is as  the  presentation  z  :  y,  triangle  T(p,q,r).  on t h e  a  It  = y  2  group  is  hyperbolic  illustration  x  the  z  = (yz)  2  w h i c h we h a v e full  plane  below  =  2  (p,q,r)  H does  shows  not  p  = (zx)*> = ( x y )  p r e v i o u s l y denoted  triangle preserve  T a n d one  of  =  r  its  group,  1  >.  (see  p.  and i t s  orientation.  33)  action  The  images.  R  example:  Let  new d e f i n i t i o n s elements Let  us c o n t i n u e  t = yz,  of orders T°(p,q,r)  v  r  =  t  u v =  has 1  T°(p qjr)  denote  (0,3:p,q,r). one  of  below  its  the  the is  images =  and v = x y ;  then  t,  fixed points  orientation  generated  presentation  then  a  Fuchsian  A fundamental  with r  example.If  we make  u , and v  the  are  P , Q , a n d R.  preserving  subgroup  by t h e  elements  T°(p,q,r)  = < t,  t,  u,  u, v :  of  and v ; t  = u^  p  >. is  5  u = zx,  previous  p, q, and r w i t h  T(p,q,r.). Then T ° ( p , q , r ) and T ° ( p , q , r )  the  under  domain  reflection  group. for  Its  signature  T°(p,q,r)  in a side.  4.  66  is  consists  of  An e x a m p l e  is  T  and  shown  R  example: the  A  fundamental  ferent  ways  domain  for this  polygon genus  we  group o f a s u r f a c e  have  group  group  F  plane  with  the fundamental  A  polygon  group  the s u r f a c e  t h e images o f t h i s  polygon  under  covering  space  f o r the s u r f a c e  schematic group  drawing  with  model  of hyperbolic  equal  to  S,  geometry.  fundamental  S.  sided  the  The p l a n e - point  H is free,  H t h e n 'is t h e u n i v e r s a l  and t h e q u o t i e n t  (2,0;...)  -  effects  the f i x e d  o f the fundamental domain  signature  dif-  f o r a surface of  creates  a c t i o n o f F.  i n many  o f t h e 4g  which  discontinuous  surface  g, a c t s  version  (g,0;...),  signature  H i f g > 1.  g, and t h e a c t i o n o f t h e f u n d a m e n t a l o f edges  with  S o f genus  is a hyperbolic  called  properly  A  surface  on t h e h y p e r b o l i c  identification tiled  Fuchsian  i s shown  It i s a regular  6 7  surface for a  here  H/F  Fuchsian  i n the  8-gon w i t h  i s S.  disk  a l l angles  Fuchsian  G r o u p s and Riemann  A F u c h s i a n group with  images  thought  of  as  Since genus  g >  of  the  are  now  to  seen  the  domain  some  Fq = {  by p(q) D(F).  f  upper  space H/F the  f(q)  is,  We r e g a r d  of  orientable  fact,  they  surface  properly  closed,  In  course,  of a p o i n t }.  these  inherit  the  surfaces  the  space  are  local  interior  quotient  space  points  of  orbits  q i n H, d e n o t e d  The p r o j e c t i o n  on t h e  the  the  a  be  p l a n e H.  orientable  on H as  H/F.  of  f in F 1-1  plane H  h a l f p l a n e H.  orbit  :  tesselated  a closed,  because  by i d e n t i f y i n g b o u n d a r y  for  covering  the  plane  H is  a Riemann s u r f a c e  space  covering  of  map p : H ->  the  b  is H/F  fundamental  H / F as  b^ and  Fq,  of  the if  2  surface b  2  =  D(F)/F  f(b^)  for  theory  apply.  seen  S of The  to  genus  be  the  g >  1,  fundamental  universal so  the  group  is  covering  results the  of  group  of  transformations.  Let's  H.  the  hyperbolic  i n F.  surface  and  of  = Fq i s  In a d d i t i o n ,  three  the  acting  surfaces  the  The g r o u p F may a l s o  of  of  be R i e m a n n s u r f a c e s ,  as  obtained  group group  quotient  q i n H, where  defined  domain.  group of homeomorphisms,  structure  defined  tesselates  symmetries  a Fuchsian  The q u o t i e n t points  of  fundamental  1 is  surfaces  then,  fundamental  a group  discontinuous  complex  its  F,  Surfaces  now e x t e n d  distinct  classification  simply connected  They are  hyperbolic  the  sometimes  of  Riemann surfaces.  Riemann s u r f a c e s  called  elliptic,  are  parabolic,  surfaces.  68  C U {oO}, and  The C,  The  full  surfaces  is  group of c o n f o r m a l automorphisms of  indicated  Theorem 2.22:  The  notation  words,  group of  and  b are  complex  The c o m p a c t arise R is  i n the one  of  the  appropriate  a  PT (C)  (3)  Aut(H) =  PSL (R).  is  three  2.23:  acting  is  as  either  properly discontinuous preserve  group  of  z ->  2x2  or,  az  upper  in  + b,  other  where  a  not  simply  quotient  connected  spaces  Riemann s u r f a c e s  R/G,  all  where  and G i s  the  R.  (1)  S is  conformally  C U {<*?}, ( 2 )  C, or  (3)  H ;  group of complex b i l i n e a r Also,  G is  isomorphic  to  1T~(S),  S.  Riemann surfaces  are  also  called  elliptic,  parabolic,  and  except  the  respectively.  the  Riemann surfaces  2-sphere  and  torus,  of  seen,  connected  For  is  which are  Every Riemann s u r f a c e  group of  is  projective  on R.  the  theorem  2  nonzero.  simply  to R / G , where R  fundamental  2  non-zero determinant,  we h a v e j u s t  that  hyperbolic  2  the  of  these  = PSL (C)  Riemann s u r f a c e s  transformations  These  U  bilinear transformations  and a  group  Theorem  and G i s  Aut(C) =  matrices  manner  equivalent  (2)  for  of  theorem.  Aut(C  2  complex  following  (1)  PT (C) stands  triangular the  i n the  each  R = H,  course  a  the the  relevant  appropriate  this  paper,  Riemann s u r f a c e  hyperbolic plane.  Fuchsian  to  In  that  group.  69  R in  case,  for  this  the  group G  The A c t i o n  on a_ R i e m a n n S u r f a c e  Tucker finally,  proves  the  the  origin  following  of  basic  group a c t i o n s  the  theorem  f o r Riemann s u r f a c e s .  the  theorem  from a v a r i e t y  Theorem  2.24:  automorphisms F.  Then  of  S/G is  conformal  let  automorphisms. of  f i n F,  The e l e m e n t formally,  Gf :  Fuchsian  g >  S/G  surface  2g  = F/K acts  sequence  means  of  that  :  the c o n t e n t  of  a^  G be  a normal subgroup  of  on S / G as  of  is  = (S/G)/(F/G).  of F / G .  