On the structure of the centralizer of a braid

a Xi :ma h/0305156 2 [ma h.GT] 12 Sep 2003 On he s uc u e o he cen alize o a b aid Juan Gonz´ alez-Meneses1and Be Wies Dp o. de Ma em´a ica Aplicada I, E.T.S. A qui ec u a, Uni e sidad de Se illa, A da. Reina Me cedes, 2. 41012 Se illa, Spain; [email p o ec ed] and IRMAR (UMR 6625 du CNRS), Uni e si ´e de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, F ance; [email p o ec ed]ni - ennes1. Abs ac The mixed b aid g oups a e he subg oups o A in b aid g oups whose elemen s p ese e a gi en pa i ion o he base poin s. We p o e ha he cen alize o any b aid can be exp essed in e ms o semidi ec and di ec p oduc s o mixed b aid g oups. Then we cons uc a gene a ing se o he cen alize o any b aid on ns ands, which has a mos k(k+1) 2 elemen s i n= 2k, and a mos k(k+3) 2elemen s i n= 2k+ 1. These bounds a e shown o be sha p, due o wo k o N.V.I ano and o S.J.Lee. Finally, we desc ibe how one can explici ly compu e his gene a ing se . AMS Classi ica ion 20F36; 20E07, 20F65. Keywo ds b aid, cen alize , Nielsen-Thu s on heo y. 1 In oduc ion and s a emen o he esul s In 1971, Makanin [25] ga e an algo i hm o compu ing a gene a ing se o he cen alize Z(β) o any gi en elemen βo he n-s ing b aid g oup Bn. His me hod, howe e , ends o yield e y la ge, and highly edundan gene a ing se s. One hin ha much smalle gene a ing se s could be ound came om he expe imen al esul s o Gonz´alez-Meneses and F anco, which we e ob ained wi h a adically imp o ed e sion o Makanin’s algo i hm, based on new heo- e ical wo k [16]. Also, i has p obably been clea o specialis s o a long ime ha Nielsen-Thu s on heo y could be used o imp o e upon Makanin’s esul s. Howe e , he e seems o be no such esul in he li e a u e, and he aim o he p esen pape is o ill his gap. Al hough ou main in e es was o compu e, o any gi en β∈Bn, a small gene a ing se o Z(β), we succeed in desc ibing his cen alize in e ms o semidi ec and di ec p oduc s o mixed b aid g oups (see [26, 27]). These 1Pa ially suppo ed by MCYT, BFM2001-3207 and FEDER. 1 g oups a e de ined as ollows: le X={P1,...,Pn}be he base poin s o he b aids in Bn. Gi en a pa i ion Po X, he mixed b aid g oup BPconsis s o hose b aids whose associa ed pe mu a ion p ese es each cose o P. The well known classi ica ion o mapping classes o a punc u ed su ace in o pe iodic, educible and pseudo-Anoso ones, yields an analogous classi ica ion o b aids. I βis educible, hen one can decompose i , in a ce ain sense, in o a ubula b aid b β, and some in e io b aids β[1],...,β[ ], all o hem ha ing less han ns ands. The main esul o his pape is he ollowing: Theo em 1.1 Le β∈Bn. One has: (1) I βis pseudo-Anoso , hen Z(β)≃Z2. (2) I βis pe iodic, hen Z(β)is ei he Bno isomo phic o a b aid g oup on an annulus. (3) I βis educible, hen he e exis s a spli exac sequence: 1−→ Z(β[1])× · · · × Z(β[ ])−→ Z(β)−→ Z0(b β)−→ 1, whe e Z0(b β)is a subg oup o Z(b β), isomo phic ei he o Z2o o a mixed b aid g oup. No ice ha Z≃B2=B{{1,2}} , also Bn=B{{1,...,n}}, and inally he b aid g oup o e an annulus on ks ands is isomo phic o B{{1,...,k},{k+1}} ⊂Bk+1. Hence all hese g oups can be seen as mixed b aid g oups. Then, by ecu ence on he numbe o s ands we deduce he ollowing: Co olla y 1.2 Fo e e y β∈Bn, he cen alize Z(β)can be exp essed in e ms o semidi ec and di ec p oduc s o mixed b aid g oups. Using he abo e s uc u e we shall cons uc , o any b aid β∈Bn, a gene a ing se o Z(β) ha ing e y ew elemen s. Mo e p ecisely, we ob ain: Theo em 1.3 I β∈Bn, hen he cen alize Z(β)can be gene a ed by a mos k(k+1) 2elemen s i n= 2k, and a mos k(k+3) 2elemen s i n= 2k+ 1. We will p esen an example, communica ed o us by S. J. Lee, showing ha he abo e bound is sha p. Tha is, we will de ine, o e e y posi i e in ege n, a b aid in Bnwhose cen alize canno be gene a ed by less han k(k+1) 2elemen s i n= 2k, o less han k(k+3) 2elemen s i n= 2k+ 1. (The i s o obse e 2 ha he numbe o gene a o s o he cen alize may g ow quad a ically wi h he numbe o s ands was N.V.I ano [21].) Howe e , he abo e bound e e s o he wo s case, and one could be in e es ed in he minimal numbe o gene a o s o a pa icula b aid. We shall gi e a gene a ing se which is in some sense he smalles “na u al” gene a ing se o he cen alize o a b aid. Howe e , we shall also gi e an example ha illus a es he di icul y o inding he absolu ely minimum possible numbe o gene a o s. Le us men ion ha , o he special case o educible b aids conjuga ed o a gene a o σi, i s cen alize has al eady been desc ibed in [14]. The plan o he pape is as ollows: in sec ion 2 we se up no a ion and some s anda d machine y, and gi e he men ioned example by S. J. Lee. In sec ion 3 we s udy Z(β) in he case whe e βis pe iodic, sec ion 4 deals wi h he pseudo- Anoso case, and sec ion 5 he educible one, which is he mos in ol ed. In sec ion 6 we de ine a gene a ing se which is no la ge han he s a ed uppe bound. In sec ion 7 we desc ibe a gene a ing se which is as small as possible while s ill e lec ing he geome ic s uc u e o he Nielsen-Thu s on decompo- si ion. We also gi e an example o show ha by algeb aic icke y, e en smalle se s can be ob ained. Finally in sec ion 8 we discuss how he gene a ing se ha we de ined can be ound algo i hmically. 2 P e equisi es om Nielsen-Thu s on heo y We deno e by D he closed disk o adius 2 cen e ed a 0 in he complex plane. Fo any n∈N, he disk D, oge he wi h any choice o ndis inc poin s in i s in e io , is deno ed Dn, and he dis inguished poin s a e called he punc u es. We shall use di e en choices o he exac posi ion o he punc u es a di e en imes - hey may be lined up on he eal axis, o egula ly dis ibu ed on a ci cle o adius 1, o again one o hem may be in he cen e while he emaining n−1 a e dis ibu ed o e he ci cle o adius 1. In mos ins ances, he posi ion o he punc u es is i ele an , and we shall lea e i unspeci ied. We ecall ha he b aid g oup Bnis he g oup o iso opy classes o home- omo phisms ixing (poin wise) he bounda y and pe mu ing he punc u es o Dn. He e he iso opies mus ix poin wise he bounda y and he punc u es. Al e na i ely, Bncould be de ined as he g oup o iso opy classes o disjoin mo emen s o he punc u es, s a ing and ending wi h he con igu a ion o Dn. Ye ano he de ini ion o Bnis as he se o iso opy classes o b aids wi h n s ings in he cylinde D×[0,1], whe e he s a and end poin s o he s ings 3 a e exac ly he punc u e poin s in Dn× {0}and Dn× {1}. We shall use all h ee poin s o iew. We shall o en wo k wi h a ce ain quo ien o he g oup Bn, a he han wi h Bni sel . We ecall ha he cen e o Bnis isomo phic o he in ege s, and gene a ed by he ull wis ∆2(whe e ∆ is Ga side’s hal wis ). Geome ically, he g oup p ojec ion Bn→Bn/h∆2iis gi en by smashing he bounda y cu e o Dn o a punc u e, so ha Bn/h∆2iis na u ally a subg oup o he mapping class g oup o he sphe e wi h n+ 1 punc u es. In o de o keep no a ion manageable, we shall use he same le e s o elemen s o he b aid g oup Bn and o hei image in he quo ien Bn/h∆2i. This abuse o no a ion should no cause con usion. We say ha an elemen β∈Bnis pe iodic i he elemen o Bn/h∆2i ep e- sen ed by βis o ini e o de . Equi alen ly, βis pe iodic i he e exis s a k∈N such ha in Bnwe ha e ha βkis equal o some powe o ∆2. We say an elemen βo Bnis educible i he e exis s a nonemp y mul icu e Cin Dn(i.e. a sys em o disjoin simple closed