University of Wrwik institutionl repository: http://go.wrwik..uk/wrp A Thesis Sumitted for the Degree of PhD t the University of Wrwik http://go.wrwik..uk/wrp/3645 This thesis is mde ville online nd is proteted y originl opyright. Plese sroll down to view the doument itself. Plese refer to the repository reord for this item for informtion to help you to ite it. Our poliy informtion is ville from the repository home pge.
The Struture of Rks y Hyley Jne Ryder sumitted for the degree of Ph.D. University of Wrwik August 1993
Contents. 1. 2. 3. 4. 5. 6. l Ī. 8. Thnks nd knowledgements. Summry. Introdution. Preliminries. Rks nd opertions on rks. The opertor group nd the ssoited group. The fundmentl rk. Congruenes on rks. Blok strutures on rks. Opertor equivlene. Assoited group equivlene. Norml surks. Groups. The ssoited group. The opertor group. Expnsions nd extensions. The rtesin produt. Opertor equivlent expnsions. Complete rks. Representtions of rks. Definition of R( G, g). Definition of R(G,g, h, k, ). Trnsitive rks. Non-trnsitive rks. Coset spes nd quotient rks. Congruenes nd sugroups. Congruenes on trnsitive rks. Congruenes on non-trnsitive rks. Congruenes on the fundmentl rk. Referenes. (i) (ii) (iii) 7 7 11 13 15 16 24 27 33 38 38 43 47 48 58 62 66 67 69 72 75 79 84 84 91 98 105 '-;-',
Thnks nd knowledgements. I thnk my supervisor, Colin Rourke, for his help, support, enourgement nd supervision style. I thnk the people t the Mthemtis Institute who mde my time there very enjoyle. I thnk Thoms Cooper for useful onverstions nd for generlly eing there. I would lso like to thnk the following people for support nd friendship: Try Bguley nd Mike Smith, Pul Bguley, Peter Flnign, Aln Mgregor nd John Srgent. This thesis ws prtilly funded y SERC. Delrtion I delre tht the work ontined in this thesis is entirely my own, exept where otherwise stted. This thesis is dedited to Wendy Ptrii Ryder. (i)
Summry. In this thesis we look t the struture of rks. Chpter two looks t ongruenes on rks. We exmine opertor group equivlene nd ssoited group equivlene in detil. We show tht the fundmentl qundle of knot in S3 emeds into the knot group if nd only if the knot is prime. In hpter three we look t onditions on the ssoited group nd the opertor group of rk. We prove tht G is the ssoited group of rk only if the ssoited group of Conj(G) is isomorphi to G x N, where N is elin. We lso show tht ny group n e the opertor group of rk. Chpter four looks t expnding nd extending rks. We derive neessry nd suffiient onditions for rottion loks to form rk when used to expnd rk. We lso show tht ny rk, R, n e extended to omplete rk whih hs the sme opertor group s R. The work in hpter five is losely onneted to the work of Joye in [ J ]. We define rks whih n e used to represent ny rk. In hpter six we show tht the lttie of ongruenes on trnsitive rk is isomorphi to sulttie of the lttie of sugroups of the ssoited group. We generlize this result to non-trnsitive rks. The lst hpter looks t the fundmentl rk of knot in S3. (ii)
Introduetion. In this thesis we study the lger of rks. A rk is the lgeri distilltion of the seond nd third Reidemeister moves. It is possile to ssoite rk, lled the fundmentl rk, to ny odimension two frmed link. This rk is omplete invrint of irreduile links in S3. It is likely tht mny new, esily lulle link invrints n e derived from the rk one the lgeri struture is more fully understood. The theory of rks hs strong onnetions with the theory of groups, espeilly with the theory of onjugtion in groups. Conwy nd Writh [C-W] first studied rks, onentrting on the onnetion with group onjugtion. The nme 'rk' (or wrk using the originl spelling) ws first used y Conwy nd Writh, who desried rk s the 'rk nd ruin' of group, left when the group opertion is disrded nd only the onept of onjugtion remins. In this thesis we onentrte minly on the lger of rks. We first, in hpter two, look t ongruenes on rks. A ongruene is n equivlene reltion whih respets the rk opertion. Congruenes on rks orrespond preisely to quotient rks. As we hve sid, the fundmentl rk is omplete invrint of n irreduile link in S3. A presenttion for the fundmentl rk is very esy to write down ut is not prtiulrly useful s presenttions for rks inherit ll the prolems ssoited with presenttions for groups. Congruenes nd quotients simplify rk onsiderly nd therefore re rih soure of potentil link invrints. For exmple, the fundmentl rk of link hs quotient isomorphi to the dihedrl rk, D 3, if nd only if the link is three-olourle [F-R]. The ltter prt of the first hpter fouses on two ongruenes: opertor equivlene nd ssoited group equivlene. These ongruenes oth redue rks to (surks of) onjugtion rks whih re onsiderly esier to work with thn generl rks. Assoited group equivlene is prtiulrly useful ongruene s it does not hnge the ssoited group nd therefore n e used to simplify the lultion of the ssoited group. We show tht the fundmentl qundle of knot in S3 is ssoited group redued if nd only if the knot is prime. In other words, the fundmentl qundle of knot in S3 emeds into the knot group if nd only if the knot is prime. The ssoited group is one of two groups losely onneted with rk; the other eing the opertor group. We look t onditions on these two groups (iii)
In hpter three. We prove tht group, G, is the ssoited group of rk only if the ssoited group of Conj(G) is isomorphi to G x N, where N is elin. We lso show tht ny group n e the opertor group of rk. As we hve sid, ongruenes simplify rk. In the first prt of hpter four we look t the reverse proess: tht of expnding rk R to produe new rk, Re, whih hs quotient isomorphi to R. We study severl lsses of expnsion rks in detil, inluding opertor expnsions. Any rk is n opertor expnsion of qundle. Therefore, ny rk my e reted y using rottion loks to expnd qundle. We derive neessry nd suffiient onditions, on rottion loks, under whih the loks form rk when used to expnd rk. The ltter prt of this hpter looks t wys of extending rk y dding extr elements. We prove tht ny rk my e extended to omplete rk in whih every element of the opertor group ppers s element. Chpter five looks t wys of representing rks using groups. We generlize severl definitions nd theorems of Joye [ J ]. Given group G, nd elements g, h,..., we define the rks R(G, g) nd R(G, g, h,... ). We show tht these rks n e used to represent ny rk R, where G is the opertor group or the ssoited group of R. In hpter six we use the results of hpter five to re-exmine ongruenes on rks. We show tht the lttie of ongruenes on trnsitive rk is isomorphi to sulttie of the lttie of sugroups of the ssoited group or the opertor group. This result is speil se of the result for nontrnsitive rks whih sttes tht the lttie of ongruenes only equting elements within the sme orit is isomorphi to sulttie of the produt lttie, Iln Li, where L; is isomorphi to the lttie of sugroups of the ssoited group or the opertor group nd n is the numer of orits of R. In the finl hpter we look t the fundmentl rk of knot in S3. We show tht the fundmentl rk of frmed knot in S3 n e defined y onsidering the tion of the overing trnsformtions on omponents of the frming urve in the universl over. This desription llows us to desrie ertin ongruenes on the fundmentl rk geometrilly. (iv)