For  a group  F o r a p o i n t p i n S,  the  p.  orbit  by G f ( G p ) =  a group o f  G(f(p)).  automorphisms of  a  a F u c h s i a n g r o u p F and a the k e r n e l o f e,  have a ~ x  More  G(f(p)).  K = ker(e),  The g r o u p K = k e r ( e )  l  a group  the 1  standard  b " ... a L  So F a n d K a c t  [  b  g  g  on t h e  a  is  then  is a  presentation  g ~  l  b  g  _  1  =  1  >•  upper h a l f p l a n e H  on H / K .  1 This  there  F/K = J.  We may s u m m a r i z e exact  g  stated  conformal  follows.  Gp t o  is  generators.  g  S/F  defined  g r o u p and w i l l  1  Let  as  orbit  is  group o f  then the o r b i t of  J such that  K = < a,,b ,...,a ,b  and J  the  1 if  F ->  with  Of c o u r s e ,  Gp i s  a g i v e n group J  genus  free  in S/G.  We h a v e  examine  and F / G a c t s  G f be an e l e m e n t  S / G ->  epimorphism e : torsion  S.  Furthermore  Gf takes  Conversely, of  a discrete  F / G on S / G a r i s e s  let  explains,  perspectives.  a Riemann s u r f a c e  Gp be a p o i n t  surface  F be  which  on s u r f a c e s . We w i l l  a Riemann s u r f a c e ,  The a c t i o n element  Let  of  theorem,  the  s i t u a t i o n by s a y i n  g  there  is  a  groups > ker(e) — i — > F --e--> J  i  that  i s a monomorphism, e  >  1.  i s an e p i m o r p h i s m , and  70  short  ker(e) = image(i). condition of  is  In  met,  fact,  that  we  is,  may  say  whenever  that  there  whenever is  this  a short  algebraic  exact  sequence  groups 1  > K — i—>  with F a Fuchsian group,  F — e—>  then  J  acts  of  the  J  —>  1  on H / K as  a  group  of  conformal  automorphisms. A direct of  consequence  a hyperbolic  fundamental  triangle  domain of  presentation  of  Theorem  for  2.25:  This the  a  finite  surface the  2T{  (2j  group  last  is  G.  If  -2)  a  us  to  Now s i n c e H / F by T u c k e r ' s  a Fuchsian  the  e:  give  for  area  the  of  the  from  area  the  is  just  last  a  >.  description  of  on a R i e m a n n  signature  an e p i m o r p h i s m f r o m F  subgroup K = ker(e) (g,0;...),  hyperbolic  domain  for  F,  + (1its -  its  l/m(k))  hyperbolic 2)  branched we may  is  a  on  plane.  area >.  to  Fuchsian  then G = F/K acts  the  (2g  theorem,  + (l-l/m(k))  H is  K, t h e n  H/K gives  w,ith  G be  say,  + ...  21T"{  + ...  fundamental  a concrete  group  normal  signature  any  automorphisms  F ->  H / K , where  for  of  allow  l/m(l))  domain  area  + (l-l/m(l))  fundamental  + (1-  theorem,  formula  F have  2)  with signature,  is  group  -  and l e t  surface  D(F)  fundamental  F  m(k)),  quotient If  will  Fuchsian  a group of c o n f o r m a l  group  hyperbolic  hyperbolic  2TT< ( 2 j  Suppose  ( j , k ; m( 1 ) ,  the  Then the  is  of  the  Bonnet  a F u c h s i a n g r o u p F c a n be c o m p u t e d  Let  theorem  action  surface.  F  that  -  F.  (j ,k;m(l)...m(k)). domain  is  Gauss  is  If  area,  given  D(K)  is  according  by a to  the  >. covering invoke  of  the  71  H/F,  with  Riemann -  (H/K)/G Hurwitz  =  formula  to  2 This  get  2g  gives  =  |G| =  Because images hence just of  of  the  the  is  of  area  H / K , seen  surface  of us  the  images  most  :  n  S^  of  that  every  f i n i t e genus,  ...  y,  of  genus  1  where  n>l.  n,  words,  1  n  i s o m o r p h i c to F acts  In a d d i t i o n ,  H / K = S2.  This  same  angles,  formula.  the  of  tiling  images  of  and  Therefore  up a c o m p l e t e action  all  tesselation  G on t h e  of  |G| is  the  Riemann  surface,  in  D(F).  finite  group G does  Let  every  in fact  f i n i t e group  a presentation  F be t h e  act  of  the  on a  does  form  fundamental group of  a  so :  n  a ^ ^  h : F ->  h o m o m o r p h i s m h . As we k n o w ,  other  the  showing that  i s o m o r p h i c to F / K ,  is  l/m(k))).  preserving action,  And t h e  S u p p o s e G has  a homomorphism  and F  -  D(F) m a k i n g  by t h e  (1-  D(F)).  Bonnet  is  ... -  B u r n s i d e Genus  F = < a ,b ,...,a ,b  then  of  angle  concretely,  a B u r n s i d e genus.  surface  S^.  or  Gauss -  steps,  now show  o f some  <gj,...,g  l/m(l)) -  D(K))/(area  by t h e  number o f  Let  G is  (1-  a conformal,  number o f  Existence  Define  -  f u n d a m e n t a l d o m a i n D(F) h a v e  same  finite  have  (area  2j  f u n d a m e n t a l domain D(K).  surface a  F  its  the  |G|( 2 -  the  G  S^  is  establishes  a  the  1  1  ...a b a n  n  n  1  b  1  n  =1  by h ( a )  = g^ and h(b^)  K is  k e r n e l of  the  covered  group of  also  b  k  where  on H , a n d t h e  H/K i s  1  by' t h e  covering  quotient  surface, desired  = g^  it  result.  7 2  -1  the  upper h a l f p l a n e H,  transformations.  space  call  >.  H/F is S2,  the  In  surface  and F / K a c t s  on  Improved  H u r w i t z Bounds  Let's a  recall  surface  S o f genus  | G | _< 8 4 ( g Groups the  that  1).  i f G i s a group  g>l,  then  bound, few  soon  groups.  Theorem 2.26:  group  if  (2,4,5)  or  the  if  1),  7 < r  Suppose  that  ([30]),  has i m p r o v e d  "close"  be q u o t i e n t s  to the  of a  certain  result.  Then  S o f genus  |G| < 84(g -  then G i s a q u o t i e n t  orientation  | G | >^ 2 4 ( g -  (2,3,r),  proof:  his  with order  must  of  "Finite  of a Group"  groups  automorphisms.  | G | >^ 12(g -  T°(p,q,r),  Furthermore,  Here's  entitled  L e t G a c t on a s u r f a c e  of conformal  addition,  that  t o be d e f i n e d p r e c i s e l y ,  triangle  group  i n a paper  a n d t h e Genus  H u r w i t z bound by s h o w i n g  automorphisms  t h e H u r w i t z bound on G g u a r a n t e e s  T . W. T u c k e r ,  A c t i n g on Surfaces  of conformal  -  1),  preserving  then  g > 1 1).  the t r i p l e  In  o f some  subgroup  of  (p,q,r)  as a  triangle T(p,q,r).  must  be  < 11.  