cu es in Dn, none o hem iso opic o he bounda y o enclosing a single punc u e) which is s abilized by β, i.e. such ha β(C) is iso opic o C. No e ha βmay pe mu e di e en componen s o he mul icu e C. The ollowing de ini ion is aken om [8] (see also [20]). To e e y educible b aid β∈Bnone can associa e a canonical in a ian mul icu e: i s canonical educ ion sys em, which by de ini ion is he collec ion o all iso opy classes c o simple closed cu es which ha e he ollowing wo p ope ies: i s ly, cmus be s abilized by some powe o β, and secondly any simple closed cu e which has non-ze o geome ic in e sec ion numbe wi h cmus no be s abilized by any powe o β. Fo ins ance, le us conside he punc u ed disk D6, whe e he 6 punc u es a e a anged uni o mly on he ci cle o adius 1 a ound 0. Then he o a ion o he punc u es a ound he ci cle by an angle o 2π 3is a pe iodic elemen o B6(o pe iod 3), i is also educible (e.g. he h ee simple closed cu es enci cling punc u es 1 and 2, 3 and 4, and 5 and 6 espec i ely o m an in a ian mul icu e), bu i s canonical educ ion sys em is emp y. This example, howe e , is somewha un ypical: i a non-pe iodic b aid is educible, hen i s canonical educ ion sys em is nonemp y (see [20]). I Cis an in a ian mul icu e o a educible b aid β, hen we de ine he ubula b aid induced by βand C o be he b aid on ewe s ings ob ained om β by emo ing om Dnall he disks bounded by ou e mos cu es o C, and collapsing each ou e mos cu e o C o a punc u e poin . I should be s essed ha his b aid is only de ined up o conjugacy. 4 An al e na i e way o look a he same de in ion is he ollowing: le us conside again βas an iso opy class o ndisjoin s ings in D×[0,1] wi h ex emal poin s a he punc u e poin s o Dn× {0}and Dn× {1}, such ha each disk D× { }in e sec s each s ing exac ly once. Now ou pic u e can be comple ed by embedded cylinde s in D×[0,1] which a e disjoin om each o he and om he s ings o he b aid, each o which in e sec s each disk D× { }in exac ly one ci cle, and whose bounda y componen s a e exac ly he ou e mos cu es o Cin D×{0}and D×{1}. We can in e p e he solid cylinde s bounded by hese cylinde s as “ a s ings”, and he esul ing b aid wi h some a s ings is exac ly he ubula b aid de ined abo e. The in e io b aids induced by βand Ca e he b aids on ewe s ings in- duced by βa he in e io o he discs bounded by he ou e mos cu es o C. They can be hough o as he b aids ‘inside’ he ubes o he ubula b aid. The e o e, o e e y educible b aid β, and e e y in a ian mul icu e C, we can decompose βin o one ubula b aid and some in e io b aids – as many as he numbe o ou e mos cu es in C. Finally, we ha e he no ion o a pseudo-Anoso elemen o Bn, o which we e e o [13] o [20]. Roughly speaking, β∈Bnis pseudo-Anoso i i is ep esen ed by a homeomo phism o Dnwhich p ese es wo ans e se measu ed olia ions on Dn(called he “s able” and he “uns able” olia ion), while scaling he measu e o he uns able one by some ac o λwhich is g ea e han 1, and he measu e o he s able one by 1 λ. Thu s on’s heo em [32, 13] s a es ha e e y i educible elemen o Bnis ei he pe iodic o pseudo-Anoso . We end his sec ion wi h he p omised example, due o S. J. Lee, ha should be help ul o unde s anding he ela ionship be ween he Nielsen-Thu s on decomposi ion and he cen alize subg oup o a b aid β∈Bn. This example was also ound independen ly by N. V. I ano and H. Hamidi-Teh ani [22]. Example 2.1 Suppose ha n= 2m, and deno e by σi he s anda d gene a o o Bn, in which he i h and he (i+ 1)s punc u es pe mu e hei posi ions in a clockwise sense. We de ine β=σ1σ2 3σ3 5···σm 2m−1. The canonical educ ion sys em o βconsis s o mci cles, he i h one enclosing he punc u es 2i−1 and 2i. The co esponding ubula b aid is he i ial b aid o Bm, and he in e io b aids a e, espec i ely, σ1,σ2 1, . . . , σm 1(no ice ha all o hem a e non-conjuga e, since conjuga e b aids ha e he same exponen sum). 5 Le D(1),...,D(m)be he disks bounded by he abo e ci cles. As we shall see, any b aid ha commu es wi h βhas o send each disk D(i) o i sel (since he in e io b aids a e non-conjuga e). A gene a ing se o he cen alize subg oup o βis gi en by (i) o each i∈ {1,...,m}, he b aid σ2i−1, whose suppo is con ained in D(i), (ii) any gene a ing se o he pu e b aid g oup on ms ings Pm– all he gene a o s he e ac as he iden i y on D(1) ∪...∪D(m), and can be seen as a pu e ubula b aid on ms ings ( ubes), whe e he i h ube s a s and ends a D(i). I can be easily shown ha , in his case, Z(β)≃Zm×Pm. The essen ial obse a ion now is he ollowing: i can be deduced by he p esen a ion gi en in [6], ha he abelianiza ion o Pmis isomo phic o Zm(m−1)/2(see also [1]). Hence, he abelianiza ion o Z(β) is isomo phic o Zm×Zm(m−1)/2. The e o e, a leas m+m(m−1) 2=m(m+1) 2gene a o s a e needed o he cen alize o he b aid β. The case when n= 2m+ 1 is analogous. The b aid p oposed by S. J. Lee is: β=σ2σ2 4σ3 6···σm 2m. This ime he i s s and is no enclosed by any cu e o he canonical educ ion sys em o β, and one has: Z(β)≃Zm×Pm+1 . Hence, in his case he minimal possible numbe o gene a o s is m+m(m+1) 2=m(m+3) 2. By p o ing heo em 1.3, we will show ha he abo e examples a e he wo s one can ind. 3 The pe iodic case We ha e o s a by desc ibing he pe iodic elemen s o Bn. In o de o s a e his classi ica ion esul , which is classical, we need o de ine wo b aids. I Dnis he disk wi h npunc u es a anged egula ly on he ci cle o adius 1, hen he b aid which we shall call δ(n)is ep esen ed by a clockwise mo emen o all punc u es on his ci cle by an angle 2π n. I no con usion is possible, we shall simply w i e δ, wi hou indica ing he numbe o s ands (no e ha his b aid is he Ga side elemen o he Bi man-Ko-Lee s uc u e o Bn[7]). Simila ly, i we hink o Dnas ha ing one punc u e in he cen e, and n−1 punc u es a anged ci cula ly a ound i , hen we de ine γ(n)∈Bn o be he b aid gi en by a ci cula mo emen o he n−1 punc u es by an angle o 2π n−1, 6 while lea ing he cen al punc u e ixed. Again, o simplici y we shall o en only w i e γins ead o γ(n). The esul ha classi ies pe iodic b aids, which is due o Eilenbe g [11] and de Ke ´ekj´a ´o [23] (see [10] o a mode n exposi ion) is: Lemma 3.1 E e y pe iodic b aid in Bnis conjuga e o a powe o δ(n)o γ(n). Thus we only need o conside he cen alize subg oups o δk (n)and γk (n) o all n, k ∈Z, since he cen alize s o conjuga e elemen s a e isomo phic by an inne au omo phism o Bn. This p oblem has been sol ed by Bessis, Digne and Michel [4], on he wide con ex o complex e lexion g oups. We shall explain hei esul in he pa icula case o b aid g oups: We suppose i s ha β=δk (n)whe e, wi hou loss o gene ali y, k⩾0. Le d= gcd(n, k). Fo u= 1,...,n, we will deno e Pu=ei2πu/n he punc u es o Dn, so β=δk (n)sends Pu o Pu+k o e e y u( he indices a e aken modulo n). Hence he pe mu a ion induced by βhas do bi s (cycles) o leng h =n d, ha we deno e by C1,...,Cd. See in igu e 1 an example whe e n= 12, k= 9, d= 3 and = 4: he b aid δ(12) and he h ee o bi s o δ9 (12) . Figu e 1: The b aid δ∈B12 , and he h ee o bi s o δ9(in black, whi e and g ey). I > 1 ( ha is i d < n), conside he once punc u ed disc D∗=D {0}, and he −shee ed co e ing θ=θ :D∗→D∗de ined by θ(aei ) = aei = aei n/d . The o bi s C1,...,Cda e sen by θ o he poin s Q1,...,Qd, whe e Qu=ei2πu/d . I we conside he hal -line L={aeiπ/d, a ∈]0,2]}(no ice ha Lpasses be ween Qdand Q1), hen D∗ Lis a undamen al egion o θ(see igu e 2). 