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Chpter One - Preliminries. Bsi definitions nd results. The definitions nd results in this hpter re tken from [F-R], exept where stted s otherwise. Rks nd opertions on rks. 1.1 Definition A rk is set X together with mp A, from X to S x, the symmetri group on X, whih stisfies the rk identity. Nottion. We use exponentil nottion to indite the rk tion. In other words, refers to the result of A( ) ting on. We use the onvention: The rk identity is s follows: There is seond, equivlent, form of the rk identity given y: (A()) -1 A()A() = -\( ). Nottion. Let R e rk. R; refers to the set of elements of Rnd r* refers to A(r). We often use upper rs to denote inverse elements. 1.2 Definition The rdinlity of the set R; is the order of the rk R. 1.3 Definition The elements of the sugroup of the symmetri group SR" ' generted y the elements r*, we ll opertors. A trivil opertor, on rk R, is n opertor uj whih is suh tht " = for ll in R. 1.4 Definition Surks, quotient rks, rk homomorphisms nd rk utomorplusuis re defined in the nturl wy. - I
Definitions nd exmples of some ommon rks. We often use digrms, similr to Cyley tles, to desrie rks. The elements re written down the left hnd side nd the orresponding opertors re written ross the top. The trivil rk. The trivil rk of order n, written Tn, is the rk with n elements, ll of whih re trivil s opertors. Exmple. The following is the trivil rk of order four. d d d d d d The rottion rk. A rottion rk is non-trivil rk in whih ll the elements re equl s opertors. If rottion rk hs order n nd the opertors re given y single yle permuttion, we ll the rk the yli rk of order n, written t; Exmple. The following is the yli rk R4. d d d d d 8
The onjugtion rk. Let G e group. We n define rk s follows: Elements: the elements of G. Ation: =. This is rk s we hve: = nd C (- )() = e = = 'o. The rk defined ove is lled onjugtion rk nd is written C onj(g). Exmple. The following is the onjugtion rk Conj(S3). id 2 2 id id id id id id id 2 2 2 2 2 2 2 2 2 2 2 2 2 2 The dihedrl rk. Any omplete union of onjugy lsses in group, G, forms surk of C onj(g). In prtiulr, the set of refletions in the dihedrl group, D 2 n, forms surk of Conj(D 2 n ). We ll this rk the dihedrl rk of order n, written D,«. 9
ExmpIe. The following is the dihedrl rk D 3. 1.5 Definition A qundle, defined y Joye in [ J ], is rk in whih = for ll in R. Trivil rks nd (surks of) onjugtion rks re lwys qundles. Cyli rks re never qundles. Produts of rks. The rtesin produt. The rtesin produt of two rks, Rnd 5, is defined y: Elernents: ll ordered pirs of the form (" s), where r is n element of R nd s is n element of 5. Ation: I ') I I ( ("s) T,s = (,T,SS). The disjoint union. The disjoint union of two rks, Rnd 5, is defined y Brieskorn in [ B ]. It hs the disjoint union of the sets R; nd 5 s given y: _ -, E R or, E 5. s elements with the tion E R, E 5 or E5. ER. 10
Quotients of rks. 1.6 Definition A ongruene on rk is n equivlene reltion whih respets the rk opertion. In other words, ongruene, rv, is n equivlene reltion suh tht r-;» ' nd rv ' implies tht rv ' ' A ongruene rv on rk R enles us to form the quotient rk, written ~, whose elements re the rv equivlene lsses nd in whih the rk opertion is derived from tht of the originl rk. We use the nottion [] ""' to refer to the equivlene lss of n element, of rk R, under the ongruene r-«, We omit the susript if the ongruene is ovious or unspeified. We often use the sme nottion, [] ""', to refer to the pproprite element of the quotient rk ~. Orits nd stilizers. 1.7 Definition The orit of n element in rk R, written orr( ), is the set of ll elements whih re suh tht there exists n opertor w with " =. If R is ovious or unspeified, we simply write or(). 1.8 Definition A rk with single orit is lled trnsitive rk. 1.9 Definition Let S e suset of the set of elements of rk. The orit of S, written otr(s), is the set of ll elements whih re suh tht there exists n opertor wnd n element s in S with SW =. We write ot( S) if R is ovious or unspeified. 1.10 Definition The stilizer of n element of rk R, written st R (), is the set of ll opertors w whih re suh tht W =. If R is ovious or unspeified, we simply write st(s). The opertor group nd the ssoited group. 1.11 Definition The group of opertors of rk, R, we ll the opertor group of R, written Op(R). 11
There is seond group onneted with rk defined s follows. 1.12 Definition Any rk R hs presenttion of the form: (,,,... I =,... ). The ssoited group, of the rk R presented s ove, is defined s the group given y the presenttion: (,,,... I =,... ). We write the ssoited group of rk R s As(R). A stndrd presenttion for the ssoited group of R is presenttion derived from rk presenttion s ove. Note. A rk does not hve unique presenttion; therefore the ssoited group does not hve unique stndrd presenttion. 1.13 Lemm of R. The ssoited group of rk R is well defined invrint The mp F, Iroiu the tegory of rks to the tegory of groups, given y: R -+ As(R), is funtor nd is left djoint to the funtor sending group G to the onjugtion rk Conj(G). If R is ny rk nd G ny group, then there is nturlidentifition etween the sets Horn(As(R), G) nd Hom(R,Conj(G)). Therefore, given ny rk homomorphism f : R -+ Conj(G), there exists unique group homomorphism f' : As(R) -+ G whih mkes the following digrill COIllII1ute: R T/ -+ As(R) 11 1f' Conj(G) (r7 nd id re the nturl mps). id -+ G The ssoited group of rk R, with presenttion 0 (,,,... I =,... ), n lso e defined s the quotient of the free group on elements of R y the norml sugroup, K, generted y ll words of the fonn. where equls (/ in R. The opertor group is equl to the ssoited group quotiented y the nonnl sugroup. N, onsisting of elements whih t trivilly when we 12