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 h a v e 2 where  2g = | G | ( 2 m(i) i s  projection Case  I:  Case  II:  no b r a n c h Case  III:  satisfies  2q  the order  p:  S ->  If 2 -  - ( 1 - 1 / m( 1 ) ) - ( 1 - 1 / m( 2) ) - . . . - ( 1 - 1 / m( k ) ) ) , o f the i  S/G.  2q < 0,  Consider then  If 2 -  2q = 0, t h e n  points  then  If  2 -  k > 3,  t  2 -  2 2 -  2g = 0,  n  branch  2 -  o f the c o v e r i n g  the f o l l o w i n g cases. 2g < - | G | . 2g<-|G|/2, contrary  2q = 2 a n d i f t h e n u m b e r then  point  2g < - | G | / 6 .  because  i f there are  to our assumption. of branch  This  points  is easily  73  k  verified.  Therefore, if  2 -  2g > - | G | / 6 ,  containing give  taking  2 -  or  exactly  2g  the  | G | > 12(g  three  > 0.  contrapositive,  branch  Hence G must  -  1),  we h a v e  so  then S/G is  points,  because  a  two of  far  shown  sphere  branch  be  the  quotient  a  -  1),  an e x a m i n a t i o n  that  points  triangle  group  T°(p,q,r). If  in addition,  candidates  for  the  shows  the  only  that  The r e s u l t group G, that surface  of  addition, T°(2,4,5)  is  a  or  Hurwitz  we  cases  proof:  Riemann -  the  the  the  Hurwitz  theorem.  is  on s u r f a c e  relation  #  B u r n s i d e genus  S of genus  1)  i f and o n l y  7 < r  of  the  -  2g  of  the  g , b u t on no  2.28:  If  |G|/84 + G acts  Therefore,  then  =  =  t  #  G is  a  a quotient  comes  -  1).  In  of  from further  a partial  u v  1 =  =  (1-1/p) in  finite  work  abq>ut F u c h s i a n g r o u p s .  G has  (p,q,r)  the  ... -  By R i e m a n n  (1-1/q)  theorem  of  G  presentation  >.  quotient  If  of  -  -  (1-1/r)).  reveals  that  T°(2,3,7),  in  then  1.  on a s u r f a c e  2 -  theorem  |G|( 2 -  1).  | G | < 84(g  11.  the  r  Then  i f G is  known r e s u l t s  triples -  <  of  = u^ = v  p  = g > 1.  B  -  | G | > 24(g  genus (G) = B  t  in  when g  genus (G)  T°(p,q,r),  2  Theorem  0.  of  get  Examination  is  Let  duplicates  :  the  of  genus.  The " i f " p a r t  u, v  in  those  restated  T°(2,3,r),  that  t,  are  when G a c t s  | G | > 24(g  quotient  G = <  all  is,  be  24(g (p,q,r)  ones  2.27:  proof: Tucker's  triple  can  lesser  Theorem  |G| >  2g =  S of  |G|( 2 -  genus  (1-1/2))  g, -  74  and  the  genus  (1-1/3)) -  of S/G  (1-1/7))).  The  result  follows.  Therefore, upper bound,  n  >^ 168.  a Hurwitz group,  is  Conder  a homomorphic  [4]  has  Hence,  Theorem  #  shown  Tucker  2.29:  result  bounds that  A  ([30]) B  to  as  me  importance  is  n  =  n  a quotient  the  next  step  require  in its  the  84(g  -  1)  T°(2,3,7). of  T°(2,3,7)  for  result.  n ! / 1 6 8 + 1,  one  to  which a t t a i n s of  gets  genus (A )  seems  has  image  that  A c o n c l u s i o n reached Hurwitz  one  i n the  for  n >  proof  special  own r i g h t ,  168.  of  the  emphasis.  so  I  will  improved It's  state  a it  as  a  theorem.  Theorem S of genus p:  S ->  Genus  2.30:  g >1,  S/G is  Formulas  Let and l e t  for  from  and  I  the  the  think  three  it  major  work  The  first  is  Theory  of  1,  work  a n d 2.  Groups of  is  solved  Burns i d e ' s  0,  -  covering  Burnside  1)  genus  group G act 1).  on a R i e m a n n  Then  S/G is  with exactly  three  surface  a sphere, branch  and  points.  Groups  being completely  describe  finite  | G | > 12(g  a branched  Determining difficult,  the  or fair  to  for  all  results  the  I find Finite  action  in  say  the  reported  Order  of  that  the  finite  first  a group problem  groups.  is  very  is  far  We w i l l  now  field.  determination it  genus  of  all  groups  by B u r n s i d e published in  75  of  in his 1896.  action text, However,  the  author's  footnotes  be c r e d i t e d footnotes tion  of  to  Dyck,  hint  that  g r o u p genus  Burnside's  version  The g r o u p s groups These  of  addition  I.  early  the  A  = B  2  |G|  A  action  A  =  2(dg  2  =  3  = B 2  out  4  -  ef).  = 1  -  =  (AB)  d  2  8  = 1  3  2  = 4(d  =  2  f  e  de +  e  +  I  e ). 2  = 1  4  = 1 e ). 2  to  be  of order  relations.  (AB) (BC)  d  3(d  and  (ABC)  (AB)  say  of  the  work  1882.  should  Additional  about  this  will  give  1 = 1  just  Z , n  defini-  into  Here  cyclic  2n; A ^ ; S^;  C a y l e y genus  i n v o l v e a number o f  =  (AB ) (BAB) |G|  that  =  2  (AB ) (BAB)  III.  turn  1 fall  e  =  0  B u r n s i d e genus  d  |G|  genus  G of  = C  and  1893.  those  2  of  results.  generators  2  as  i n c l u d e d among  = B  3  as  to  all  (AB) (BC)  II.  at  least  all  i n 1880  something  presentation the  or  H u r w i t z had  n  are  to  some  n; D , d i h e d r a l groups  The g r o u p s classes  that  who p u b l i s h e d i t  of  of  of order groups  suggest  four  0.  distinct  parameters they  and A ^ .  are.  in  IV.  A  = B  6  = (AB)  3  (A B ) (AB A) 2  2  |G|  =  Burnside parameters example,  d  2  6(d  these  in class  =  e  1 1  + de  2  observes  in  =  2  +  e ). 2  that  for  a  presentations  I,  if  dg -  ef  few  special  the  g r o u p G may h a v e  is  a prime,  groups  G, up t o  values  of  then G i s  the genus  0.  For  a dihedral  group. There of  are  Burnside  I.  A  II.  genus  =  4  A  1,  B  = B  8  2,  2  A  = B ,  IV.  A  8  = B  3  2  =  Maclachlan's  The  second  ([18])  Riemann groups  upper h a l f p l a n e surface  the  AB = A  B =  work  -  1  3  2  (A B) 4  of  "Abelian  In  this  2  =  1.  C. M a c l a c h l a n , Groups of  paper  the  The methods  results H.  course,  follows,  AB = A  3  isomorphism of  work  determined.  important  _ 1  - 1  BA =  _ 1  entitled  Surfaces." is  B  (AB)  2)  is  B  2  1,  4  as  = A ,  =  A  four  given  2  III.  paper  these  exactly  are  based  genus  g > 2,  then  hyperbolic  plane  H , and R i s  its  Burnside  that  if  in a  Automorphisms genus  of  of  R is  universal  representable  groups  all  as  to  obtain  on  the  Riemann  space  H / K , where  77  abelian  acting  a compact covering  1965  Compact  M a c l a c h l a n employs  on F u c h s i a n  As we now k n o w ,  of  reported  is  the  K is  a  torsion group  F  to  is  free of  Fuchsian  calls  a homomorphism  an a b e l i a n  group  G a  that  surface  there  group.  