7 Figu e 2: The co e ing map θ=θ4associa ed o δ9 (12) . Now no ice ha e e y b aid in Bd(D∗) can be li ed, by θ−1, o a b aid in Bnin a na u al way. The esul ing b aid is a 2πd n-symme ic b aid, ha is, i is in a ian unde a o a ion by an angle o 2πd n. Bu hen i is also in a ian unde a o a ion o angle 2πk n; in o he wo ds, he esul ing b aid commu es wi h β. Hence we ha e a na u al homomo phism: θ∗:Bd(D∗)→Bnwhose image is con ained in Z(β). Then one has Theo em 3.2 ([4]) The na u al homomo phism θ∗:Bd(D∗)→Z(δk (n))is an isomo phism. In o he wo ds, e e y elemen in he cen alize o β=δk (n)can be seen ( ia θ) as a b aid on a once punc u ed disc, ha is, a b aid on an annulus. No ice ha i = 1 ( ha is, i kis a mul iple o n), hen βis a powe o δn (n)= ∆2 (n). In his case θis he iden i y map, and he undamen al egion is he whole Dn. Hence he cen alize o βis he whole Bn, as one should expec . Since we a e in e es ed in minimising he se o gene a o s, we obse e ha i d=n( hus = 1), hen Z(β) = Bnis gene a ed by wo elemen s, namely A in’s σ1and Bi man-Ko-Lee’s δ. In a simila way, i 1 < d < n, hen he b aid g oup Bd(D∗) is gene a ed by jus wo elemen s, namely δ(n)=θ∗(δ(d)) and he b aid θ∗(σ1) shown in igu e 3(a). No ice ha his case con ains he abo e one, whe e θ∗is he iden i y. Finally, i d= 1 hen B1(D∗) is cyclic. Thus we ha e: P oposi ion 3.3 I kand na e cop ime, hen Z(δk (n))is gene a ed by a single elemen , namely δ(n). I , by con as , gcd(k, n)⩾2, hen Z(δk (n))is gene a ed by wo elemen s: δ(n)and he b aid θ∗(σ1). 8 Figu e 3: Gene a o s θ∗ 3(σ1) and ¯ θ∗ 3(σ1) o he cen alise s o δ4 (12) and γ4 (13) . I is clea ha he gene a ing se gi en by p oposi ion 3.3 is indeed minimal. Nex we s udy he cen alize o β=γk (n), s ill ollowing he wo k in [4]. This ime we call d= gcd(n−1, k), and = (n−1)/d. I d < n −1, he abo e map θinduces a na u al homomo phism ¯ θ∗=¯ θ∗ :Bd(D∗)→Bn, whe e his ime he cen al poin o Dis conside ed as a punc u e. Hence, he cen al s and o e e y b aid coming om Bd(D∗) is i ial. We obse e ha he image o his homomo phism is con ained in Z(β), and in ac one has: Theo em 3.4 ([4]) The na u al homomo phism ¯ θ∗ :Bd(D∗)→Z(γk (n))is an isomo phism. By con as , i d=n−1, hen βis a powe o γn−1= ∆2, so θ∗ = 1, Z(β) = Bn and e e y hing wo ks as abo e. Hence we ha e P oposi ion 3.5 I kand n−1a e cop ime, hen Z(γk (n))is gene a ed by a single elemen , namely γ(n). I , by con as , gcd(k, n−1) = d⩾2, hen Z(γk (n)) is gene a ed by wo elemen s: γ(n)=¯ θ∗(δ(d))and he b aid ¯ θ∗(σ1). See igu e 3(b) o an illus a ion o he b aid ¯ θ∗(σ1). We summa ize all he esul s in his sec ion as ollows: Co olla y 3.6 The cen alize o any pe iodic b aid in Bnei he equals Bn o is isomo phic o Bd(D∗), o some d < n. In pa icula , i can be gene a ed by a mos wo elemen s. We end wi h a esul ha will be help ul la e : Co olla y 3.7 I kis no a mul iple o n, hen Z(δk (n))∼ =Z(γk (n+1)). 9 Figu e 7: How o mo e β[i] om Ci,4 o Ci,2, when i= 4. 5.2 Cen alize o a b aid in egula o m We will now s udy he cen alize o β, assuming ha βis in egula o m. Recall ha he only non- i ial in e io b aids o βa e deno ed β[1],...,β[ ], and ha b βis he ubula b aid associa ed o βand R(β). In his sec ion we will show ha he e is an exac sequence: 1→Z(β[1])× · · · × Z(β[ ])g −→ Z(β)p −→ Z0(b β)→1, whe e Z0(b β) is a subg oup o Z(b β). La e on we will see ha his sequence spli s. Fo i∈ {1,..., }, conside he cen alize Z(β[i]) in Bmi. We de ine a map gi:Z(β[i])→BR(β)as ollows: gi en γ∈Z(β[i]), gi(γ) is he b aid α∈BR(β) sa is ying bα= 1, αj,k = 1 o j6=i, and αi,k =γ o k= 1,..., i. We need o show he ollowing: P oposi ion 5.2 The map gide ined abo e is an injec i e homomo phism, and i s image is con ained in Z(β). P oo The map giis gi en by he diagonal homomo phism Z(β[i])→Z(β[i])× ...×Z(β[i]) ( i ac o s), ollowed by he homomo phism induced by an inclu- sion o icopies o an mi- imes punc u ed disk in o idisjoin subdisks (each con aining mipunc u es) o Dn. By he esul s o [28] we can deduce ha gi is indeed an injec i e homomo phism. I emains o show ha o e e y γ∈Z(β[i]) one has α=gi(γ)∈Z(β). Since bαis i ial, α−1βα =bα−1b βbα=b β. So we jus need o show ha he in e io b aids o α−1βα and βcoincide. Fo j6=i, he b aids αj,k a e i ial o e e y k, so α−1βαj,k =βj,k . Now, o k6= i, one has α−1βαi,k = 16 α−1 i,k βi,k αi,k+1 =γ−11γ= 1 = βi,k . Finally, since γcommu es wi h β[i], one has α−1βαi, i=α−1 i, iβi, iαi,1=γ−1β[i]γ=β[i]=βi, i. The e o e α−1βα =β, so he image o giis con ained in Z(β). P oposi ion 5.3 The map g:Z(β[1])× · · · × Z(β[ ])−→ Z(β)de ined by g(γ1,...,γ ) = g1(γ1)···g (γ )is an injec i e homomo phism. P oo Gi en γ∈Z(β[i]), he only non i ial s ands in gi(γ) a e hose inside he ubes Ci,1,...,Ci, i. Hence i i6=j,γ∈Z(β[i]) and δ∈Z(β[j]), hen gi(γ) and gj(δ) commu e. Since e e y giis a homomo phism, his shows ha gis also a homomo phism. Bu we know by he p e ious p oposi ion ha giis injec i e o i= 1,..., . Using an a gumen simila o he p oo o p oposi ion 5.2, one can deduce ha gis also injec i e. Now we will ela e Z(β) and Z(b β). E e y b aid in Z(β) p ese es he canonical educ ion sys em o β(see [20]), so i mus p ese e R(β). Tha is, Z(β)⊂ BR(β). Le p:BR(β)→Bmbe he homomo phism which sends α o bα, he ubula b aid induced by αand R(β). I we ake α∈Z(β) hen β=α−1βα, so p(β) = p(α−1βα) = p(α)−1p(β)p(α). Hence p(α) commu es wi h p(β) = b β. The e o e, i we es ic p o Z(β) we ge p:Z(β)→Z(b β). Un o una ely, nei he p:BR(β)→Bmno i s es ic ion p:Z(β)→Z(b β) a e su jec i e, bu we shall see ha he elemen s in he image o pin ei he case can be easily cha ac e ised by he pe mu a ion hey induce. No ice ha pinduces a bijec ion ep om R(β) o {P1,...,Pm}, he punc u es o Dm. We deno e by τ he in e se o ep. De ini ion 5.4 Le η∈Bm, and le πηbe he pe mu a ion induced by η on he punc u es o Dm. We say ha πηis consis en wi h R(β)i , o i= 1,...,m,τ(Pi)and τ(πη(Pi)) enclose he same numbe o punc u es. P oposi ion 5.5 An elemen η∈Bmis in he image o p:BR(β)→Bmi and only i πηis consis en wi h R(β). P oo I ηis in he image o p, le α∈BR(β)wi h p(α) = η. Then, o e e y i= 1,...,m,τ(Pi) and τ(πη(Pi)) a e he op and bo om ci cles o a ube de e mined by α. Hence hey mus enclose he same numbe o punc u es ( he numbe o s ands inside he ube). Con e sely, suppose ha πηis consis en wi h R(β). Take i∈ {1,...,m}and suppose ha τ(Pi) = Cj,k . Then ake he i h s and o ηand conside i as a 17 ube, enclosing he i ial b aid on mjs ands. Do his o e e y i= 1,...,m. The esul ing b aid, ψ(η), is well de ined since πηis consis en wi h R(β), and i belongs o BR(β). Mo eo e , p(ψ(η)) = ηby cons uc ion. The homomo phism ψin oduced in his p oo will play a p ominen ˆole in wha ollows: i η∈Bm, hen ψ(η) is he b aid in BR(β)whose ubula b aid equals η, and whose in e io b aids a e all i ial. All he elemen s in Bm ha shall be conside ed om now on will ha e pe - mu a ions consis en wi h R(β). Hence, by abuse o no a ion, we will iden i y Ci,k =ep(Ci,k) and Ci=ep(Ci) i i does no lead o con usion. We s ill need o cha ac e ise he elemen s in he image o p:Z(β)→Z(b β). We jus know ha hei pe mu a ions mus be consis en wi h R(β), bu his is no su icien . Recall ha he pe mu a ion induced by βon he componen s o R(β) has o bi s, C1,...C . The key obse a ion now is ha e e y elemen α∈Z(β) p ese es hese o bi s se wise, hough i could pe mu