let the ssoited group t on the rk elements in the ovious wy. other words, we hve the ext sequenes: In nd J( ~ FG(R s ) ~ As(R), N ~ FG(R s ) ~ Op(R) ~ ~ As(R) ~Op(R). The fundmentl rk. The fundmentl rk of odimension two frmed link is defined s follows: Elements: hornotopy lsses of pths from point on the frming urve to se point, where the strting point of the pth my move round in the frming urve throughout the homotopy. There is mp, '\, from these lsses to the fundmentl group given y: where mens 'trvel from the se point to the frming urve long the reverse of pth in []' nd rn is meridin loop round the pproprite ornponent of the link, strting t the end point of. Ation: = 0,\(). If the link is unfrrned, the strting points of the pths my move round in tuulr neighourhood of the link during the homotopy. The resulting rk is qundle, defined y Joye in [ J ], lled the fundmentl qundle. A presenttion for the fundmentl rk. It is very esy to write down presenttion for the fundmentl rk of n oriented link in S 3, given projetion of the link. The rs in the projetion re leled with letters,,,,..., nd, t eh rossing point, sy / ;/ we write down the reltion: =. 13
1.14 Definition lssil rk. If R is the fundmentl rk of link in S3, then R is Bkground mteril. During this thesis we use some si, well known group theory, lttie theory, knot theory nd overing spe theory. Detils of this n e found in [ R ], [D-P], [Ro] nd [ M ] respetively. We lso use some well known ommuttor identities whih n e found in [L-8]. Misellneous nottion. We write [, ] for the ommuttor, where nd re elements of group. We write IAI for the rdinlity of A, where A is set nd Igl for the order of 9 where 9 is n element of group. We write A \ B for the set of elements whih re in A ut not in B, where A nd B re sets. We write (,,... ) for the sugroup of G whih is generted y the elements,,... of G. 14
Chpter Two - Congruenes on Rks. A ongruene on rk is n equivlene reltion whih respets the rk opertion. Congruenes enle us to study the struture of rk in simplified form, They orrespond to quotient rks nd divide the elements of the rk into sets whih move s 'loks' under the tion of the opertors. Looking t quotients of rks enles us to study the wy the 'loks' move, without worrying out wht hppens inside them. In this hpter we first formlize the ide of loks nd show tht lok strutures re equivlent to ongruenes. We then look t two ongruenes in detil. The first of these is opertor group equivlene. This ongruene equtes elements whih re equl s opertors nd redues rk to ( surk of) onjugtion rk. We show tht rk redues to trivil rk under opertor equivlene if nd only if the opertor group is elin. We generlize this result to rks with nilpotent opertor groups. The seond ongruene we study, ssoited group equivlene, equtes elements whih re equl when onsidered s elements of the ssoited group. This ongruene lso redues rk to ( surk of) onjugtion rk. Assoited group equivlene is more useful thn opertor group equivlene euse it redues rk to ( surk of) onjugtion rk without ltering the ssoited group. We lso show tht the fundmentl qundle of knot in S3 is ssoited group redued if nd only if the knot is prime. In the lst setion of this hpter we define norml surks nd look t onnetions etween surks nd ongruenes. 2.1 Definition tht: A ongruene rv on rk is n equivlene reltion suh rv ' nd rv ' implies tht rv»: Let rv e ongruene on rk R. The quotient rk, ~, is quotient of R defined s follows: the elements of R given y: "" rt> the rv equivlene lsses nd the rk opertion is [F-R]. 15
Congruenes on rk R re in one-to-one orrespondene with surjetive homomorphisms of R. Any ongruene, rv determines homomorphism, f rv : R ~ ~, defined y: frv() = nd ny homomorphism f orresponds to ongruene, r-;»f, defined y: [F-R]. [] rvf if nd only if f() = f() Nottion. When using rk digrms to desrie rk R, we often use lnk lines to indite ongruenes on R. For exmple, the following digrm shows the ongruene r-;» f, on D 3 x R 2, where f : D 3 x R 2 ~ R 2. 2.2 Exmple x y z x z y x z y z y x z y x y x z y x z x y z Blok strutures on rks. We my think of ongruene on R s follows: ongruene on rk R is prtition P of R s into sets PI, P 2,. where, for ll in Pi nd for ll in P j, is n element of Pv, k depending only on i nd j. As we n see from exmple 2.2, ongruenes on rk R n e illustrted y dividing digrm for R into 'loks'. We refer to the wy in whih 16
ongruene on R n e represented y dividing digrm for R into loks s lok struture on R. We often wnt to refer to individul 'loks' in digrrn with lok struture. 2.3 Definition Let P e prtition orresponding to ongruene, rv on rk R. A lok is triple of sets in P, (Pi, r; Pk), suh tht ptj = Pk, together with the tion of P j on Pi. The elements of Pi re lled the lok primries, the elements of P, re lled the lok opertors nd the elements of Pk re lled the lok seondries. Exmple. In the digrm elow, single lok hs een highlighted. d e f f f d d e e f f d d d e e e d d f f f e e In this exmple, the elements nd re the lok primries, the elements f' nd f re the lok opertors nd the elements nd d re the lok seondries. Exmples of ongruenes. Any rk hs t lest two lok strutures or ongruenes: r-:» for ll nd in R nd the trivil ongruene, given y r-:» for ll in R. A non-trivil ongruene tht does not equte ll the elements in R is lled proper ongruene. 17
The prtition into orits, written s <«. gives third exmple of ongruene if R is non-trnsitive rk. Certin loks our frequently in the study of rks nd for tht reson we mke the following definitions. 2.4 Definition A rottion lok is lok in whih ll the lok opertors t equivlently. For exmple, in the following rk, ll the loks re rottion loks. d e f d d d d d d d e e e e e e e f f f f f f f 2.5 Definition The loks tht our long the leding digonl of the rk digrm re lled digonl loks. Note. The digonl loks formed under f"v o re surks. The loks fnned under the prtition into orits, f"v o, re lled orit loks nd the digonl loks formed under f"v o re lled orit surks. The digonl loks re nlogous to osets of norml sugroup of group. The individul loks form the elements of the quotient rk in the sme wy s the osets of norml sugroup form the elements of the quotient group. The nlogy is not perfet. The set of osets of norml sugroup ontins one distinguished element, the norml sugroup itself, whih is group. This, of ourse, eomes the identity, distinguished element of the quotient group. There is no nlogy of n identity element in generl rk (lthough 18