exists  again  2(g  -  the  right  is  can  be  =  2(q  fundamental will  determining detailed will  of  i n one  that  Z  some  a  X  Z  Maclachlan groups  F such  the  of  all  fundamental  if  K =  Fuchsian  homomorphism  (l-l/m(l))  of  group ker(f)  groups  F  F o n t o G.  ( g , k ; m( 1 ) , . . . , m( k ) ) .  set  of  to  minimize  Fuchsian  groups  homomorphism.  F with  groups  ,  n(r); n  L  S  of  his  quotient  This  G whose  G is  where  n(i)  is  the  for  G.  The number  three  there e x i s t s  will  groups  a surface  his  results,  determined divides  r,  Z Q) n  the  i n any  cases  in  show  final  canonical  cyclic  generators  methods  this  work.  group  the  in  and  with his  of  considers  the  group  (l-l/m(k)).  necessary  Maclachlan's  abelian  abelian  of  is  +  area.  nature  T  + ...  surface-kernel  Together  n(r)"  it  over  Fuchsian  all  ...  number  that  +  G by a  finite  generators minimal  homomorphism  considers  equation  case.  The g e n e r a t o r s  the  1)  least  n(2),  ***  X  -  qualitative  n(l),  n(2)  canonical  the  G from a Fuchsian  with signature  i n d i c a t i o n of  genus  Recall  G.  also  the  the  X  He t h e n  this  finding  give  convey  n(l)  of  d o m a i n has  work  invariants Z  side  to  F ->  f i x e d group G,  mapped o n t o  equivalent  I  K is  relation  1)/|G|  hand  f:  a surface-kernel  To m i n i m i z e g f o r a  that  fact,  surface-kernel  F be a F u c h s i a n g r o u p  Consider  F  In  Maclachlan  Let  the  group.  R.  a Fuchsian  such  surface  all  by  n(i + l)  will of  be  possible  called  the  presentation  78  and G =  representation  rank  kernel  its  group, o f G.  Fuchsian  homomorphism  is  h : F ->  G. Case  I consists  m( 1 ) , . . . , m ( k ) ) . k generators groups  F  groups.  with  with  III  consider  examine  must  h(a^)=a_£,  also  were  h(b^)=b^,  Fuchsian  least  the  and  then  Xi ^  =  are  of  surface having a  that  Theorem  [F,F]  n  F ->  If  kernel, Here,  generators  be e x a c t l y  then of  is  a  to n  d  s  if  its  the k e r n e l  of h,  torsion-free.  the commutators reduces  to  F  i n G.  m ( i ) , because  ^ ^ would, belong  v  of  (g,k;m(l),...,m(k)).  Z.i>"*»y_k"  d  First,  f o r the F u c h s i a n group  the k e r n e l  relation  multiple  2.31:  d  y  condition.  a  will  h(yj)=£j.  i.i .2*"i.i-l£i+l'". 4c»  an e p i m o r p h i s m h :  signature  relation  y^  are a l l t r i v i a l ,  just  XlX2"'^k must  o  and  =  divide  the  { m( 1) , m( 2 ) , . . . , m( i - 1 ) , m( i + l ) , . . . , m ( k ) } .  c o n d i t i o n on the p e r i o d s  subgroup  Fuchsian  G, w i t h a t o r s i o n - f r e e  o f y_^ m u s t  fact  defining  common  L.C.M.  consists  now, b u t t o do s o we  by the c o r r e s p o n d i n g  d(i) < m(i), the  closely  F with  the group G i s a b e l i a n ,  Therefore  II  M a c l a c h l a n has worked o u t .  the d e f i n i n g  the o r d e r  contradicting If  Case  are  a^ , b ^ , . . . ,a_g >b_gJ  be s a t i s f i e d  Furthermore, order  These  I more  group  generators  In a d d i t i o n ,  be  Case  a Fuchsian  have  course,  given.  (0,k;  i s 0 and t h e r e  of " m i x e d " F u c h s i a n groups  i s an e p i m o r p h i s m h : F ->  G must  This  orders  two p r e l i m i n a r y t h e o r e m s  there  of these groups  (g,0;...).  consists  F with signature  ( g , k ; m( 1 ) , . . . , m ( k ) ) .  We w i l l  the  genus  the f i n i t e  signature  Case  signature  need  The o r b i t  of those groups  of  the group F,  necessary  G with torsion-free  M a c l a c h l a n proves  L e t F be a F u c h s i a n  of F i s a Fuchsian  the  kernel, following  group.  surface  group  79  for there  will  be c a l l e d  theorem.  The commutator i f and o n l y  to  if  the  periods  m(i)  The  of  next  methods  of  this  that  Then  there are  2  w  ^ ^  x,,...,x ;  (1-  l/w(D) Now  2  =  t  G.  result  will  abelian x  (  y,,...,y  t  x  1 that  =  addition,  be  needed  whose  subclass That  is,  < (1-  to  sketch  the  (  2  =  )  ...  =  x  y^y ...y 2  the  be  q  p  q  (  q  elements =  )  1.  = 1 and  t  same s u b g r o u p  w(i+l),  group  n(i + l). 0,  by  of  p  l/p(D)  abelian  genus  F^  2  x^,...,x  y ,  w  ^  as  while  + ... + (1-  l/p(q)).  argument.  divides  quotient  x  divides  main  with orbit  =  )  G let  satisfying  w(i)  l/w(t))  n(i)  1  group  generate  G be a g i v e n  where  D e f i n e the  F/[F,F]  ^ ^  W  Let  groups  condition.  p  Maclachlan's  m(l),...,m(k)), G.  t  (  1 and  elemnts  in  I:  n( l ) , . . . , n ( r ) Fuchsian  =  q  an  + ... + ( 1 -  for  Case  In  x ...x  and  q  L.C.M.  paper.  = ... = y  2  the  technical  2.32:  such  X l  satisfy  rather  Theorem  = y  F  a  with  C o n s i d e r the  that  is  with  torsion-free  FQ to  be  abelianization  invariants  those  class  signature normal  F  i n F^  (0,k;  subgroup  i n FQ such  o f , groups  FQ of  is  that  gives  the  g r o u p G. Take (0,k;  a  group  F  m(l),...,m(k)),  G is  then  in  Fj,  with  having  h:  d e f i n e d by t h e  F  ->  generators G the  generators  x^,...,x  abelianizing  x^ = h ( x ^ ) ,  and  k  signature  homomorphism.  1 < i £ k,  where  m(i ) the  order  and  ^...x^  o f x^ =  is  1.  therefore  -2  technical  theorem  with  the  + (1  m(i),  and the  The v a l u e -  relations  of  of  l/m(l))  relations  the  quantity  + ...  + (1  Maclachlan's y ,  u  (  1  )  = y  2  u  (  x^  -  above, 2  )  =  ...  ^  we w i s h  l/m(k)).  y  t  u  80  to But  G has =  =1,  1 < i <  k,  minimize  is  by  generators (  t  )  the yj,...y  = y y ...y l  2  t  =  t  1,  and  also,  u(i)  (1-  l/m(l))  divides  + ... + (1-  generators  y^,...,y  domain  F.  for  Since  u(i)  that  u(t)  = u(t-l).  