e hem. The e o e, o i= 1,..., , one has α(Ci) = Cj o some j. In he same way, o any η∈Z(b β) one has α(Ci) = Cj o some j. Lemma 5.6 Le α∈Z(β). I α(Ci) = Cj o some i, j ∈ {1,..., }, hen β[i]=β[j]. P oo Since α(Ci) = Cj, he wo o bi s ha e he same leng h, which we shall deno e ; hus = i= j. Now β is a b aid ha p ese es Ci,k and Cj,k o e e y k, and is such ha (β )i,k =β[i]and (β )j,k =β[j]. Now since α commu es wi h β, hen i also commu es wi h β . Suppose ha αsends Ci,1 o Cj,k . Then β[j]= (β )j,k = (α−1β α)j,k = (αi,1)−1(β )i,1αi,1= (αi,1)−1β[i]αi,1. The e o e β[i]and β[j]a e conjuga e, and since βis in egula o m, β[i]=β[j], as we wan ed o p o e. Lemma 5.6 imposes ano he condi ion o a b aid in Z(b β) o be in p(Z(β)): De ini ion 5.7 Le η∈Z(b β). We say ha πηis consis en wi h βi i is consis en wi h R(β)and, u he mo e, o e e y i, j ∈ {1,..., }such ha η(Ci) = Cj, one has β[i]=β[j]. De ini ion 5.8 Z0(b β)is he subg oup o Z(b β)consis ing o hose elemen s whose pe mu a ion is consis en wi h β. 18 Then lemma 5.6 can be es a ed as ollows: I α∈Z(β) hen p(α)∈Z0(b β). Mo eo e , we can p o e he ollowing: P oposi ion 5.9 The homomo phism p:Z(β)−→ Z0(b β)is su jec i e. P oo Le η∈Z0(b β). We shall cons uc a p eimage o ηunde pin wo s eps. Since πηis consis en wi h β( hus wi h R(β)), we can, as a i s s ep, conside he b aid ψ(η)∈Bn. We hen ha e p(ψ(η)) = η; bu ψ(η) does no necessa ily commu e wi h β, since he in e io b aids o ψ(η)−1βψ(η) could di e om hose o β. Ac ually, since he in e io b aids o ψ(η) a e all i ial, conjuga ing βby ψ(η) jus pe mu es he in e io b aids o β. Mo e p ecisely, he b aid ψ(η)−1βψ(η) equals β, excep ha , o each i∈ {1,..., }, i may no be he ube Ci, iwhich con ains he non i ial in e io b aid β[i], bu some o he ube om he amily Ci. Ou aim in he second s ep is hus o ill he ubes o ψ(η) wi h mo e sui able in e io b aids, in o de o ob ain a b aid ha commu es wi h β. Fo e e y i∈ {1,..., }, we know ha ψ(η) sends Ci o some Cj. Le ki∈ {1,..., i}be such ha ψ(η) sends Ci,ki o Cj, j, and conside he b aid µ(i, ki) de ined a he end o Subsec ion 5.1. I we conjuga e βby µ(i, ki) we mo e β[i] om Ci, i o Ci,ki. I we u he conjuga e by ψ(η), hen β[i]goes o Cj, j. Bu ηis consis en wi h β, so β[i]=β[j]. Hence, he in e io b aids in Cja e p ese ed. We can do his o i= 1,..., , so we ob ain ha he b aid Y i=1 µ(i, ki)!ψ(η) commu es wi h βand i s ubula b aid is η, so i is in p−1(η)∩Z(β). This shows he esul . We can inally b ing oge he all he esul s in his sec ion o s a e he ollowing: Theo em 5.10 Le β∈Bnbe a non-pe iodic educible b aid in egula o m. Then he sequence 1→Z(β[1])× · · · × Z(β[ ])g −→ Z(β)p −→ Z0(b β)→1 is exac . P oo By p oposi ion 5.3 gis injec i e, and by p oposi ion 5.9 pis su jec i e. I jus emains o show ha im(g) = ke (p). 19 By cons uc ion, e e y elemen in he image o ginduces a i ial ubula b aid, so im(g)⊂ke (p). Le hen α∈ke (p), ha is, bα= 1. Since α∈Z(β), we ha e α−1βα =β, and since βi,k = 1 o k6= i, we mus ha e α−1 i,k 1αi,k+1 = 1, so αi,k =αi,k+1 o k= 1,..., i−1. Hence αi,1=αi,2=· · · =αi, i o e e y i. Mo eo e , we ha e β[i]=βi, i=α−1 i, iβi, iαi,1=α−1 i,1β[i]αi,1, so αi,1∈Z(β[i]). The e o e, α=g1(α1,1)g2(α2,1)···g (α ,1) = g(α1,1, α2,1, . . . , α ,1). Tha is, ke (p)⊂im(g). 5.3 Finding a sec ion o p In his subsec ion we will p o e ha he exac sequence o heo em 5.10 spli s. We ecall ha b βis ob ained om βby collapsing he disks bounded by ou - e mos cu es in he canonical educ ion sys em o β o single punc u es. In pa icula , he canonical educ ion sys em o b βmus be emp y. Hence, b βis ei he pe iodic o pseudo-Anoso . We will dis inguish hese wo cases, o de ine a mul iplica i e sec ion o p, bu i s we will show an easy pa icula case. Recall ha a b aid is pu e i i induces he i ial pe mu a ion o i s base poin s. P oposi ion 5.11 I b βis pu e, he e is a homomo phism h:Z0(b β)→Z(β) such ha p◦h= 1. P oo We shall p o e ha in his case, he homomo phism ψcons uc ed in he p oo o p oposi ion 5.5 is such a sec ion. Le η∈Z0(b β). Since b βis pu e, Ci={Ci,1} o all i. Hence, i ηsends Ci o Cj hen i sends he ube Ci,1 (con aining β[i]) o he ube Cj,1(con aining β[j]=β[i], since βis in egula o m). The e o e, illing e e y ube in ηwi h he i ial b aid, ha is, de ining h(η) = ψ(η), yields indeed an elemen o Z(β). Nex we s udy he gene al case, depending whe he b βis pe iodic o pseudo- Anoso . P oposi ion 5.12 I b βis pe iodic, he e is a homomo phism h:Z0(b β)→Z(β) such ha p◦h= 1. P oo Recall ha we a e s udying βup o conjugacy. This implies ha we can also s udy b βup o conjugacy since, o e e y ξ∈Bm, i we conjuga e β by ψ(ξ) we a e conjuga ing b βby ξ. Mo eo e , a e conjuga ing by ψ(ξ), β con inues o be in egula o m (up o enaming he ci cles in R(β)). The e o e 20 we can suppose, up o conjugacy, ha b βis a igid o a ion o he disc, ha is, a powe o δ(m)o γ(m). Suppose i s ha b β=δk (m) o some k. We can suppose ha kis no a mul iple o m, since in ha case b βwould be a powe o ∆2 (m), hus i would be pu e, and his case has al eady been s udied in p oposi ion 5.11. Recall he analysis o pe iodic b aids in sec ion 3: he base poin s Q1,...,Qmo b βwill be e enly dis ibu ed along a ci cle o adius 1 a ound 0. Le d= gcd(m, k)< m and =m/d. Then b βsends Qi o Qi+k, and he e a e do bi s C1,...,Cdo leng h . The o bi Ciwill con ain he poin s Quwhe e u≡i(mod d). Since we can choose which ubes o βcon ain he in e io b aids, we will suppose ha hese a e he ubes s a ing a Qm−d+1, Qm−d+2,...,Qm, ha is, he las dpoin s o Dm. We will conside now some line segmen s in Dwhich sepa a e he poin s Q1,...,Qmin o se s o dpoin s. Le Lbe he line segmen joining he o igin wi h he bo de o D, passing be ween he poin s Qm−dand Qm−d+1 , and le L′be he segmen passing be ween Qmand Q1. No ice ha Land L′de e mine a sec o which con ains he poin s Qm−d+1, . . . , Qm, co espond- ing o he ubes o βwi h non i ial in e io b aids. Le ϕ:C→Cbe he o a ion a ound he o igin by an angle o 2πk/m ( he angle induced by b β), and deno e Li=ϕi(L). Since gcd(m, k) = d, he segmen s L0,...,L −1 di ide Din o m/d = sec o s, each one o angle 2π/ and con aining he poin s Qid+1,...,Qid+d o some i. Take he smalles in ege e > 0 such ha ϕe(L) = L′. Then one has L0=Land Le=L′. We a e in e es ed in he union o segmen s L=L1∪L2∪ · · · ∪ Le(see igu e 8 o an example). Figu e 8: The segmen s L,L′, and he union o segmen s L, o b β=δ6∈B15 . Le hen η∈Z0(b β). In o de o de ine h(η), i su ices o de ine i s in e io b aids. This is done as ollows: ecall ha , since ηcommu es wi h b β, i can be iso oped o a symme ic b aid (wi h espec o he o a ion ϕ), so we ake a symme ic ep esen a i e o η. Fo e e y base poin Qio b β(co esponding 21 o a ci cle Cj,u ), conside he s and o ηs a ing a Qi( he i h s and o η). Then we de ine he in e io b aid h(η)j,u = (β[j])L(η,i), whe e L(η, i)∈Z is he algeb aic numbe o imes ha he i h s and o ηc osses L. This is well de ined by heo em 3.2 (i you ake wo dis inc ep esen a i es o ηas a symme ic b aid, hey a e iso opic h ough symme ic b aids, so he s ands ne e ouch he o igin and he in e sec ion numbe L(η, i) is p ese ed). In o he wo ds, we de ine h(η) as ollows: we s a wi h i ial in e io b aids, and we ollow he mo emen o he s ands o η. Each ime a s and c osses a segmen o Lin he posi i e sense, we mul iply i s in e io b aid by β[j](whe e jis he index o he o bi Cjo ha s and). And e e y ime a s and