a Xi :1008.0238 1 [ma h.GT] 2 Aug 2010
Reducible b aids and Ga side heo y
Juan Gonz´alez-Meneses∗Be Wies
Augus 2, 2010
Abs ac
We show ha educible b aids which a e, in a Ga side- heo e ical sense, as simple as possible wi hin
hei conjugacy class, a e also as simple as possible in a geome ic sense. Mo e p ecisely, i a b aid belongs
o a ce ain subse o i s conjugacy class which we call he s abilized se o sliding ci cui s, and i i is
educible, hen i s educibili y is geome ically ob ious: i has a ound o almos ound educing cu e.
Mo eo e , o any gi en b aid, an elemen o i s s abilized se o sliding ci cui s can be ound using he
well-known cyclic sliding ope a ion. This leads o a polynomial ime algo i hm o deciding he Nielsen-
Thu s on ype o any b aid, modulo one well-known conjec u e on he speed o con e gence o he cyclic
sliding ope a ion.
1 In oduc ion
The e a e cu en ly wo known app oaches o he p oblem o de e mining algo i hmically he
Nielsen-Thu s on ype o a gi en b aid, i.e. deciding whe he i is educible, pe iodic, o pseudo-
Anoso [12, 7, 11]. Since pe iodici y o b aids is as and easy o de ec [19], he main di icul y is
o de e mine whe he a gi en b aid is educible.
One app oach is due o Bes ina and Handel [2], and uses he heo y o ain acks (see also [22]).
The algo i hmic complexi y o he Bes ina-Handel algo i hm is s ill mys e ious – his is pa icu-
la ly eg e able since i seems o be as in p ac ice, a leas gene ically.
The second app oach, which was ini ia ed by Bena de e, Gu i´e ez and Ni ecki [1], and de eloped
by Lee and Lee [20], uses he Ga side s uc u e, as exposed in [9], on he b aid g oup. Indeed, i is
shown in [1] ha ound educ ion cu es a e p ese ed by cycling and decycling. As a consequence,
i a gi en b aid x∈Bnis educible, hen he e is a leas one elemen o i s supe summi se [9]
which has a ound educ ion cu e, and whose educibili y is hus easy o o de ec . The d awback
o his app oach is ha he algo i hm has o compu e he comple e supe summi se o x, and
his is is e y slow [15].
In o de o ha e any hope o ob aining a polynomial ime algo i hm om he second app oach,
we would need o eplace he supe summi se o xwi h ano he se sa is ying he ollowing
p ope ies: (1) I is an in a ian o he conjugacy class o x, (2) an elemen in his subse can be
compu ed e icien ly, and (3) o e e y elemen in his subse , he educibili y o i educibili y can
be de ec ed apidly. Supe summi se s sa is y he i s wo p ope ies, bu no he hi d.
In he special case o he ou -s and b aid g oup, he supe summi se can ac ually do he job [6].
In he gene al case o he b aid g oup Bn(wi h n∈N), he ul a summi se de ined in [15] can do
∗Pa ially suppo ed unde Aus alian Resea ch Council’s Disco e y P ojec s unding scheme (p ojec numbe
DP1094072), he Spansih P ojec s MTM2007-66929, P09-FQM-5112 and FEDER.
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he job, bu only unde ce ain condi ions. I is shown in [20] ha i a b aid is educible and he
ex e nal componen is simple ( om he Ga side heo e ical poin o iew) han he whole b aid,
hen one can apidly de ec educibili y o any gi en elemen in i s ul a summi se , as e e y
elemen in his se has a ound educ ion cu e. Hence, unde his hypo hesis, he ul a summi
se sa is ies (1) and (3) abo e. I is a well-known conjec u e [3] ha i also sa is ies (2).
The aim o he p esen pape is o cons uc a subse o any conjugacy class which sa is ies (1) and
(3) abo e, and is conjec u ed o also sa is y (2), jus like Lee and Lee’s subse [20], bu wi hou
hei echnical hypo hesis. In pa icula , we p o e he exis ence o a polynomial ime algo i hm
o deciding he educibili y o i educibili y o a gi en b aid, modulo a well-known conjec u e
(Conjec u e 3.5), again conce ning (2) abo e, which we lea e open.
Whe e Bena de e, Gu i´e ez and Ni ecki alk abou ound cu es, we ha e o admi a somewha
la ge amily o educing cu es which we call almos ound cu es. Also, he subse o he
conjugacy class o which ou esul holds is nei he he supe summi se no he ul a summi
se , bu a sligh ly mo e complica ed class, which we call he m imes s abilised se o sliding
ci cui s, deno ed SC[m](x), whe e mis a posi i e in ege .
We will show ha one can conjuga e a gi en elemen xo Bn o an elemen in SC[m](x), by
applying i e a i ely a special kind o conjuga ion called cyclic sliding. This i e a ed cyclic sliding
p ocedu e is a Ga side- heo e ic ool which simpli ies ( om an algeb aic poin o iew) he b aid
wi hin i s conjugacy class, and which has al eady been used o sol e he conjugacy p oblem in
b aid g oups and Ga side g oups [16, 17].
Fu he , we will show he ollowing esul (whe e ∆ deno es he hal wis o all s ands, so ha
||∆|| =n(n−1)/2):
Theo em 3.4 Le x∈Bnbe a non-pe iodic, educible b aid. The e is some m⩽||∆||3− ||∆||2
such ha e e y elemen y∈SC[m](x)admi s an essen ial educ ion cu e which is ei he ound
o almos ound.