rks with identity re defined nd studied in [F-RJ) nd, therefore, there is no distinguished digonl lok. Digonl loks nd surks. Sometimes no digonl loks form surks, x y x y z y x y x z x x z y y z y x x y x z nd, on other osions, they ll do. p q r s p q T s r q p s r q p s s r r q q p p 19 q q p p s s T T
We my lso hve mixture. e p q r s e e e e e e e p p p r r r q q q s s s s 7' r r p p p p s s s q q q q Note. We never hve mixture of loks tht form surks nd loks tht do not form surks within n orit of rk. Elements of rk within single orit re very similr s, if W = e, then the mp w*, given y: is rk utomorphism rrying to e. Let [] e the equivlene lss orresponding to nd [e] the equivlene lss orresponding to e. For ll ' in [] we must hve 'w = e', for some ' in [], nd for ll e" in [e] we must hve e"w = ", for some " in []. Therefore elements of the two lsses re in one-to-one orrespondene nd, s w is n utomorphism, if one forms surk then the other must form n isomorphi surk. 2.6 Definition A element of rk hs q-order equl to n if n = nd " r #- for ll 7' less thn n. 20
2.7 Proposition Let e n element of or() in rk R. Then the q-order of is equl to the q-order of. Proof Sy = ", where hs q-order n. Then " n =. Therefore, we hve: = " =. Therefore the q-order of is not greter thn the q-order of. Similrly, the q-order of is not greter thn the q-order of nd we hve the result. <> 2.8 Proposition The equivlene lss of n element, in rk R, under ongruene rv, forms surk if nd only if the imge of in the quotient hs q-order one. Proof Let [] e the equivlene lss ontining. The equivlene lss of the lement forms surk if nd only if ll elements ontined in [], s opertors, orrespond to ijetions on the set []. This is equivlent, in the quotient, to [][] = []. <> 2.9 Corollry All the equivlene lsses given y ongruene rv on rk R Ioriu surks ifnd only ifthe quotient is qundle. These surks re ll isomorphi if the quotient is trnsitive qundle. <> We now look t the effet whih ongruene on rk R hs on the opertor group nd the ssoited group of R. Tking ongruene on rk simplifies the rk. the ssoited group nd the opertor group. 21
2.10 Proposition Let R e rk nd rv ongruene. The opertor group of the quotient rk nd the ssoited group of the quotient rk re quotients of the opertor group nd the ssoited group of R. Let G e the opertor group of R nd let N e the norml sugroup of the opertor group generted y elements of the form * *, where rv in R. The opertor group of the quotient rk is quotient of ~ y sugroup ontined within the entre of ~. If the quotient rk hs no equivlent opertors, then the opertor group of the quotient rk is ~ quotiented y tlle entire entre of ~. Proof We write 5 for ~ We first show tht the ssoited group of the quotient is quotient of the ssoited group of R. Let R; = {,,... }nd let [] e the equivlene lss of under r-:. Let - (,,... I 0 0 0,... ) e stndrd presenttion of the ssoited group of R nd let ( [ ], [],... I [ ] 0 m0 [ ] 0 [ ],... ) e stndrd presenttion for the ssoited group of the quotient. Let ]( e the sugroup of the free group on elements of R s generted y ll elements of the form rs 0 "if 0 s 0 r, where rnd s re elements of R. The ssoited group of R is the free group on elements of H; quotiented y!(. We define f to e the surjetive mp, from the free group on elements of R; to the free group on elements of 5 s, tht tkes r to [r]. Let 9 e the omposition of f with the quotient Inp, e, from the free group on elements of 5 s to the ssoited group of 5. As 9 is the omposition of two surjetive mps, it is surjetive. If rs 0 S 0 r 0 os is trivil in R, then [r s ] 0 [s ] 0 [ r ] 0 [s] is trivil in 5; therefore!{ lies in the kernel of g. This mens tht 9 ftors through the ssoited group of R to give surjetive mp h from the ssoited group of R to the ssoited group of the quotient rk. In other words, the ssoited group 22
of the quotient rk is quotient of the ssoited group of the originl rk. K = K er(q).ker(g) f q 9 As(R) h ---~.,.. As(S) We now look t the opertor groups. Composing h with the quotient mp, 7r, from the ssoited group of the quotient to the opertor group of the quotient gives nother surjetive mp, j, mpping the ssoited group of R to the opertor group of S. If n element of the ssoited group of R orresponds to trivil opertor on R, then [] is trivil opertor on S. Therefore, j ftors through the opertor group of R to give surjetive mp, k, from the opertor group of R to the opertor group of S. Therefore, the opertor group of the quotient rk is quotient of the opertor group of the originl rk. K =Ker(7r')C]{er(j) As(R) h ---~~ As(S), rr Op(R) --~~ Op(S) We now onsider the opertor group of S. If ~ in R, we must hve o; equl to ; in S. Therefore, ll elements of N eome trivil in the opertor group of S. Sy w ts trivilly in S. We must hve " ~, for ll in R. Therefore we hve w*w* in N, for ll in R. In other words, w is in the entre of ~. If w is in the entre of ~, then, in S, we must hve (o,w)* = 0,*. Therefore, if S ontins no equivlent opertors, w must t 23
11 GjN tnvi y nd the opertor group of S is Z(GjN)' <> As we hve seen, when we quotient rk we quotient the opertor group twie; y the sugroup N nd y sugroup of the entre of ~. The former dels with opertors whih eome equivlent nd the ltter ours euse, in the quotient rk, we hve fewer elements for the opertors to t upon. The next two setions look t two speifi ongruenes in more detil. The first, opertor equivlene, equtes ll elements equl s opertors. This ongruene is prtiulrly interesting s it redues ny rk to (surk of) onjugtion rk. Opertor equivlene. We define the ongruene opertor equivlene, <o«, s follows. 2.11 Definition <o; if nd only if nd re equl s opertors. This is ongruene sine, s n opertor, is equl to ; therefore, if ' rvo p ' nd rvo p ', we hve <o; '. This ongruene hs very simple effet on the opertor group. 2.12 Proposition Quotienting rk y opertor equivlene quotients the opertor group y its entre. Proof By proposition 2.10, we know tht the opertor group of.-v~p IS quotient of the opertor group of R. Let e in the kernel of the quotient mp, i, in other words, ts trivilly on the quotient rk. Therefore, in R, we must hve for ll. In other words, in the opertor group of R, we hve = for ll : therefore is in the entre. Elements of the entre of the opertor group of R n only mix rk elements whih re equl s opertors; therefore the entre is in the kernel of f <> 24