c osses Lin he nega i e sense, we mul iply i s in e io b aid by β−1 [j]. We ha e hus de ined a map h:Z0(b β)→BR(β). To show ha his a homo- mo phism, i su ices o see ha he in e io b aids o ηξ a e he p oduc o hose o ηand ξ, o η, ξ ∈Z0(b β). Suppose ha he i h s and o ηgoes om Qi(co esponding o Cj,u ) o Qi′(co esponding o Cj′,u′). Hence η sends Cj o Cj′, and since η∈Z0(b β), i ollows ha β[j]=β[j′]. One also has, by de ini ion, L(ηξ, i) = L(η, i) + L(ξ, i′). The e o e (ηξ)j,u = (β[j])L(ηξ,i)= (β[j])L(η,i)(β[j])L(ξ,i′)=ηj,uξj′,u′, so his a homomo phism. We mus inally show ha , wi h his de ini ion, h(η)∈Z(β), o e e y η∈ Z0(b β). We will de ine i s some special b aids. Fo e e y i, j ∈ {1,...,d} such ha i < j and β[i]=β[j], de ine he symme ic b aid Si,j =Sj,i = θ∗ (σi···σj−2σj−1σj−2···σi) (see igu e 3 in sec ion 3 o ecall he de ini ion o θ∗ , and igu e 9 he e o an example). The b aid Si,j commu es wi h b β(since i is symme ic), and i pe mu es he o bi s Ciand Cj, p ese ing he o he s. Hence Si,j ∈Z0(b β). Mo eo e , i s s ands do no c oss L, so by de ini ion o h one has h(Si,j) = ψ(Si,j) ( he in e io b aids a e i ial). Figu e 9: The b aid S1,3, o b β=δ6∈B15 (assuming ha β[1] =β[3] ). Bu h(Si,j) commu es wi h β, since he only ubes i pe mu es a e hose o 22 he o bi s Ciand Cj; among hese ubes, he only wo wi h non- i ial in e io b aids a e exchanged, and hei co esponding in e io b aids a e equal (β[i]= β[j]). Hence he in e io b aids o βa e p ese ed by ψ(Si,j) = h(Si,j), so h(Si,j)∈Z(β). Take hen an a bi a y η∈Z0(b β). We mus show ha h(η)∈Z(β). Suppose ha ηsends Ci o Cj o some i, j. Then β[i]=β[j], so Si,j is de ined, and he b aid ηSi,j p ese es he o bi Ci. We can con inue his way, un il we ob ain a b aid ηSi1,j1···Sik,jk ha commu es wi h b βand p ese es e e y o bi Ci, o i= 1,...,d. Since h(Si,j)∈Z(β) o e e y i, j, and his a homomo phism, in o de o show ha h(η)∈Z(β) i su ices o show ha h(ηSi1,j1···Sik,jk)∈ Z(β). The e o e, we can suppose ha ηp ese es e e y o bi Ci. Deno e α=h(η). We need o show ha he in e io b aids o α−1βα coincide wi h hose o β. Since ηp ese es all o bi s, we will conside jus he ubes o C1, he o he ones being analogous. Suppose ha αsends he ci cle C1,u o C1, . Then i mus send C1, o C1, −u o e e y ( he indices a e aken modulo ). We will iden i y he poin s Q1,...,Qmwi h hei co esponding ci cles Ci, . Fo e e y = 1,..., , le b be he s and o ηs a ing a C1, . Since η is symme ic, we ha e ϕ(b ) = b +1 . Suppose ha b c osses imes he segmen Li, whe e i∈ {0,..., −1}. Then b +1 will c oss imes he segmen ϕ(Li) = Li+1 . The e o e, i b c osses l imes L, and i i c osses l0 imes L0 and le imes Le, hen b +1 c osses l−le+l0 imes L. I 6= and 6=u, hen b nei he s a s no ends a C1, . Then i c osses L0 and Le he same numbe o imes. Hence, b and b +1 c oss L he same numbe o imes, say l. The e o e, i 6= , u, one has (α−1βα)1, −u= (α1, )−1β1, α1, +1 =β−l [1] 1βl [1] = 1 = β1, −u. I u= = , hen b s a s a ends a C1, . Hence, as abo e, i c osses L0 and Le he same numbe o imes, so b =b and b +1 =b1c oss L he same numbe o imes, say l. We hen ha e (α−1βα)1, −u= (α−1βα)1, = (α1, )−1β1, α1,1=β−l [1] β[1]βl [1] =β[1] =β1, =β1, −u. Hence, i u= , we ha e al eady seen all he possible cases. We will hen suppose ha u6= . I = , hen b s a s (bu does no end) a C1, . Hence, i c osses Leone mo e ime (in he posi i e sense) han i c osses L0. The e o e, i b =b c osses l imes L, hen b +1 =b1c osses i l−1 imes. One has: (α−1βα)1, −u= (α−1βα)1, −u= (α1, )−1β1, α1,1=β−l [1] β[1]βl−1 [1] = 1 = β1, −u=β1, −u. Finally, i =u hen b ends (bu does no s a ) a C1, . In his case, i c osses Leone less ime (in he posi i e sense) han i c osses L0. Hence, i 23 b =buc osses l imes L, hen b +1 =bu+1 c osses i l+ 1 imes. One hen has: (α−1βα)1, −u= (α−1βα)1, = (α1,u)−1β1,uα1,u+1 =β−l [1] 1βl+1 [1] =β[1] = β1, =β1, −u. The e o e, in e e y possible case we ha e (α−1βα)1, −u=β1, −u, o e e y . This means ha he in e io b aids o (α−1βα) and o βcoincide, ha is, α=h(η) commu es wi h β, as we wan ed o show. This comple es he p oo o p oposi ion 5.12 in he case b β=δk (m), and i only emains o deal wi h he case when b β=γk (m). As abo e, we can suppose ha kis no a mul iple o m−1, since in ha case b βwould be pu e, and his case has al eady been ea ed in p oposi ion 5.11. Hence, he only ixed poin in he pe mu a ion induced by b βis he o igin. The e o e, e e y ηcommu ing wi h b βmus ix he o igin. This means ha , o e e y η∈Z0(b β), we can ill i s cen al ube wi h he i ial b aid, and he o he ubes in he same way as abo e (de ining L, and coun ing he numbe o imes each s and c osses L). This de ines a homomo phism h:Z0(b β)→Z(β) which is a sec ion o p. The p oo is he same as abo e. I emains o s udy he case when b βis pseudo-Anoso . P oposi ion 5.13 I b βis pseudo-Anoso , hen he e is a homomo phism h:Z0(b β)→Z(β)such ha p◦h= 1. P oo In his case, we know ha Z(b β) is a ee abelian g oup o ank 2, gen- e a ed by a pseudo-Anoso and a pe iodic b aid. Hence, Z0(b β) is an abelian g oup o ank one o wo. No ice ha ∆2 (m)∈Z0(b β), because his b aid com- mu es wi h b βand because π∆2is i ial, and hus consis en wi h β. Hence Z0(b β) con ains a leas one pe iodic elemen . On he o he hand, b βbelongs i sel o Z0(b β), since πb βis clea ly consis en wi h β. Hence in Z0(b β) he e a e also pseudo-Anoso b aids. Since all powe s o a pe iodic b aid a e pe iodic, and all powe s o a pseudo-Anoso b aid a e pseudo-Anoso , i ollows ha Z0(b β) has in ac ank wo. Mo e p ecisely, Z0(b β) = hηi × hρi, whe e ηis pseudo-Anoso and ρis pe iodic. In pa icula , we ha e b β∈ hηi × hρi, and he h ee b aids b β,ηand ρa e mu ually commu ing. Ou aim is o de ine wo commu ing b aids h(ρ) and h(η) in Z(β) which a e p eimages o ρ espec i ely ηunde p. The de ini ion o h(ρ) is e y simple: we ake an a bi a y p eimage o ρunde p– his is possible since pis su jec i e by p oposi ion 5.9. I emains o cons uc h(η). 24 Lemma 5.14 Suppose α∈BR(β), ha is, he b aid αp ese es he se o ou e mos cu es in he canonical educ ion sys em o β. Suppose also ha µ, ν ∈Z(bα). Suppose ha ιµ∈BR(β)is a b aid wi h i ial ubes (i.e. bιµ= 1) such ha ψ(µ)·ιµ∈Z(α). Finally, suppose ha µand νinduce he same pe mu a ion. Then we ha e as well ha ψ(ν)·ιµ∈Z(α). In o he wo ds, i wo ubula b aids commu e wi h bα, i hey induce he same pe mu a ion, and i some “ illing” o one o hem commu es e en wi h α, hen he same illing o he o he will also commu e wi h α. P oo o lemma 5.14 Conjuga ing αby ψ(ν)·ιµ∈Z(α) yields a ce ain b aid α′; we ha e o check ha α′=α. Fi s ly, we ha e an equali y o ubula b aids b α′=bα, because ν, he ubula b aid o ψ(ν)·ιµ, commu es wi h bα. Mo eo e , since µand νinduce he same pe mu a ions, we ha e o i= 1,...,m ha he i h ube o α′con ains he same b aid as he i h ube o (ψ(µ)·ιµ)−1·α·(ψ(µ)·ιµ). Since ψ(µ)·ιµcommu es wi h α, his is in u n he same as he i h ube o α. In summa y, αand α′ha e he same ubula b aids, and co esponding ubes con ain he same in e io b aids, which implies ha α=α′. Nex we ha e o hink in de ail abou he o bi s uc u e o b β. Le us choose a bi a ily a punc u e Po he disk Dm(on which b βac s), and le O(b β, ρ) be he o bi o ha punc u e unde he ac ion o he subg oup hb βi × hρio Z0(b β). Le O(b β, ρ, η) be he o bi o Punde he ac ion o he g oup hρi × hηi(no e ha his g oup is also isomo phic o Z2, and con ains b β). We a e going o suppose wi hou