actually  generated  1  t-1.  <  i  _<  o f G, and so  Because by  +  (1-  +  from  the  direct  sum  of  in  FQ\F^.  domain  for  quotient nature  The  of  presented Let's into genus  the  next  +  r  ...  +  h  that of  the  the  G is  y^ ^^  =  u  1,  generators  m i n i m u m we s e e k  l/u(t))  l/n(r))  implies  = 1,  t  canonical  e  of  condition  the  (1-  the  relations  Hence,  (1-  is  t  <  fundamental  y^y2...y  the e  use  l/u(t))  +  =  (1-  l/n(r)).  minimum can  abelian  is  group  be  computed  G as  the  groups.  shows  to  that  the  to  find  is  groups  then  a  + (1-  multiple  L.C.M.  relation with  = n(i).  result  step  Fuchsian  this  the  representation  final  G, and  of  area  common  the  l / u ( D ) + ... +  this  cyclic  Maclachlan  and so  y^,y2> — > y t - i  l/n(l))  canonical  least  :  (1-  The m e a n i n g o f  the  + ...  therefore  minimum -  y^,y2>»->yt -i  = r and u ( i ) -2  -2  u(i+l),  Therefore t-1  the  u(t-l),  is  l/u(l))  We may  seeking  divides  u(l),u(2),...,u(t-l)  and ( 1 -  l/m(k)).  in  t  u(i+l),  F  take  remaining  for the  work  is  minimum i s the  cases  the  minimum II  and  III,  minimum over  all  similar  to  the  same f o r area  three I  F  fundamental  where  work  groups  F again cases.  has The  have  here. look  at  the  discussion,  equal  to  final  since  results.  every  Cyclic  cyclic  groups  group  will  not  o b v i o u s l y has  0.  81  enter Burnside  Here order  we p r e s e n t  less  than 2  X Z )  =  genus (Z  2  X Z^)  = 1  genus (Z  2  X Z  genus (Z  3  X Z3)  Now  the  B  B  B  with  for  2  | G | > 10,  X Z )  2  =  let  (r  =  next  taken  Case  2:  (r  even,  =  1 + (|G|/2)  +  -  l/n(r  (1  In taken  the  over  3:  genus (G)  =  B  +  (1  -  (r  -  (  1  Let  G be  an  X Z  )  all  n  (  2 )  groups  of  finite  X ...  X Z  n  (  abelian r  )  ,  where  group r  >  2  i .  r  that  >  + (1  0 <  2/n(2)).  that  and t h e  minimum  0 < 2p  <  r.  1)  -  l/n(l))  2)  min{  p is  -  a parameter  p such  2p))  case,  p such  Case  ....  last  n  l/n(l)  p is  over  genus (G) B  non-cyclic  2)  case,  is  +  for  = 1 +(|G|/2)(1 -  indicated  ...  G = Z  1:  the  four  3  result.  Case  In  the  4.  general  and  =  2  d i v i d e s n(i + l)  B  of  2  n(i)  genus (G)  genus  10:  genus (Z B  and  the  2(p -  -  l/n(r  + (1 -  a parameter 2p <  2p))  + (1  -  l/n(2))  }.  and t h e  minimum i n d i c a t e d  is  r.  odd)  1 + (|G|/2) l/n(r  -  2p))  min{  2(p  +  -  (1  -  1)  l/n(r  + -  (l-l/n(l)) 2p))>.  82  + (1 -  l/n(2))  +  3)  Glover The  of  the  third  groups,  after  after  field  the  very  early  the  in calculating work l a t e  the  genus  i n the  problem for  of  the  the  groups  special  linear  group of  genus  century,  and  abelian groups,  PSL (q)  2 X 2 matrices  is  by G l o v e r  and  a prime, is  the  2  n  2  Burnside  last  The g r o u p P S L ( q ) , where q = p , and p i s  over  the  finite  GF(q). In  a  Riemann  1985  paper  surface  genus  of  Recall special  of  least  that  gives  .  2  A = h(S),  From group i s  the  also  T°(2,3,p).  It  is  PSL (p) 2  that  integers. = (ST)  2  that  It  or  has  the  calculate  projective  a presentation  where  PSL (p),  2  a  paper.  = 1>,  3  first  and,  2  of  S =  reducing c o e f f i c i e n t s  P S L ( Z ) ->  = < A,B,C : B = h(ST),  -  A  2  = B  and C =  given presentation  as  usual,  = C  3  h(T  for  the e p i m o r p h i c image  mod p  therefore,  on  the  surface  T h e genus  D/4X1/6  -  of  1/p),  the  _ 1  the  of  the  it  p(p  as  2  is  short  — f —>  surface  ...  >,  clear  that  Fuchsian triangle a  H/ker(f)  1,  ). 2  we h a v e  where  = ABC =  p  PSL (p),  k e r ( f ) — i— > T ° ( 2 , 3 , p )  acts  automorphisms. 2  and S j e r v e  modular group,  known  h:  Therefore, 1 ->  1 + (p(p  the  PSL^Cp) on  presentation: PSL (p)  where  the  < S,T : S  an e p i m o r p h i s m  following  is  2  group over  j  "Representing  genus," Glover  PSL (Z)  2  ^  entitled  We now d i s c u s s  2  form PSL (z) = /  ([7])  PSL (p)»  linear  and T =  and  achievement  determination of  projective  the  major  work  Maclachlan's s o l u t i o n of  Sjerve.  the  and S j e r v e ' s  exact 2  H/ker(f) -  l)/2  83  =  is  group  sequence:  P S L ( p ) ->  a group of  this  1  conformal  known  to  |PSL (p)|. 2  be  Therefore this  the  upper  Burnside  for  determine and  then  if  the  PSLjCp) i s  G l o v e r and S j e r v e  which the  there  genus  is  of  less  procedure  class  of  if  it  quotient  is  PSL (p).  will  is  the  epimorphism  than  or  equal  to  yield  This  the  unique  presentation  of  Glover  and S j e r v e  prove  groups  are  the  is  to  T°(r,s,t)  o n l y ones  any  didn't  groups  the  the  all  triples ->  PSL (p), 2  Hurwitz formula,  group  sufficiently  groups.  that  f:  of  consider  abelian  such  find  Riemann -  genus  problem  of  to  triples.  T°(r,s,t)  genus  indeed  f r o m the  such  unnecessary  2  determining  all  groups  purpose,  an  would l i k e  H/ker(f)  minimize over  This  of  bound.  Ideally, (r,s,t)  genus  In  for  because  of  to  for  groups  the F  whose  Maclachlan the  theorem  orientation needed  other  only  2  large  arise  the  PSL (p)  essentially  to  follow,  preserving  calculate  in  triangle  the  genus  of  PSL (p). 2  First PSL (p)  is  2  then  the  observe  that  realized  as  surface  S/PSL (p)  is  converse,  given  2  map p :  S ->  has  2  i n the If  the 2  of  genus  covering  is  S is  is  a  some  surface  triangle  0 and  the  is  3  2  a branched  has  branch is  of  least  genus  covering  group  what  a Riemann s u r f a c e S/PSL (p)  arising  projection  with exactly  f o l l o w i n g theorem,  surface  S/PSL (p)  S = H/ker(f)  quotient  S/PSL (p)  2.33:  then  2  a  a branched  Theorem PSL (p),  if  0 and  T°(r,s,t), map p :  points.  