2.13 Corollry The opertor group of rk R is elin if nd only if R is trivil rk. "'0 p <> 2.14 Definition elin rk. We ll rk with n elin opertor group very Note. Aelin qundles re defined in [ J ] y Joye s qundles tht stisfy the elin entropy ondition: All very elin qundles re elin qundles ut not ll elin qundles hve elin opertor groups. Quotienting rk y r-...jo p rk. redues rk to ( surk of) onjugtion 2.15 Proposition group of R. Proof ~ is surk of Conj(G), where G is the opertor <o «We define the mp, A, from.e: into Conj(G) s follows: <o» The mp A is rk homomorphism s: = *** = A([])'\([]). The mp A is injetive y definition of <o«. <> Often R is rk with equivlent opertors nd we n pply the on-, rr o» gruene <o; gin. We define the following nottion. Nottion. We write the result of quotienting rk R y <o«n times s The ove orollry generlizes to rks with nilpotent opertor groups. Detils of the following two definitions my e found in Roinson [ R ]. 23
Definition. The upper entrl series for group, G, is defined indutively s follows: Zl(G) = Z(G), Definition. A group, G, is nilpotent if the upper entrl series for G termintes with G fter finitely mny steps. The nilpotent lss of nilpotent group is the numer of steps in the upper entrl series. 2.16 Proposition The opertor group of rk R is nilpotent of lss n if nd only if tile rk redues to trivil rk fter quotienting y <o; n times, with the rk non-trivil fter n - 1 quotients. Proof We first show tht Op(R) y indution on.s. If.s is equl to one, the result follows from proposition 2.12. Assume, for ll i less thn s, tht the opertor group of R/ "'~p to is isomorphi Op(R) The quotient R/"'op is equl to R/-i: quotiented y "'op; therefore, y proposition 2.12, the opertor group of R/"'op is isomorphi to the opertor group of R/-t; quotiented y its entre. By indution, s r-;» Op(R)/ZS-l{Op(R)} Op(R/"'op) = Z{Op(R)/ZS-l(Op(R))} By definition of the upper entrl series, ZAG) ~ z( G ) Zs-l(G) Zs-l(G). 26
Therefore, nd, y the seond isomorphism theorem, Op(R)jZs-l {Op(R)} z, {Op(R)} j Zs-l { Op(R)} Op(R) By definition, if G is nilpotent group, then there exists n n suh tht Zn( G) = {id}; therefore we hve the result. The reltion rvo p is useful ongruene s it redues rk to ( surk of) onjugtion rk. Conjugtion rks nd surks of onjugtion rks re esier to work with thn generl rks. However, rvo p often lters the ssoited group of the rk nd is of little use if we wish to study the ssoited group. We now look t seond ongruene on rks. This ongruene equtes elements equl in the ssoited group rther thn in the opertor group. Agin this ongruene redues rk to ( surk of) onjugtion rk. However, s we show, this ongruene does not lter the ssoited group. Assoited group equivlene. 2.17 Definition We define rvas y: rvas if nd only if nd re equl s elements of the ssoited group. This is ongruene sine, y definition, in the ssoited group; therefore, if rv As ' nd r-:» As ', we hve rvas,' 2.18 Theorem The ssoited group of rk, R, quotiented y rv As IS isomorphi to the ssoited group of R. Proof We refer to elements of R s using the symols,..... We write S for R quotiented y rva8' S is quotient R nd we write [] for the imge of n element of R under the quotient mp. Ss is equl to { [ l- []... }. The ssoited group of R is isomorphi to quotient of the free group on {,,... } nd the ssoited group of S is isomorphi to quotient of the
free group on { [] 1 [ ],... }. We ll the kernels of these quotient mps ]{ nd K' respetively. The quotient mp, f : R --+ 5 1 is suh tht I---t [] 1 for ll in R. This indues f : ~ As(R) --+ As(5). We define g : As(5) --+ As(R) s follows: we hve the sequene: where i is the injetive mp, rnpping the free group on 5 s into the free group on R s, tking [] to, nd 7f is the quotient mp from the free group on R; to the ssoited group of R. The omposition 7f i is mp from FG(5 s ) to As(R). 1(', the kernel of the mp from FG(5 s ) to As(5), is given y: ( [ ] TT ~ [ ] I [], [ ] E 5s, [ ] = [] [ ) in 5). Clim: 1(' is ontined in the kernel of 7f i. -- Proof: we need to prove tht [] [ ] [ ] [ ], where [] is s ove, is in the kernel of 7foi for ll [] nd [] in 5. We hve: i([]otto[]o[j) =. As [] = [F) in 5, we hve r-:» As in R. Therefore, y definition of rvas, we hve = in the ssoited group of R. In other words is in the kernel of tt. Therefore, = is in the kernel of 7f nd - - [ ] [ ] [] [ ] is in the kernel of 7f i. As 1(' is in the kernel of 7f i, the mp 7f i As(5) to As(R). I(er(i 7f) ftors to give mp from ]{' )>-------?>FG(5) --~~As(5) ~ f 9 FG(R s ) --~~ As(R) ~ We define g to e this mp. The mps f nd g re inverse mps giving the result. o 28
Wp ll rk ssoited group redued if the mp from the rk elements to the ssoited group is injetive. 2.19 Proposition '"V~B is surk of Conj(G) where G is the ssoited group of R. Proof We define the mp, '\, from f"v~b into Conj(G) y: '\([J) = f-1(), where f-1 is the mp from R to As(R). The mp,\ is rk homomorphism s:,\([ ] [ J) = x( [ ] ) = f-1()f-1()f-1() = '\([])'\([ J ). The rnp,\ is injetive y definition of I"VAs. <> 2.20 Proposition The opertor group of rk R quotiented y I"VAs is the ssoited group of R quotiented y its entre. Proof We define the tion of elements of the ssoited group on elements of R in the ovious wy. The opertor group of R is isomorphi to As~R), where N is the sugroup of the entre of the ssoited group of R generted y elements orresponding to trivil opertors. Therefore, the opertor group of R quotiented y I"VAs is quotient of As~R). If w is in the entre of the ssoited group of R, then ww = for ll elements of the ssoited group. Therefore, in ~, we must hve " = f"v A B for ll, sine no two elements of JL orrespond to the sme element of f"v A B the ssoited group of R. In other words, ll elements of the entre of the ssoited group of R t trivilly. Elements of the ssoited group tht t trivilly on the rk elements re in the entre; therefore elements of the ssoited group of R quotiented y I"V As whih orrespond to trivil opertors re preisely those ontined in the entre of the ssoited group of R nd we hve the result. <> 29