loss o gene ali y ha O(b β, ρ, η) con ains all punc u es o Dm, and we shall speci y how he ubes o ηco esponding o his o bi shall be illed – indeed, i he e a e o he o bi s, hen hese can be ea ed in same way, independen ly. Special case: Le us s a by conside ing he simple special case ha O(b β, ρ, η) = O(b β, ρ), i.e. ha he ac ion o ηp ese es he (b β, ρ)-o bi . In his case we ha e Lemma 5.15 The e exis in ege s kand lsuch ha ηand b βk·ρlinduce he same pe mu a ions on O(b β, ρ). P oo o lemma 5.15 One can choose kand lsuch ha b βkρl(P) = η(P), simply because η(P) is in he o bi o Punde he ac ion o b βand ρ. Now i 25 ⩽S+ 1 2−m−1 2+b+m 2 =S(S+ 1) 2+b+m−1 = S(S+ 1) 2+b+u+ ⩽S(S+ 1) 2+S+ =S(S+ 3) 2+ . I = 0 hen b= 0 and k=S, so |G′|⩽S(S+3) 2=p(n). I = 1 hen n= 2S+2 and k=S+1. Then |G′|⩽S(S+3) 2+1 = (S+1)(S+2) 2= p(n). I = 2 hen n= 2S+ 3 and k=S+ 1. Then |G′|⩽S(S+3) 2+ 2 = S2+3S+4 2< (S+1)(S+4) 2=p(n). Finally, i ⩾3 hen n= 2S+ + 1 so k⩾S+ /2. Hence p(n)⩾(S+ /2)(S+ /2 + 1) 2=S2+ ( + 1)S+ ( + 2)/4 2 ⩾S(S+ 3) 2+S/2 + /2>S(S+ 3) 2+ ⩾|G′|. The e o e, in e e y case |G′|⩽p(n), and heo em 1.3 is p o ed. Recall ha , in example 2.1, we de ined b aids o any numbe o s ands whose cen alize could no be gene a ed by less han p(n) elemen s. The e o e, he bound gi en by heo em 1.3 is he bes possible one. 7 Small gene a ing se s We saw in he p e ious sec ion an uppe bound o he numbe o gene a o s o he cen alize o a b aid β, in e ms o i s numbe o s ings. Bu one could ob ain a be e bound i mo e in o ma ion abou βis gi en. In his sec ion we will de ine a new gene a ing se G o Z(β), which is in mos cases smalle han he se G′de ined be o e. I is also he smalles possible “na u al” gene a ing se , in he sense ha each gene a o belongs o one o he + 1 ac o s in he semidi ec p oduc decomposi ion in heo em 1.1(c). Thus in a philosophical sense, Gis he “ igh ” gene a ing se , e en hough i is no in gene al he smalles possible one, as we shall see a he end o his sec ion. I βis pe iodic o pseudo-Anoso , we al eady know a minimal gene a ing se , wi h a mos wo elemen s. We also know a minimal gene a ing se o any 32 mixed b aid g oup (see he p oo o p oposi ion 6.1). Hence we can de ine Gby induc ion on he numbe o s ands, when βis a educible, non-pe iodic b aid. We can also suppose ha βis in egula o m. We ecall ha he in e io b aids a e deno ed β[1],...,β[ ], and he ubula b aid b β. De ini ion 7.1 We will say ha i, j ∈ {1,..., }a e pe mu able i he e exis s some η∈Z0(b β)such ha η(Ci) = Cj. Rema k ha pe mu abili y is an equi alence ela ion, and he de ini ion o Z0(b β) says ha i iand ja e pe mu able hen β[i]=β[j]. Le hen {i1,...,i } ⊂ {1,..., }be cose ep esen a i es o pe mu abili y. Le Gikbe a minimal se o gene a o s o Z(β[ik]), and G0be a minimal se o gene a o s o Z0(b β). Then we de ine G=gi1(Gi1)∪ · · · ∪ gi (Gi )∪ h(GH). No ice ha G⊂G′, and hey coincide i and only i he e is no pai o pe mu able indices. P oposi ion 7.2 Gis a gene a ing se o Z(β). P oo F om he exac sequence o heo em 5.10 i ollows ha , i Giis a se o gene a o s o Z(β[i]), hen a se o gene a o s o Z(β) is G′=g1(G1)∪ · · · ∪ g (G )∪h(G0). Hence, we jus need o show ha i j∈ {1, . . . , } {i1,...,i }, hen e e y elemen in gj(Gj) can be w i en as a p oduc o elemen s in G. Take hen jas abo e. The e mus be some ikpe mu able wi h j, so β[j]=β[ik] and he e is some η∈Z0(b β) such ha η(Cik) = Cj. No ice ha Gjis a se o gene a o s o Z(β[j]) = Z(β[ik]), so e e y γ∈Gjcan be w i en as a p oduc o elemen s in Gik. Hence he b aid α=h(η)−1gik(γ)h(η) can be w i en as a p oduc o elemen s in G. Mo eo e , one has bα=d h(η)−11d h(η) = 1, and he only non i ial in e io b aids in αa e hose co esponding o Cj. Since he in e io b aids h(η)ik,l o e e y la e jus powe s o β[ik]=β[j], and γ commu es wi h β[j], i ollows ha o e e y l,αj,l =γ. The e o e α=gj(γ), so e e y elemen in gj(Gj) can be w i en as a p oduc o elemen s in G, hus Gis a gene a ing se o Z(β). The gene a ing se we ha e jus de ined is, un o una ely, no always he small- es possible one: Example 7.3 Conside he i e s ing b aid β=σ3σ4σ2σ3σ1σ2σ2σ3σ4σ1σ2σ3 – he canonical educ ion sys em o his b aid has wo ound ci cles, one con- aining punc u es numbe 1, 2 and 3, he o he punc u es numbe 4 and 5; he 33 ubula b aid is jus a ull wis o he wo a s ings: b β=σ2 1. Mo eo e , he in e io b aids o each ube is i ial. Acco ding o heo em 1.1, he cen alize o his b aid is Z(β)∼ =(B3×B2)⋊PB2∼ =(B3×Z)⋊ Z and he gene a ing se cons uc ed in his sec ion has ou elemen s: wo o B3, and one o each ac o Z. We now claim ha his gene a ing se is no as small as possible. Indeed, B3×Zcan be gene a ed by only wo elemen s (and hus Z(β) can be gene a ed by h ee elemen s). To see his, ecall ha he 3-s ing b aid g oup is isomo phic o he g oup o he (2,3)- o us kno . Thus B3has a p esen a ion hy, z |y3z−2= 1i(wi h y=σ1σ2and z=σ1σ2σ1). Mo eo e , he ac o Zis gene a ed by σ4. Now he wo gene a o s (y, σ4) and (z, σ4) gene a e B3×Z, because (1, σ4) can be w i en as (y, σ4)3(z, σ4)−2. 8 Some algo i hmic aspec s The aim o his sec ion is o p esen he essen ial ing edien s o an algo i hm which, o any gi en b aid, inds a gene a ing se o i s cen alize subg oup ha ma ches he desc ip ion o he p e ious sec ions. Since, o any b aid βand any k∈Z, he cen alize subg oups o βand β∆2kcoincide, we can always assume ha βis posi i e. We s a by men ioning ha algo i hms ha pe o m he Nielsen-Thu s on classi ica ion, and gi e he in a ian oli a ions in he pseudo-Anoso case (in he o m o ain acks), a e a ailable – no ably, he e a e Bes ina-Haendel’s [5] and o Los’ [24] algo i hms; and compu e implemen a ions a e a ailable on he web [9, 19]. We ecall b ie ly he idea o he wo au oma ic s uc u es on b aid g oups ha a e ele an o us: o he i s one, gi en by Ga side [17] and Thu s on [33] (and e ined by El-Ri ai and Mo on [12]), we hink o Dnhas ha ing he n punc u es lined up on he eal line in he disk D. Fo he second one, gi en by Bi man, Ko, and Lee [7], we hink o Bnas ha ing he npunc u es egula ly spaced on he ci cle o adius 1. Apa om ha , he s uc u es a e exac ly analogue. In he Ga side-Thu s on s uc u e, he e is a canonical way o w i e βas a p oduc o di iso s o ∆, namely by pushing each c ossing be ween wo s ings in o a ac o as a o he le as possible. This no mal o m is called he le g eedy no mal o m. Fo ins ance, in his no mal o m all ac o s which 34 a e equal o ∆ (no jus di iso s o i ) a e g ouped oge he a he e y le o he p oduc decomposi ion. Analogously, Bi man-Ko-Lee w i e each b aid as a p oduc o di iso s o δin a le -g eedy way. I βis a posi i e b aid, hen i s supe summi se is he subse o all elemen s αo i s conjugacy class which sa is y he ollowing condi ions: (i) αis posi i e, (ii) he w i ing o αin le g eedy no mal o m has as ew ac o s as possible among all elemen s sa is ying (i), (iii) he w i ing o o αin le g eedy no mal o m has as many ac o s on he le as possible equal o ∆ (o δ), among all elemen s sa is ying (i) and (ii). Two posi i e elemen s o Bna e conjuga e i and only i hei supe summi se s coincide. Gi en β∈Bn he e is an algo i hm, gi en in [15] (which is an imp o emen o he algo i hm in [12]), o compu e i s supe summi se . I is as ollows: i s we epea edly cycle β(i.e. mo e he i s ac o di e en om ∆, espec i ely δ, o he end and calcula e he le g eedy o m o he esul ing b aid), un il his p ocess uns in o a loop. A