proof  instructive,  of  this  important  and we w i l l  present  theorem it  is  genus  the  interesting  84  The  for  projection  with exactly  here.  S  required.  3  points. The  when  and  branch  ->  proof: the If  Let  surface the  G be  S i n the  projection  orders  p:  2 -  2g =  g =  1 + (|G|/2)(2h Since  genus  This  2h -  2 +  (1  -  -  are  no b r a n c h  and  S/PSL2(p)  quotient  of  is  genus  Z X Z,  the  projective  special  abelian.  Therefore  surface  of  Our  genus  + (1  number o f If  covering, is  and  fundamental  -  (1  + ...  +  group  get  the  inequality  . . .  +  (1  -  h =  1 quickly  leads  if  groups  one  acts  Therefore,  fundamental  h =  even  PSL2(p)  group  to  2  a  1/6  Hence  there  points  required torus.  the  2  1/p.  1.  fixed  abelian  = S ,  of  contradiction.  is the  or  -  >^ 2.  without  non -  0 and S / P S L ( p )  <  m(i)  of  gives  on a s u r f a c e  h =  PSL2(p)  are  acts  l/m(k))  S/G.  l/m(k))).  our  h = 0 or  1.  -  of  with  l/m(k))), (1  genus  surface  y^,...,}^  formula -  either  is  on  to  But  and Z X Z 2-sphere  be  S, a  the is or  the  now:  l/m(l))  then  clearly  -  the  the  0.  branch  k = 0,  ...  of  points  forces  linear  inequality -2  +  Then  has  genus  g is  that  we  satisfied  points.  -  l/m(l))  seen  1/p),  that  not  the  Riemann-Hurwitz  -  l/m(l))  Suppose  branch  l/m(D)  already  assumption  inequality  this  -  inequality clearly  The  S / G has the  (1  PSL2(p).  and h i s  2 + (1  1 + (|G|/2)(l/6  The  The  S -> then  we h a v e  2h -  group  theorem  m(l),...,m(k); |G|(2 -  the  points the  + (1  here  automorphism  -  S  be k <  p:  group  since  l/m(k))  must  projection  impossible  group,  + ...  S -> of  is  the  < 1/6  -  1/p.  4.  S/G is cover  a sphere  w h i l e we know f r o m c o v e r i n g  85  an  unbranched  S is  a n d has space  PSLjCp). a  But  trivial  theory  that  the  automorphism subgroup  of  group the  Suppose  of  the  cover  fundamental  k =  1.  is  group  Denote  the  p:  S \  unbranched  covering,  again  PSL (p).  sphere  minus  a  \  p  which an  is  acts  2  short  on  exact  contradiction If  there  inequality  are  is  is  2 -  it  we  must  1/2  -1/3  is  =  Hence  2 -  T/6,  Knowing  that  -  sufficient  class  Glover  Sjerve  and  a  a  the  regular  group of S  Z ->  -  class  1 £  genus  also  group  branch  Z,  this  our  -l/m(4)  of  1/6  this >^ 2.  2  >  cover, Since  is  the a  Z.  basic  <  but  the  we g e t  image  PSL (p)  -  1/p,  gives  g =  Therefore  2 -  1/2  -1/2  inequality.  points. groups  form epimorphic triples  of  But  -l/m(4)  of  86  #  T°(r,s,t) images  (r,s,t)  p:  group  be Z .  is  1.  then  basic  triangle  all  group  i <^ 4 ;  l/m(3) our  three  from which to determine  = 2,  Then  2 <  fundamental  an e p i m o r p h i c  l/m(3)  the  unbranched  a  ->  k =>4,  is  group.  must  2  -  l/m(2) -  of  Z,  cover  be y ^ a n d y  has  PSL (p)  points,  action  exactly  of  the {yi>,  fundamental  not  by m ( i )  \  regular  a  fundamental  l/m(2)  the  l/m(l)  the  Then  fundamental  the  is  branch  l/m(l)  are  is  again  s  which c o n t r a d i c t s  there  y,.  because  subgroup  2  known t h a t  have  a  Z ->  w h i c h c o u l d be s a t i s f i e d 1 and  *-  so  PSL (p)  four  {y^}  be,  whose  1 ->  because  of  space.  two b r a n c h p o i n t s  and  cover,  sequence  point  w i t h two p u n c t u r e s  which  the  \  trivial  {yjj}^}  infinite cyclic, group  base  automorphism  a  the  \  S  cannot  has  Now a s p h e r e  infinite  PSL (p)  S  ->  the  also  and l e t  iy\)Y2^  covering.  and  point,  the  to a q u o t i e n t  9  (yp)  This  2  T r y k = 2, S  {p  of  branch  —1 projection  isomorphic  is  of  > 2 so  a  PSL (p),  that  2  there  -  exists and  an  then  extension calculate  1 -> the  g = m i n <1 with  the  great which as  a  of  We w i l l the If or  of  p,  let  so  (p+l)/2  we w i l l  b)  1 + p(p -l)/40 2  2  and d > 2  d >  extend of  ponents  2  is  f:  T°(r,s,t)  ->  a  exactly PSL (p), 2  groups  PSL (p),  p ^  2  5.  > 7 and e i t h e r  k  divides  (p-l)/2  1/p)  )  is:  if  p =  5,7,11  p >  13,  p = +1  (mod5),  if  p >  13,  p not  +1  if  p > 13,  p not  +1  (mod8),  (mod5),  p = +1  (mod8),  p = +1  (mod5),  p = +1  (mod8),  9 2  paper  their  PSL (q),  There  work.  the  ([8])  previous  where  -  1/d)  "The  genus  results  q = p , n  in a l l  for  of  other  cases.  PSL (q)," 2  by d e t e r m i n i n g all  1  l/t)>,  (r,s,t).  if  ( (p -l)/4)(l/6  1987  -  12  1 + p(p -l)/80  P  1/s  15  1 + p(p -l)/48  the  -  2  1 +  k  2  (p(p -l)/4)(l/6  and  -  needed.  PSL (p)  B  1 +  Sjerve  of  ->  2  involved in determining  the  genus  PSL (p)  }.  a)  In  the  1/r  epimorphism omit  —f—>  as  triples  theory  an  genus (  and d >  genus  allow  -  such  d = min < k:  Theorem 2.34:  e)  all  following d e f i n i t i o n is  k divides  d)  over  T°(r,s,t)  group  (1  2  now p r e s e n t  p > 13,  c)  the  + (p(p -l)/4)  (r,s,t)  function  —i—>  g of  d i f f i c u l t group  triples  First  genus  minimum taken  deal  ker(f)  primes  n.  87  Glover the  p and a l l  and  Burnside ex-  Recall  that  symmetries  of  euclidean  triples  the  plane  plane P i f  T°(r,s,t)  1/r  spherical  P if  1/r  + 1/s  (r,s,t)  is  of  +  an  plane 1/s  + 1/t  orientation-preserving  +  < 1.  positive  P if 1/t  1/r  =  +  1,  +  and o f  As b e f o r e ,  integers  1/s  such  the  1/t  the  >  1,  of  of  the  hyperbolic  task  that  group  is  there  to  find  exists  the  an  epimorphism f: from  the  triangle acts  group  PSL (q)  then  Hurwitz  formula gives  2  on  over  To s t a t e (1)  If  (2)  results, d=d  k>7  (b)  not d i v i d e  n and If  R  the  -  k divides  any o f  the  1/r  (r,s,t)  the  the  free.  and t h e  Riemann  the  conditions (p +l)/2  integers  (p  (p +l)/2  +  gives  following  least  one  1/s  the  integers* 2  m  2  definitions. k satisfying  integers  -l,  1/t).  genuSg(PSL (q)).  