2.21 Definition Let "- nd rv' e two ongruenes on rk R. We sy tht rv is less thn -:' nd tht -:' is greter thn rv if: rv implies tht -:'. Let rv e ny ongruene on rk R whih is less thn rvas. Then rv does not hnge the ssoited group of R nd my e used to simplify the lultion of the ssoited group. The following ongruene is n exmple. 2.22 Definition rv q C if there exists n interger n with = n. 2.23 Proposition ":«is ongruene. Proof We hve: rv q nd rv q d implies tht = n nd d = rn = n rv C q This is the ongruene whih redues rk to its ssoited qundle, defined in [F-R]. It is less thn r--;»as, does not hnge the ssoited group nd the opertor group of R is quotient of the opertor group of R, lying etween ""q the opertor group of R nd the opertor group of R quotiented y its entre. The quotient of rk y rv q is lwys qundle. Therefore, if G is the ssoited group of rk, then it is lso the ssoited group of qundle. However r-:»q is not equivlent to rvas nd it is often possile to redue the rk further without hnging the ssoited group. We now show tht the fundmentl qundle of the onneted sum of two non-trivil knots is not ssoited group redued. o The geometri signifine of rv As on lssil rks. 2.24 Definition A projetion, P, of knot. K: relizes n element r of the fundlnentl rk of ]{ if there is n r in P leled with 1'. 30
If knot, K, is onneted sum of two non-trivil knots, K' nd K", with onneting rs in projetion leled nd, then nd re two distint elements of the fundmentl rk whih mp to the sme element of the ssoited group, [F-R]. We now show tht, if nd re two distint elements of the fundmentl qundle tht mp to the sme element of the ssoited group, then there is projetion P of K whih relizes nd s onneting rs. some definitions. s onneted sum We first need two results nd 2.25 Lemm Given ny finite set, 5 = {t,...,n}, of elements of lssil rk R, then there exists projetion P, of the link orresponding to the rk, whih relizes ll the elements in 5. Proof It is suffiient to prove tht given fixed projetion, sy P, of the link L, nd n element r of the fundmentl rk of L, then there exists new projetion, P", whih relizes ll elements relized y P together with 7'. Let P e ny projetion of the link with rs leled using the symols { t,...,t}' Then r n e written s i w where w is word in the elements {j}. The element r orresponds to pth from the r leled, to the se point, whih we put 'ove' the projetion. By gring little piee in the entre of the r leled ; nd pulling it through the projetion, long the pth given y r, we rete new projetion, P", whih relizes r. This proess only uses the seond Reidemeister move nd this move, of, sy, n r leled under n r leled, leves prts of the originl rs, nd, unhnged in the projetion. o In the proof of the following result we use the Annulus Theorem. The version elow nd the definitions elow re tken from Jo, [J]. Definitions. A 2-lnnifold, T, properly emedded in 3-mnifold,.J.~I, is ouipreseile in M if either T = 52 nd T ounds 3-elL or T #- 52 nd there exists dis D in M suh tht DnT = 8DnT nd is nonontrtile urve in T. Otherwise, T is inompressile. A ompt orientle. irreduile 3-nlnifold is lled Hken-mnifold if it ontins two-sided inompressile surfe. A 3-nlnifold pir, (.::\1. T). is lled Hken-pir if 1\1 is Hken-rnnifold 31
nd T is n inompressile 2-mnifold in 8M. A mp of pirs, 1 : (X, Y) ----t (M, T) is essentil if 1 is not homotopi to mp 9 : (X, Y) ----t (M, T) suh tht g(x) is ontined in T. A mp I: (51 x 1,5 1 x 81) ----t (M,T) is nondegenerte if 1 is essentil nd 1* : 7r1 (51 X 1) ----t 7r1 (M) is injetive. Annulus Theorem. Let (M, T) e Hken-mnifold pir. Suppose tht 1 : (51 x 1,5 1 x 81) ----t (M, T) is mp of pirs. If 1 is nondegenerte, then there exists n emedding 9 : (51 X 1,5 1 x 81) ----t (M, T) tht is nondegenerte. Furthermore, if 11(51 x8i) is n emedding, then 9 my e hosen so tht 11(51 X8I) = gl(5 1 x8i)' o 2.26 Theorem Let nd e elements of the fundmentl qundle, Q, of link L in 53. Then I"VAs if nd only if nd re pths strting on rs in the seine omponent,!(, of Lnd K is onneted sum with the rs lelled nd s onneting rs. Proof I"VAs. Therefore nd re oth in the sme orit of Q nd must e on the sme omponent of L. By lemm 2.25, we my tke projetion P whih relizes nd. We emed L so tht L projets 'downwrds' onto P, with the se point * 'ove' L. The entres of the rs leled nd n now e 'pulled up' so tht they pss right next to *, wy from the rest of the link. Nottion. We refer to the pths orresponding to the elements nd in the fundmentl qundle s p nd p nd we refer to the loops in the fundmentl group equl to these elements s opertors s i nd z. The elements nd re equivlent s elements of the ssoited group; therefore we must hve i homotopi to z se point. with the homotopy fixing the This homotopy gives us mp, I, of (51 x 1,5 1 x 81) into M. The mp 1 is ertinly essentil. As nd re not equl s elements of the fundmentl qundle, the omponent 1{ is non-trivil nd 7r1 (1{) emeds into 7r1 (M), where M is the omplement of L. Therefore 1 is nondegenerte. (M, 8M) is Hken-mnifold pir. We n hoose 1 so tht 11(51 x8i) is n emedding; therefore, y the Annulus Theorem, there exists n emedding, 9 : (51 x I, 51 x 81) ----t (M, 8M) whih grees with 1 on the oundry. (51 x 1,5 1 x 81) is sphere with two holes. Therefore we hve n emedding of sphere with two holes into the omplement of 32
the link with the r leled going through one hole nd the r leled going through the other hole. As the rs, long with the se point, were pulled wy from the rest of the link, no other rs go through these holes. By gluing diss onto these holes suh tht the rs leled nd only interset eh dis in one point, we produe n emedding of 2-sphere into 53. Let p e pth on this ernedded sphere going from the point where the r orresponding to intersets the sphere, to *, nd then to the point where the r leled intersets the sphere. By thikening the imge of the sphere, we n ut the rs leled nd where they interset the imge of the sphere nd join the piee of the r leled tht is outside the sphere to the piee of the r leled tht is outside the sphere, long the 'outside' of the pth p. We n do the sme thing inside the sphere. We now hve two knots, seprted from one nother y the imge of sphere. As p is not homotopi to p, the two knots re non-trivil nd the originl omponent K is their onneted sum, with the rs leled nd s onneting rs. o 2.27 Corollry The fundmentl qundle of knot in 53 emeds into the ssoited group if nd only if the knot is prime. o We now onsider wys of defining norml surk. Norml surks. We wish to define norml surk. We wnt the definition of norml surk to e nlogous to the definition of norml sugroup. However, for resons whih will eome pprent, it is not possile to produe ompletely nlogous definition. There re two equivlent definitions of norml sugroup. Definition (i). A norml sugroup, N, of group G, is ny sugroup suh tht: hnh is n elernent of N for ll n in Nnd h in G. Definition (ii). A norml sugroup of G is the kernel of homomorphism frorn G. We first onsider definition (i). There is no onept of onjugtion in generl rk. Conjugtion in the opertor group or the ssoited group 33