his poin we a e gua an eed o ha e achie ed condi ion (ii) abo e. Then we epea edly decycle (i.e. mo e he las ac o o he on and calcula e he le g eedy o m o he esul ing b aid) un il we un in o a loop. Then all elemen s o his loop belong o he supe summi se . A e wa ds, all o he elemen s o he supe summi se can be ound ecu si ely by conjuga ing al eady known elemen s by (sui able) di iso s o ∆ ( espec i ely δ), and e aining he esul i i belongs o he supe summi se . This algo i hm o compu ing he supe summi se is necessa y o ou pu - poses. Now suppose we a e gi en a b aid β∈Bnand we wan o compu e i s cen alize . Fi s we need o de e mine i βis pe iodic, educible o pseudo- Anoso , and hen we can use he esul s in his pape . Rema k 8.1 Ve y ecen ly, V. Gebha d [18] p esen ed a be e algo i hm o he conjugacy p oblem in b aid g oups. He de ined he ul a summi se , which is in gene al much smalle han he supe summi se desc ibed he e. Pe iodic elemen s Deciding whe he a gi en elemen βo Bnis pe iodic is e y easy: one calcula es he n−1s and he n h powe o β. Then βis pe iodic i and only i one o he wo esul s is a powe o ∆2. 35 I βn−1= ∆2k o some k∈N, hen βis conjuga e o γk (n)(as can be easily seen om lemma 3.1), and a conjuga ing elemen can be ound explici ly using ei he o he wo s anda d algo i hms. Simila ly, i βn= ∆2k, hen βis conjuga e o δk (n), and ei he algo i hm yields an explici conjuga ing elemen . In ei he case, one can ind explici ly a gene a ing se o he cen alize subg oup wi h only wo elemen s, using p oposi ions 3.3 o 3.5. Finding educing cu es o educible elemen s A e es ablishing ha an elemen βo Bnis no pe iodic, we need o check whe he i is educible, and i i is, we wan o ind explici ly an in a ian mul icu e. This is, in ac , a s anda d pa o Bes ina-Haendel’s [5] and o Los’ [24] algo i hms. We wan o poin ou one pa icula y elegan al e na i e, which is due o Be- na de e, Gu ie ez and Ni ecki [3] (see also [2]). We hink o Dnas ha ing he npunc u es lined up ho izon ally, and we look a Ga side-Thu s on’s le g eedy no mal o m. The key obse a ion om [3] is he ollowing: suppose ha Cis an in a ian mul icu e o a b aid β, and ha he no mal o m o β is β=β1·...·βk, whe e β1,...,βk∈Bna e di iso s o ∆. Mo eo e , suppose ha all componen s o Ca e ound (i.e. ac ual geome ic ci cles in Dn). Then we ha e no only ha β1·...·βn(C) = C, bu also ha all componen s o all he mul icu es β1·...·βi(C) a e ound o i= 1,...,k. As ema ked in [3] his implies as a co olla y ha in a ian mul icu es a e isible as ound cu es in he supe summi se o β, and in pa icula he educibil y o a b aid is easily de ec able om he supe summi se . To p o e he co olla y we no e ha βhas a conjuga e in which all componen s o he cu e sys em Ca e ound; mo eo e , βand i s conjuga e ha e he same supe summi se . Now cycling and decycling his conjuga e does no change he ac ha he e is a ound in a ian cu e sys em, by he key obse a ion abo e. A he end o he cycling/decycling p ocedu e we ha e ound elemen s o he supe summi se which con ain he desi ed ound in a ian cu es. Now i is shown in [3] how o de e mine i a gi en b aid p ese es a sys em o disjoin ound cu es. And he e is a ini e numbe o hese sys ems. Mo eo e , since o each elemen o he supe summi se we know how i can be conjuga ed o ob ain β, we can ind explici ly all cu es ha belong o a educ ion sys em o β. We can hen easily de e mine, by i s de ini ion, which o hese cu es belong o he canonical educ ion sys em o β. Tha is, we can compu e he canonical educ ion sys em o β. 36 By he esul s in his pape , Z(β) is hen a semi-di ec p oduc o wo g oups ha can be compu ed by induc ion on he numbe o s ings. Hence, i only emains o s udy he case when βis pseudo-Anoso . Pseudo-Anoso elemen s: commu a ion wi h δk (n) Suppose ha ou b aid β ails he es s o pe iodici y and educibili y, hence i is known o be pseudo-Anoso . We need o check i i commu es wi h a pe iodic b aids o he han powe s o ∆2. We shall hink o Dnas ha ing i s npunc u es uni o mly dis ibu ed o e he ci cle o adius 1, and we conside Bi man-Ko-Lee’s le -g eedy no mal o m. We wan o decide algo i hmically whe he βis conjuga e o a b aid αwi h he p ope y ha αcommu es wi h δk (n) o some posi i e in ege k < n. I i is, we wan o know he conjuga ing b aid explici ly. The ollowing esul yields such an algo i hm. P oposi ion 8.2 Suppose ha a pseudo-Anoso b aid βhas a conjuga e which commu es wi h δk (n) o some in ege k. Then he e exis s an elemen α o he supe summi se o βwhich has he p ope y ha α, and in ac e e y ac o o he le g eedy no mal o m o α, commu es wi h δk (n). P oo Le β′be a conjuga e o βwhich commu es wi h δk (n). I β′=β′ 1·...·β′ is he le -g eedy no mal o m o β′, hen each ac o β′ iis a di iso o δ(n)which is 2πk n-symme ic. This ollows om he ac ha he e y de ini ion o he le - g eedy no mal o m is comple ely o a ion symme ic. Mo e p ecisely, he ac ha wo consecu i e ac o s β′ iβ′ i+1 de e mine a le -g eedy no mal o m is no modi ied by o a ing hem. Hence, he p oduc (δ−k (n)β′ 1δk (n))···(δ−k (n)β′ δk (n)) is in le -g eedy no mal o m. Since his p oduc equals δ−k (n)β′δk (n)=β′, whose le - g eedy no mal o m is β′ 1···β′ k, we ob ain ha δ−1 (n)β′ iδ(n)=β′ i, o i= 1,..., . Using he same a gumen induc i ely, we see ha he cycling and decycling p ocedu e only e e c ea es b aids in le g eedy no mal o m in which all ac o s a e 2πk n-symme ic. Now we no ice ha i is e y easy o decide i a gi en di iso o δ(in he Bi man-Ko-Lee con ex ) is in a ian unde a gi en o a ion. Hence one can de e mine i a b aid commu es wi h an (explici ly compu able) conjuga e o δk (n)by looking a he elemen s o i s supe summi se . 37 Pseudo-Anoso elemen s: commu a ion wi h γk (n) Now we wan o de e mine i a gi en pseudo-Anoso b aid commu es wi h a conjuga e o γk (n), o a gi en posi i e in ege k < n −1. This is only possible i he e is some index i∈ {1,...,n}such ha βp ese es Pi, as can be easily seen by looking a he co esponding pe mu a ions. Call Pi={{Pi},{P1,...,Pi−1, Pi+1,...,Pn}}, a pa i ion o {P1,...,Pn}. Then βshould belong o BPi. The e is a na u al map i:BPi→Bn−1which consis s o o ge ing he i h s ing. No ice ha , i a b aid αcommu es wi h γk (n)(whe e P1is conside ed o be he cen al poin o D(n)) hen 1(α) com- mu es wi h 1(γk (n)) = δk (n−1) . Hence we ha e a necessa y condi ion ha mus be sa is ied. I βp ese es a punc u e Pi, hen we conjuga e i o some α ha p ese es P1, and we es whe he a conjuga e o 1(α) commu es wi h δk (n−1) o some k < n −1. I his does no happen, o i= 1,...,n, hen no conjuga e o βcommu es wi h γk (n). This necessa y condi ion is o cou se no su icien . A su icien and es able condi ion is now gi en by he ollowing esul . Recall ha , by co olla y 3.7, he e is an isomo phism χ= (¯ θ∗)−1θ∗ om Z(δk (n−1)) o Z(γk (n)), gi en by adding a i ial s ing a he cen e o Dn−1. No ice ha , i ζ∈Z(γk (n)), hen χ( 1(ζ)) = ζ. Then one has: P oposi ion 8.3 Suppose ha α∈Bnp ese es P1, and eα= 1(α)com- mu es wi h δk (n−1) . Then he ollowing wo s a emen s a e equi alen . (i) αis conjuga e o an elemen ζo Bnwhich commu es wi h γk (n), and he conjuga ing homeomo phism p ese es P1. (ii) αis conjuga e o χ(eα). P oo The implica ion (ii)⇒(i) is immedia e, by choosing ζ:= χ(eα). Fo he implica ion (i)⇒(ii), we suppose ha (i) holds, ha is, he e is an elemen η∈BP1such ha η−1αη =ζ, whe e ζ∈Z(γk (n)). We can apply 1 o all hese elemen s, deno ing eη= 1(η) and e ζ= 1(ζ). This yields (eη)−1eαeη=e ζ, whe e eα, e ζ∈Z(δk (n−1)). I we show ha eη∈Z(δk (n−1)), hen we can apply χ o all ac o s, ob aining χ(eη)−1χ(eα)χ(eη) = χ(e ζ) = ζ, hence χ(eα) is conjuga e o ζwhich is conjuga e o α, and he esul ollows. 