integer  of  +  2 +l m  2 -l,  the  2 +l  n  n  where m  -l)/2,  any o f  n  m  the  let  (a) (c)  k_>7  p  m  the  n  (b)  k does  where  integers  d=d (p) denote k divides  not  divide  m d i v i d e s n and - l , p +l  where  m  least  one any  of of  l<m<n  integer the  the (d)  k  does  2m d i v i d e s n a n d  l<2m<n. This  d e f i n i t i o n of  2,3,4,5,7,9,11. For  the  -  surface:  2  triples  torsion  P/ker(f)  (|PSL (q)I/2)(1  an odd p r i m e ,  n  divide  of  denote  (p -l)/2,  not  ker(f)  surface  we n e e d  integers  r a  2  Km<n.  p is  k satisfying  1 +  relevant  let  (a)  k does  divides  the p=2  conditions (c)  the  genus  PSL (q)  with  quotient  the =  ->  T°(r,s,t),  the  genus(P/ker(f)) Minimizing  T°(r,s,t)  others  the  The c a s e s we  integer  d does  q = 5,7,11 are  not  a p p l y when q =  covered  have:  88  by T h e o r e m  2.34.  Theorem  2.35:  (a)  If  q = 2,3,4  (b) Glover theorems genus  and S j e r v e  by i n d i c a t i n g  of PSL (q) 2  Theorem (2,3,d)  triple  from  2.36:  If  present which  following  p=2 and n>3.  (2)  p>2 a n d n = 4 , 5 , 7 , 8 , . . .  determined  Suppose  b y an ( r , s , t )  p>3.  triple,  (3,3,4)  i f p = +1  (mod 5 ) ,  (3)  (2,3,d)  i n a l l other  Suppose  PSL (q)  ) =  10.  by an ( r , s , t )  triple,  (2,5,7)  i f p = +1  (2)  (2,3,d)  i n a l l other  Suppose  by an ( r , s , t ) (2,5,7)  i f p = +2  (2)  (2,3,d)  in a l l other  determine  the  Burnside  formula.  of PSL (q) 2  (r,s,t)  p = +3  is  genus  determined  by a  of PSL (p  )  2  is given  (mod 8)  The B u r n s i d e where  as  is  follows:  and d > 1 2 .  (r,s,t)  p = +2,+3  genus is  of PSL (p  )  2  given  (mod 7)  as  is  follows:  and d > 1 0 5 .  cases.  p>2.  (1)  i n the f o l l o w i n g  and d>9.  (mod 5 ) ,  triple,  2  cases.  p>2.  (1)  determined  genusg(  triples  where  (2)  2.39:  then  The B u r n s i d e  (mod 5)  Theorem  0.  .  i f p = +2  determined  ) =  2  cases:  (2,4,5)  2.38:  PSL (q)  main r e s u l t s  genus  (1)  Theorem  genusg(  the Riemann-Hurwitz  The B u r n s i d e  i n the  2.37:  their  (r,s,t)  (1)  Theorem  q = 9  then  The B u r n s i d e where  (mod 5 ) ,  (r,s,t)  p = +2, +3  cases.  89  genus  of P S L ( p )  is given (mod 7)  2  as  6  is  follows:  and d M 0 5 .  Conelus ion At the  this  point  C a y l e y genus  group G =  we s h o u l d d e m o n s t r a t e  of  X  a group need  w i t h the G = < x,  A Cayley graph of  genus  0.  for  Therefore,  be  the  the  B u r n s i d e genus  same.  Consider  and  the  presentation  y :  G is  not  that  x  -  y =  shown b e l o w , genus^C  Z  xyxy clearly  X Z^  2  =  )  =  1  >.  embedded i n a  surface  0.  *1  However, is  not  B u r n s i d e genus  of  this  i s o m o r p h i c to Z , D , A ^ , S^, n  paper  "On the  genus  of  Z  2  definitions are  the  Genus o f  X Z^ is of  or A ^ .  n  a Group" ([33]),  1 and  that  g r o u p genus  group  are  not  0,  us  that  work s h o w s  relations  between  Theorem 2 . A O :  genus (G)  <  them.  It  is  in  the  it.  fundamentally d i f f e r e n t ,  few e s t a b l i s h e d  because  A. T . W h i t e ,  informs  Burnside's  is  his action  The and  known,  it  two  there  however,  that  This that in  if  the  c  f o l l o w s from the  a group G acts  on a  next  genus (G). B  theorem,  surface  due t o T u c k e r ,  then a Cayley graph  surface.  90  w h i c h shows f o r G embeds  Theorem preserving S/G  2.41:  Suppose G a c t s  orientation.  Let  have k b r a n c h p o i n t s  g r a p h Kp(G) f o r Fuchsian  G embeds  group  with  on t h e  g be  the  of o r d e r s i n S,  genus  of  the  surface  S / G and  m( 1 ) , . . . , m ( k ) .  where  signature  orientable  let  S  p:  S  ->  Then a C a y l e y  presentation  (g,k;m(l),...,m(k)),  P is  that  of  a  with additional  relations. In a d d i t i o n , Cayley  genus  consider  T u c k e r has  often  seems  to  1.55  shows  of  T(2,3,7),  that  subgroup  is  an  2  Theorem  2.28 B  =  a  index  showed  then genus (G)  a  finite  proper  mutually  quotient  of  1).  exclusive,  quotient  group  as  T°(2,3,7).  Then  T ( 2 , 3 , 7 ) ->  G, but  next  an  index  is,  if  finite the  of  G is  the  Burnside  a  group G i s  image  G,  then  finite  of  genus.  the  a  Let's  T°(2,3,7)  quotient  1 +  of  |G 1/168.  T°(2,3,7),  T(2,3,7), In  fact,  there  any is  composition  the  we w o u l d h a v e  however,  be s e e n  Since  well,  then  as  two  follows.  these  Therefore,  all  two  of  (genus^CG)  Let  f:  index 2 i s  is  in G is  T°(2,3,7)  conditions  an e p i m o r p h i s m  image of T ° ( 2 , 3 , 7 )  2 subgroup.  the  subgroup of  also of  a quotient  g:  always T(2,3,7)  whole of G,  we c a n d e t e r m i n e  Theorem (genus (G)  -  2.42:  If  G is  a  finite  1) > ( 1 / 2 ) ( g e n u s ( G ) g  quotient 1).  91  of  T°(2,3,7)  1)  T°(2,3,7)  an e p i m o r p h i s m the  -  are  is  the  theorem.  c  the  proper  genus^CG) =  simultaneously  which can  the  a if  G were  G be an e p i m o r p h i s m .  not  half  that  |G|/84.  group  -  if  subgroup  that  = 1 +  (l/2)(genuSg(G)  ->  empirical observation  be a b o u t  that  quotient  and  the  that.  Theorem  If  made  then  a -> fog:  This ing  i s the problem  t h e C a y l e y genus  quotient  There that  orientable  This  (Tucker  surface  genus  genus  of S  result  that  surface  T°(2,3,7) But  this  S  is Tucker's  i s the o n l y  n!/168  f o r n > 168.  index  genus  i s Theorem 2.29  f l  is a  without  n>168.  literature  G acts  on an  preserving  of lesser  genus,  orienthen the  of T(2,3,7)  part  n  that the  n  of S , i t n  must of S  is a quotient  = n!/168  again.  92  He u s e s  f o r n > 168.  o f the a c t i o n ^  genus (A )  on the  + 1 f o r n >^ 168.  2 subgroup  B  w o r k done  determination  Therefore,  Hence,  A  + 1,  i n the  If a group  n!/168  preserving  + 1.  > n!