orresponds to the rk opertion. definition (i) to rks results in: Therefore. the nturl dpttion of Definition. A norml surk, N, of rk R, is ny surk whih is suh tht: n TEN for ll n in Nnd r in R. Any surk whih stisfies this definition onsists of the union of omplete orits. In other words, this definition is simply generliztion of the definition of n orit surk nd is redundnt. We therefore turn our ttention to the seond definition of norml sugroup. There is no identity in generl rk nd the tegory of rks hs no zero ojet; therefore there is no onept of the kernel of rk homomorphism. However, the digonl loks formed y ongruene re, in some sense, nlogous to kernel. We hve seen tht sometimes no digonl loks form surks. Even when the digonl loks do form surks, they re often not isomorphi. Exmple. p q r x y p p p p q q q p p q q q q r q q 7' 7' r 7' p p p r 7' :r :r :r x x x x y y y y y y y y y x,r Therefore we hve no hope of finding one-to-one orrespondene etween ongruenes on rks nd ertin kinds of surks of R. However, we often 34
find tht the digonl loks do form surks nd we mke the following definition. 2.28 Definition A surk, 5, of rk R, is norml surk if there exists ongruene on R with the elements of 5 forming the lok primries, the lok seondries nd the lok opertors of single lok. The norml surks orresponding to ongruene re sid to form omplete set if every digonl lok forms surk. Every ongruene does not orrespond to norml surk nd some ongruenes orrespond to severl non-isomorphi norml surks. However, mny ongruenes do orrespond to omplete set of isomorphi surks. Let 5i nd 5 j e two equivlene lsses, eh ontining n element from the sme orit. By the note ove definition 2.6, S, forms surk if nd only if 5 j forms surk nd, under these irumstnes, the surks re isomorphi. This gives the following result. 2.29 Proposition Let R e rk nd ~ ongruene on R. Then ~ orresponds to omplete set ofisomorphi norml surks if the quotient is trnsitive qundle. Assoited rks. Although we n not lwys ssoite omplete set of norml surks to ongruene, we n ssoite rks (whih re not neessrily surks of R) to ongruenes. 2.30 Definition Given rk R, n element of R nd ongruene r-;» on R, we define the ~ ssoited rk t, written s ~ R, s follows. Elements: elements of [] f"v A CIon: ti =. 2.31 Proposition The ove is well defined rk. Proof We first need to prove tht if ~, then is n element of [] f"v \ l Ci i\..s~,we lve 0 < =0,. Nottion. We use (0,)( ) in r-;» R., nd for the rk opertion in R. to indite the rk opertion (s defined ove) 35
The ove stisfies the rk identity s we hve: = ooooooo = ooo. = ooooooooooo ooo =. <> The following is n exmple in whih oth ssoited rks re isomorphi to D 3. p q r p T' q p q q p T' q p q p T' q p T' p q T' 2.32 Proposition Let nd e two elements in the sme orit of rk R. Then rv R is isomorphi to rv Ri. Proof Let w e suh tht " =. We define A : rv R; ~ rv R y is in 07'( ). Therefore, the rdinlity of rv R of rv R nd A is ijetion. is equl to the rdinlity 36
Clim: A is rk homomorphism. Proof: = rwoworowowosow rosow =r. <> 2.33 Corollry A ongruene r-:» on rk R s omplete set of isomorphi ssoited rks if te quotient is trnsitive qundle. <> 37
Chpter Three - Groups. In this hpter we look t the ssoited group nd the opertor group. We first look t the ssoited group of rk nd show tht group G is n ssoited group of rk R only if the ssoited group of the onjugtion rk, C onj(g), is isomorphi to G x N, where N is elin. In the seond setion, we show tht ny group n e the opertor group of qundle. Finlly we look t onditions on the opertor group of trnsitive qundle. The ssoited group. A group, G, is n ssoited group if nd only if G hs presenttion of the form (S In) where S is set of genertors nd n is set of reltions ontining only reltions of the form: where, nd re elements of S. =, This ondition is presenttion dependent; therefore it is not prtiulrly useful. We now prove tht group, G, is n ssoited group only if the ssoited group of the onjugtion rk, C onj(g), is isomorphi to G EB N, where N is elin. 3.1 Theorem A group G is n ssoited group only if the ssoited group of Conj(G) is isomorphi to the diret produt of G with n elin group N. Proof By theorem 2.18, without loss of generlity we my ssume tht G is the ssoited group of n ssoited group redued rk R. Therefore, y proposition 2.19, R is surk of ConjAs(R). We need to prove tht there re two norml sugroups, Nnd H, of AsConjAs(R) with N elin, H isomorphi to the ssoited group of R, AsConjAs(R) = H N nd the intersetion of H with N trivil. We first set up SOUle nottion to enle us to desrie the elements of AsConjAs(R). We lel the elements of Ii., with the letters,,,..., use juxtposition to indite multiplition in the ssoited group of R nd upper rs to indite inverses in the ssoited group of R. In other words, the elemerits of the ssoited group of R re written s words in the symols {,, i, L., C,... }. The ssoited group of C onjas(r) is generted y these words (sujet to ertin reltions). We use irle, 0, to indite 38
