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].
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41