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,
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- The genus of a group
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
The genus of a group Medalen, David Norman 1989
pdf
Page Metadata
Item Metadata
Title | The genus of a group |
Creator |
Medalen, David Norman |
Publisher | University of British Columbia |
Date Issued | 1989 |
Description | This article is a survey of the two definitions of the genus of a finite group that are prominent in the mathematical literature. We call them the Cayley genus and the Burnside genus of a group, but we do not intend to suggest that either mathematician was responsible for the original definitions. Each definition gives a scheme for classifying finite groups by topological criteria. The Cayley genus is determined by embedding a certain graph for the group in a surface of least possible genus. The genus of this surface is then defined as the Cayley genus of the group. The Burnside genus of a group is determined by showing that the group is isomorphic to a group of conformal homeomorphisms of a surface of finite genus, and is not isomorphic to such a group for any surface of lesser genus. The genus of the surface is defined to be the Burnside genus of the group. This paper tries to present a broad mathematical background for each definition of group genus. We attempt to show most of the ideas and methods employed in the research, and most of the significant known theory and results. Proofs are sporadically included, but only to convey a sense of the nature of the research. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0079496 |
URI | http://hdl.handle.net/2429/27596 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1989_A6_7 M42.pdf [ 3.66MB ]
- Metadata
- JSON: 831-1.0079496.json
- JSON-LD: 831-1.0079496-ld.json
- RDF/XML (Pretty): 831-1.0079496-rdf.xml
- RDF/JSON: 831-1.0079496-rdf.json
- Turtle: 831-1.0079496-turtle.txt
- N-Triples: 831-1.0079496-rdf-ntriples.txt
- Original Record: 831-1.0079496-source.json
- Full Text
- 831-1.0079496-fulltext.txt
- Citation
- 831-1.0079496.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0079496/manifest