/336  and the o n l y  is a quotient  n  determin-  t o be g.  is exactly  n  the o r i e n t a t i o n  o f genus  n  g, p o s s i b l y  terminology,  symmetric  represent  c  n o t s o a c t on a n y s u r f a c e  o f a group  from  not a proper one.  genus (A )  genus.  of G is defined  ([29])  Conder has shown t h a t  d e f i n i t i o n o f group  genus  n  Tucker  is certainly  S o f genus  i s Tucker's  Because A  exactly:  ([29])  symmetric  Conder's  prevents  to the B u r n s i d e  but does  symmetric  n  but i t  i s another  is related  tation,  of A  of T(2,3,7),  Theorem 2.43  that  n  on a  of  + 1 f o r n > 168.  REFERENCES [I]  S. R. A l p e r t  and J .  L. Gross,  coverings of current (1976), 283-303. [2]  A. F. B e a r d o n , Verlag,  The g e o m e t r y  New Y o r k ,  Components  graphs,  J.  of  branched  Cominatorial  of d i s c r e t e  groups,  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 , U n i v e r s i t y Press, Cambridge, England, 1911.  [4]  M. D. E . C o n d e r ,  [5]  Generators  for alternating  J . London Math. Soc. (2),  22  H . S. M . C o x e t e r a n d W. 0 . J . M o s e r , relations f o r d i s c r e t e groups, V e r l a g , New Y o r k , 1972.  [6]  H. M. F a r k a s Verlag,  and 1.  New Y o r k ,  Springer  symmetric  75-86.  Generators  fourth  K r a , Riemann  and  (1980),  and  ed.,  Springer  -  surfaces,  Springer  -  1980.  H . G l o v e r a n d D. S j e r v e , Representing s u r f a c e of l e a s t genus, L'Enseignement (1985), 305-325.  [8]  H . G l o v e r a n d D. S j e r v e , a n g e w . M a t h . 380 ( 1 9 8 7 ) ,  [9]  L. Greenberg, Conformal transformations o f Riemann s u r f a c e s , A m e r i c a n J . M a t h . 82 ( I 9 6 0 ) 7 4 9 - 7 6 0 . L. Greenberg, 12 ( 1 9 6 0 ) ,  [II]  -  Cambridge  [7]  [10]  20  1983.  [3]  groups,  T h . B,  Discrete  PSL2(p) on a Riemann M a t h e m a t i q u e 31  The genus o f P S L ( q ) , 59-86. 2  groups  of motions,  J.  Canad.  reine  J.  Math.,  415-426.  J . L . G r o s s a n d S. R. A l p e r t , T h e t o p o l o g i c a l t h e o r y o f graphs, J . C o m b i n a t o r i a l T h . B , 17 ( 1 9 7 4 ) , 218-233.  current [12]  J. L. Gross  a n d T . W. T u c k e r ,  Generating  a l l graph  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 , M a t h . 18 ( 1 9 7 7 ) , 2 7 3 - 2 8 3 .  Discrete  [13]  G . A . J o n e s and D. S i n g e r m a n , T h e o r y o f maps o n o r i e n t a b l e s u r f a c e s , P r o c . L o n d o n M a t h . S o c . ( 3 ) , 37 ( 1 9 7 8 ) , 2 7 3 - 3 0 7 .  [14]  G. A. Jones  and D. S i n g e r m a n ,  Complex  a l g e b r a i c and g e o m e t r i c v i e w p o i n t , P r e s s , C a m b r i d g e , E n g l a n d , 1987.  functions:  Cambridge  93  an  University  [15]  M. J u n g e r m a n a n d A . T . W h i t e , On t h e g e n u s o f f i n i t e a b e l i a n groups, Europ. J . Combinatorics 1 (1980), 243-251.  [16]  H . L e v i n s o n and B. M a s k i t , diagrams,  [17]  J.  Special  Combinatorial  A . M. M a c b e a t h , crystallographic  The c l a s s i f i c a t i o n groups,  embeddings  T h . B 18  Canad.  J.  (1975)  of  of  Cayley  12-17.  noneuclidean  Math.  6 (1967),  1192-  1205. [18]  [19]  C. M a c l a c h l a n , A b e l i a n groups o f automorphisms o f compact R i e m a n n s u r f a c e s , P r o c . L o n d o n M a t h . S o c . 15 ( 1 9 6 5 ) , 699712. W. M a g n u s , Academic  [20]  W. M a g n u s , theory,  Noneuclidean  Press,  tesselations  New Y o r k ,  A. K a r r a s s ,  Interscience,  and t h e i r  groups,  1974.  a n d D. S o l i t a r , New Y o r k ,  Combinatorial  group  1966.  [21]  H . M a s c h k e , The r e p r e s e n t a t i o n o f f i n i t e g r o u p s , especially o f the r o t a t i o n groups o f t h e r e g u l a r b o d i e s i n t h r e e - and f o u r - d i m e n s i o n a l s p a c e , by C a y l e y ' s c o l o r d i a g r a m s , Amer. J . M a t h . 18 ( 1 8 9 6 ) , 156-194.  [22]  W. S. M a s s e y , A l g e b r a i c T o p o l o g y : An I n t r o d u c t i o n , H a r c o u r t , B r a c e , a n d W o r l d , New Y o r k , 1967.  [23]  V . K. P r o u l x , Dissertation,  [24]  V . K. P r o u l x , Graph Theory  [25]  G . R i n g e l , U b e r d r e i k o m b i n a t o r i s c h e P r o b l e m e am n d i m e n s i o n a l e n W i l r f e l und W i i r f e l g i t t e r , A b h . M a t h . S e m . U n i v . H a m b u r g 20 ( 1 9 5 5 ) , 1 0 - 1 9 . MR 1 7, 7 7 2 .  [26]  G . R i n g e l , Das G e s c h l e c h t d e s v o l l s t a n d i g e n p a a r e n G r a p h e n , A b h . M a t h . S e m . U n i v . H a m b u r g 28 ( 1 9 6 5 ) , 1 3 9 - 1 5 0 . MR 32 # 6 4 3 9 .  [27]  G . R i n g e l a n d J . W. T . Y o u n g s , Das G e s c h l e c h t V o l l s t a n d i g e D r e i - F a r b a r e n G r a p h e n , Comment. print).  [28]  T . W. T u c k e r ,  C l a s s i f i c a t i o n o f the t o r o i d a l Columbia U n i v e r s i t y , 1977. C l a s s i f i c a t i o n of 2 (1978), 269-273.  The number  o f groups  A m e r . M a t h . S o c . 258 ( 1 9 8 0 ) , [29]  the  toroidal  of a given  groups, Ph.D.  groups,  J.  des Symmetrische Math. Helv. ( p r e -  genus,  Trans.  167-183.  T . W. T u c k e r , Some r e s u l t s o n t h e g e n u s o f a g r o u p , Graph T h . 5 (1981), 337-338.  94  J.  [30]  T . W. T u c k e r ,  Finite  genus of a g r o u p , [31]  T . W. T u c k e r , irredundant  [32]  A. T . W h i t e , bipartite  groups  Hurwitz  graphs  The genus  graphs,  on s u r f a c e s  J. Combinatorial  A refined Cayley  acting  of repeated  Trans.  Amer.  T h . B 34 ( 1 9 8 3 ) ,  theorem  (pre -  and the 82-92.  f o r imbeddings  of  print). cartesian  Math.  S o c . 151  products (1970),  of 393-  404. [33]  [34]  A . T . W h i t e , On t h e g e n u s o f a g r o u p , S o c . 173 ( 1 9 7 2 ) , 2 0 3 - 2 1 4 . A. T . W h i t e , Amsterdam,  Graphs,  groups,  Trans. Amer. Math.  and s u r f a c e s ,  1973.  95  North  Holland,  

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