38 Le us hen show ha eηcommu es wi h δk (n−1) . No ice ha ζis a pseudo- Anoso b aid ha commu es wi h γk (n). Hence i p ese es a p ojec i e olia ion Fζ, which is in a ian unde a o a ion by an angle o 2πk n−1. Bu in his case e ζ also p ese es Fζ, wi h he same s e ch ac o , hence i is also pseudo-Anoso . Since eαis conjuga ed o e ζ, hen i is pseudo-Anoso as well, and we call Feα i s co esponding p ojec i e olia ion (which is also in a ian unde he same o a ion, since eαcommu es wi h δk (n−1) ). Since (eη)−1eαeη=e ζ, we ha e ha eη sends Feα o Fζ. Now conside he b aid d=eη−1δk (n−1)eη. I is conjuga e o δk (n−1) , and hence pe iodic. Mo eo e , i p ese es Fζ, so i commu es wi h e ζ. Bu he pe iodic elemen s in he cen alize o e ζ o m a cyclic g oup con aining δk (n−1) , and δk (n−1) is he only elemen ha ing exponen sum (n−2)k. Since dhas exac ly he same exponen sum, i ollows ha d=δk (n−1) . Hence eηcommu es wi h δ(n−1) , and he esul ollows. An algo i hm o es ing whe he a b aid βis conjuga e o a b aid which com- mu es wi h γk (n)is now easy o cons uc : o each o he npunc u es es whe he he punc u e is ixed by β, and whe he o ge ing his punc u e yields a b aid which is conjuga e o a b aid eα ha commu es wi h δk (n). (We know how o do his, by he esul s o he p e ious subsec ion). Fo each punc u e ha does sa is y his p ope y, es whe he χ(eα) (which is ob ained om eα by adding a “ i ial” s ing in he cen e), is conjuga e o β. I , o one o he punc u es, his is he case, hen he answe is “yes”, o he wise “no”. Pseudo-Anoso elemen s: inding oo s I emains o desc ibe a las s ep o compu ing a gene a ing se o Z(β), when βis pseudo-Anoso . We assume ha we ha e al eady compu ed he subg oup hρio pe iodic b aids commu ing wi h β. Then we can mul iply βby a sui able powe o ρ, o ob ain a b aid b ha p ese es he singula lea es o he p ojec i e olia ions co esponding o β. Then we know ha Z(β) = hαi × hρi, whe e αis he smalles possible oo o b. The las p oblem, he e o e, is o de e mine whe he a gi en pseudo-Anoso b aid bhas a k h oo , o gi en k, and o compu e ha oo . This p oblem has been sol ed in [31] (gene alised o all Ga side g oups in [30]). Mo eo e , since he numbe o possible alues o kis ini e (we a e assuming ha bis posi i e), we ha e an algo i hm o compu ing α, hus a gene a ing se o Z(β). 39 Acknowledgemen s We a e g a e ul o a numbe o people o discussions and aluable ideas ha g ea ly con ibu ed o his esea ch. The examples o Nikolai V. I ano [21, 22], which we lea ned abou h ough discussions wi h Mus a a Ko kmaz, we e an impo an inspi a ion and g ea ly helped us cla i y ou ideas. I was hus om I ano ( ia Ko kmaz) ha we lea ned ha he num- be o gene a o s may ha e o g ow quad a ically wi h he numbe o s ings, con adic ing a conjec u e in [16]. We a e e y g a e ul o Sang Jin Lee who la e , bu independen ly o I ano , came up wi h his examples, conjec u ed ha hey ep esen he wo s case, and kindly communica ed hese ideas o us by email. (Hessam Hamidi-Tehe ani ound he same examples as Lee immedia ely a e lis ening o I ano ’s alk, bu we didn’ lea n his un il e y ecen ly.) We also hank Da id Bessis o use ul discussions, and Joan Bi man o elling us abou he e e ences [2] and [3]. Re e ences [1] V. I. A nold, The cohomology ing o he g oup o dyed b aids, Ma .Zame ki, V. 5 (1969), 227-231. [2] D. Bena de e, M. Gu ie ez, Z. Ni ecki, A combina o ial app oach o educibili y o mapping classes, Con empo a y Ma h. 150 (1993), 1–31. [3] D. Bena de e, M. Gu ie ez, Z. Ni ecki, B aids and he Nielsen-Thu s on clas- si ica ion, J. Kno Theo y and i s Rami ica ions 4 (1995), 549-618. [4] D. Bessis, F. Digne, J. Michel, Sp inge heo y in b aid g oups and he Bi man- Ko-Lee monoid, Paci ic J. Ma h. 205 (2002), no. 2, 287–309. [5] M. Bes ina, M. Haendel, T ain- acks o su ace homeomo phisms, Topology 34 (1995), 109–140. [6] J. Bi man, B aids, links, and mapping class g oups, Annals o Ma h. S udies, 82, P ince on Uni e si y P ess 1975. [7] J. Bi man, K. H. Ko, S. J. Lee, A new app oach o he wo d and conjugacy p oblems in he b aid g oups, Ad . Ma h. 139 (1998), 322–353. [8] J. Bi man, A. Lubo zky and J. McCa hy, Abelian and sol able subg oups o he mapping class g oups, Duke Ma h.J. 50 (1983), 1107-1120. [9] P. B inkmann, An implemen a ion o he Bes ina-Handel algo i hm o su ace homeomo phisms, Expe imen . Ma h. 9 (2000), 235–240. Compu e p og am a ailable a www.ma h.uiuc.edu/~b inkman/so wa e/ ain/ [10] A. Cons an in, B. Kole , The heo em o Ke ´ekj´a ´o on pe iodic homeomo - phisms o he disc and he sphe e, Enseign. Ma h (2) 40 (1994) No 3-4, 193-204. [11] S. Eilenbe g, Su les ans o ma ions p´e iodiques de la su ace de sph`e e, Fund. Ma h. 22 (1934), 28–41. 40 [12] E. A. El-Ri ai, H. R. Mo on, Algo i hms o posi i e b aids, Qua . J. Ma h. Ox o d Se . (2) 45 (1994), no. 180, 479–497. [13] A. Fa hi, F. Laudenbach, V. Poena u, T a aux de Thu s on su les su aces - seminai e O say, As e isque 66–67, Soci´e ´e Ma h. de F ance, 1991. [14] R. Fenn, D. Rol sen, J. Zhu, Cen alise s in he b aid g oup and singula b aid monoid, l’Enseignmen Ma h´ema ique 42 (1996), 75–96. [15] N. F anco, J. Gonz´alez-Meneses, Conjugacy p oblem o b aid g oups and Ga - side g oups. To appea in Jou nal o Algeb a. [16] N. F anco, J. Gonz´alez-Meneses, Compu a ion o cen alize s in b aid g oups and Ga side g oups. To appea in Re is a Ma em´a ica Ibe oame icana. [17] F. A. Ga side, The b aid g oup and o he g oups, Qua . J. Ma h. Ox o d 20 (1969), 235–254. [18] V. Gebha d , A New App oach o he Conjugacy P oblem in Ga side G oups, P ep in (2003). a Xi :ma h.GT/0306199. [19] T. Hall, Compu e implemen a ion o Bes ina-Handel algo i hm a ailable a www.li .ac.uk/ma hs/PURE/MIN SET/CONTENT/membe s/T Hall.h ml [20] N. V. I ano , Subg oups o Teichm¨ulle modula g oups, T ansla ions o ma h- ema ical monog aphs ol. 115 (1992), AMS. [21] N. V. I ano , alk a he special session “Mapping class g oups and he geome ic heo y o Teichm¨ulle spaces” a he 974 h mee ing o he AMS, Ann Ha bou , MI, Ma ch 1–3, 2002. [22] N. V. I ano , Examples o cen alize s in he A in b aid g oups, P ep in (2003), a Xi :ma h.GT/0306418 [23] B. de Ke ´ekj´a ´o, ¨ Ube die pe iodischen T ans o ma ionen de K eisscheibe und de Kugel l¨ache, Ma h. Annalen 80 (1919), 3–7. [24] J. Los, Pseudo-Anoso maps and in a ian ain acks in he disc: a ini e algo i hm, P oc. London Ma h. Soc. (3) 66 (1993), 400–430. [25] G. S. Makanin, On no malize s in he b aid g oup, Ma .Sb. 86 (128), 1971, 171–179. [26] S. Man edini, Some subg oups o A in’s b aid g oup. Special issue on b aid g oups and ela ed opics (Je usalem, 1995). Topology Appl. 78 (1997), no. 1-2, 123–142. [27] S. Yu. O e ko , Quasiposi i i y es ia uni a y ep esen a ions o b aid g oups and i s applica ions o eal algeb aic cu es, J. Kno Theo y Rami ica ions 10 (2001), no. 7, 1005–1023. [28] L. Pa is, D. Rol sen, Geome ic subg oups o su ace b aid g oups, Ann. Ins . Fou ie 49 (1999), 101-156 [29] R. C. Penne , J. L. Ha e , Combina o ics o ain acks, Annals o Ma hema ics S udies, 125. P ince on Uni e si y P ess, P ince on, NJ, 1992. 41