multiplition in AsConjAs(R) nd the supersript '-1' to indite inverses. Note. The element 0 in AsConjAs(R) is equl to the genertor multiplied y the genertor. 1 whih is nother genertor. The inverse of the element in AsConj(R) IS (neessrily) equl to the element. It is not (neessrily) equl to the element equl to -I nd is not Let H e the sugroup of AsConjAs(R) generted y the elements of R s surk of C onja.s(r). Following the elements of R long the sequene: I " R~As(R)~ConjAs(R)~AsConjAs(R), we see tht H is the sugroup of AsConjAs(R) generted y the elements,,,.... This mens tht H is the sugroup onsisting of ll elements whih n e expressed in the form 1~1 0 1;2 0... I~n, where the Ii'S re single letters from the set {,,,... } nd e, = ±1. We ll these elements irle words. We hve the ommuttive digrm: ConjAs(R) li n ----+ AsConjAs(R) 17r ConjAs(R) Tf I ----+ As(R) where i is the identity mp. The elements of C onjas(r) re words in the symols,,,...,,,,..., with multiplition indited y juxtposition. These words re mpped, y n' 0 i, to the orresponding words in the group As(R). The mp "7 tkes the elements of C onjas(r) to their imges in AsConjAs(R). These imges re the genertors of AsConjAs(R) in stndrd presenttion. As i nd r7' re injetive, the only one of these words in the kernel of 7r is the word orresponding to the identity of As(R). A genertor, sy w, of AsCon) As(R) (in other words. the imge of n element of Conj A.s(R)) is mpped to the identil word in As( R). 39
The mp 7f is group homomorphism; therefore we hve nd 7f(W 0 w') = 7f(w) 07f(w'), =ww =. In other words, 7f tkes word in AsConjAs(R), 'removes the irles' nd reples inverses with rs. Exmple: 7f( 0 0 i:' 0 2 ) = 2 = 2. Clim: H, the group of irle words, is isomorphi to the ssoited group of R. Proof: the ssoited group of R is isomorphi to the free group on elements of R; quotiented y the norml sugroup, N, generted y ll elements of the form, where = in R. Let {,,... } e the genertors of the ssoited group of R orresponding to the elements of R. We define the mp 7f' : As(R) ----+ AsConjAs(R) y: n d,1'f W IS ' word : In A s (R),sy W - r e i En 1... r n, were h ri E { "... }, Ei = ±1 nd, here, r;l mens r i, then where, here, (7f'(r1))-1 mens (7f'(r1))-1. To prove tht 7f', s defined, is well defined group homomorphism from As(R) to AsConjAs(R) we need to prove tht 7f'(w) is trivil for ll W in N. In other words, we need to prove tht 7f'(n) is trivil in AsConjAs(R) for ll genertors, n of N. As we hve sid, ll genertors of N re of the form, where nd re elements of Rnd = in R. If = in R, then, s R is surk of AsConjAs(R), (,, - I,-1( ))(il-10t,'-1()) ('f-1,-1()) i 07] = t 07] C 40
in AsConjAs(R). Therefore, (TJOi,-IOTJ,-I(C))-IO(TJoi,-lor/-I())-IOTJ 0 i,-i 0 TJ,-I()oTJ 0 r:' 0 TJ,-I() is equl to the identity in AsConjAs(R), nd 7r' is well defined group homomorphism from As(R) to AsConjAs(R). As 7r' 0 7r is equl to the identity on the genertors of As(R), 7r' 07r is the identity on As(R) nd n' is injetive. By definition of the irle words, the imge of 7r' is equl to H. Therefore 7r' is n isomorphism from As(R) to H. We hve now shown tht there is sugroup of AsConjAs(R) isomorphi to the ssoited group of R. We now define N. Rell the ommuttive digrm ConjgAs(R) AsConjgAs(R) ConjAs(R) As(R). We define N to e the kernel of 7r. This mens tht the elements in ker( 7r) re preisely those elements of AsConjAs(R) whih eome trivil when we rernove the irles nd reple w- 1 with w. Exmples. The following elements re in ker( 7r) 0, 0 0 2 0 nd 0 L» «:', We hve now expliitly desried two sugroups, Nnd H. H is the sugroup onsisting of ll irle words nd N is the sugroup ontining ll words whih redue to the identity when we remove the irles nd reple w- I with w. N is norml s it is the kernel of tt. We hve lso shown tht 7r restrited to H is n isomorphism. Therefore no non-trivil element of H is in the kernel of tt nd the intersetion of H with N is trivil. We now show tht the produt of H with N is the whole of AsConjA.s(R). Clim: A.sConjAs(R) = H N. Proof: ny element of AsConj As(R) orresponds to irle word, otined 41
y repling ny ourrene of rred element, sy, with the orresponding inverse element, -I, nd putting irles etween ll the elements. For exmple, the element 0 2 0 s:' orresponds to 0 - I 0 C 0 o s:», Let W e n element of AsConjAs(R). We write w' for the irle word orresponding to w. As WW,- I is in N, the kernel of 1r, nd w' is n element of H, ny element, sy w, of AsConjAs(R), is equl to n 0 h where n equl to w 0 W'-l, is n element of ker(1r), nd h = o', n element of H. We now prove tht H is norml sugroup of AsConjAs(R) nd tht N is elin y showing tht N is in the entre of AsConjAs(R). We first need to onsider the reltions in AsConjAs(R). These re ll of the form: where wnd 11 re genertors. As n element of C onjas(r), o!", y definition, is equl to IlWIl; therefore the reltions in AsConjAs(R) re ll of the form: - -1-1 Ilw 11 = 11 0 w 0 11 We hve reltion of this form for ll pirs of genertors wnd 11 AsConjAs(R). In other words when we onjugte genertors, we n remove the irles nd reple inverse elements y the orresponding rred elements. Note. If wnd 11 in C onjas(r), is lso e genertor. re genertors of AsConjAs(R), then!", equl to IlWIl Now, let w e genertor of AsConjAs(R) nd let w' = W~I of 0 W~2 0... 0 w~n e typil word in N, with Wi genertor of AsConjAs(R) nd e, = ±1. Then: where w:- I mens Wi where pproprite. However, s w' is in the kernel of tt, z when we remove the irles nd reple inverse elements y the orresponding rred elements..o' eomes trivil. Therefore, -En -EI 0 0 WeI 0 0 Wen = W W n 0... 0 WI WI'.. n for ll genertors wnd N is in the entre of AsConj As(R). o 42
It is not known (to our knowledge) if the ssoited group of C onj(g) eing isomorphi to G x N, where N is elin, is suffiient ondition for G to e the ssoited group of rk R. We mke the following onjeture: Conjeture. If AsConj(G) rv G x N, where N is elin, then G is the ssoited group of rk R. If the ssoited group of Conj(G) is isomorphi to G x N, where N is elin, then the most likely ndidte for rk R with ssoited group G is the surk of C onj(g) onsisting of ll elements of the onjugy lsses of the genertors of the imge of G in the elinistion of G x N. We now turn our ttention to the opertor group. The opertor group. Any group n e n opertor group. We prove this y tking n ritrry group, G, nd onstruting rk, R, with opertor group isomorphi to G. In ft, s the rk we onstrut is qundle, we prove tht ny group n e the opertor group of qundle. 3.2 Theorem Any group, G, n e n opertor group. Proof Let G e n ritrry group, A nd B sets with elements in oneto-one orrespondene orrespondene with elements of G nd 0' nd j3 the ijetions nd j3 : B -----+ G. We define R s follows. The elements of R re the elements in the disjoint union of the sets A nd B. The tion is given y: r in R, in A. = 0'-1 (O'() 0 j3()) in B, in A.. i. ' in B. 43