Theo em 3.4 is elling us ha cyclic sliding no only simpli ies b aids om he algeb aic, bu also
om he geome ic poin o iew, since he educ ion cu es, which can be e ibly angled in x,
become ei he ound o almos ound a e i e a i e applica ions o cyclic slidings.
Mo eo e , we p o e ha i can be e icien ly checked whe he he e a e ound o almos ound
cu es which a e p ese ed by a b aid ylike in he s a emen o Theo em 3.4. Mo e p ecisely,
in a ian ound cu es can be e icien ly de ec ed by [1]. Fo almos ound cu es he si ua ion is
no he same: as he numbe o such cu es g ows exponen ially wi h espec o he numbe o
s ands, i is no a good idea o y o check hem one by one. To bypass his di icul y, we show
he ollowing pa icula case:
Theo em 2.9 The e is an algo i hm which decides whe he a gi en posi i e b aid xo leng h ℓ
wi h ns ands p ese es an almos - ound cu e whose in e io s ands do no c oss. Mo eo e ,
his algo i hm akes ime O(ℓ·n4).
No ice ha Theo em 2.9 canno immedia ely be applied o de ec he educ ion cu es p omised
by Theo em 3.4, o wo easons: i s ly, none o hese cu es a e necessa ily x-in a ian ( hey
may be pe mu ed by x), and secondly, e en i hey we e, he e would be no gua an ee ha hei
in e io s ands do no c oss. Mo eo e , xis no necessa ily posi i e (al hough his can be easily
achie ed jus by mul iplying xby a sui able powe o ∆2). The e is, howe e , a si ua ion which
can be educed o he cases ha can be checked using Theo em 2.9. This is he si ua ion whe e
he gi en b aid is igid [3].
Theo em 5.16. Le β∈Bnbe a non-pe iodic, educible b aid which is igid. Then he e is some
posi i e in ege k⩽nsuch ha one o he ollowing condi ions holds:
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1. βkp ese es a ound essen ial cu e, o
2. in (βk)and sup(βk)a e e en, and ei he ∆−in (βk)βko β−k∆sup(βk)is a posi i e b aid which
p ese es an almos ound essen ial educ ion cu e whose co esponding in e io s ands do
no c oss.
In pa icula , some essen ial educ ion cu e o βis ei he ound o almos ound.
The powe k⩽nin he abo e s a emen is needed o pass om in a ian amilies o cu es o
in a ian cu es, which is wha is de ec ed in Theo em 2.9, and also o assu e ha in (βk) and
sup(βk) a e e en. We hen see ha i he b aid βunde s udy is igid and admi s essen ial educ ion
cu es, we can ind hem in one o he ollowing wo ways: I one hese cu es is ound, we can
apply he well known algo i hm in [4]. O he wise, we will ind hem by applying Theo em 2.9 o
∆−in (βk)βk(whe e in (βk) is e en) and o β−k∆sup(βk)(whe e sup(βk) is e en) o k= 1,...,n/2,
as hese b aids ha e he same essen ial educ ion cu es as β.
The nex aim is o cons uc , o any gi en y∈SC[N](x), a igid b aid whose educing cu es a e
also educing cu es o y. This se es wo pu poses a once: i allows us o use Theo em 2.9 o
sea ch o educing cu es in polynomial ime, and i also gi es he key o p o ing Theo em 3.4.
In o de o do so, we will, o e e y b aid y∈SC[N](x), de ine i s p e e ed conjuga o P(y), which
commu es wi h y. We will p o e:
Lemma 5.15. Le x∈Bnbe a non-pe iodic, educible b aid. Le N=||∆||3− ||∆||2. Fo e e y
elemen y∈SC[N](x) he e is some m⩽Nsuch ha ei he ymis igid, o P(ym)is igid, admi s
essen ial educ ion cu es, and all i s essen ial educ ion cu es a e essen ial educ ion cu es o y.
F om he abo e esul s, we ob ain he ollowing algo i hm o de e mine whe he a gi en elemen
o Bnis pe iodic, educible o pseudo-Anoso :
Algo i hm 1. To de e mine he geome ic ype o a b aid.
Inpu : x∈Bn.
1. I xn−1o xnis a powe o ∆, e u n ‘xis pe iodic’ and s op.
2. Compu e an elemen y∈SC[N], whe e N=||∆||3− ||∆||2.
3. I yp ese es a amily o ound cu es, e u n ‘xis educible, non-pe iodic’ and s op.
4. Fo m= 1,...,N do he ollowing:
I ei he ymis igid o P(ym) is igid, apply he algo i hm in Theo em 2.9 o he b aids
men ioned in Theo em 5.16(2), wi h β=ymo β=P(ym), espec i ely. I an almos ound
educ ion cu e is ound, e u n ‘xis educible, non-pe iodic’ and s op.
5. Re u n ‘xis pseudo-Anoso ’.
The compu a ional complexi y o each s ep o his algo i hm is bounded by a polynomial in he
leng h and he numbe o s ands o x, wi h one excep ion: he second s ep o his algo i hm
(conjuga ing x o y∈SC[N](x)) is no cu en ly known o be doable in polynomial ime, bu i is
conjec u ed o be so (c. . Conjec u e 3.5).
The plan o he pape is as ollows. In Sec ion 2 we in oduce he basic no ions o educible
b aids and educ ion cu es, including he p oo o Theo em 2.9, and o Theo em 3.4 in he case
whe e he in e io b aid is i ial. In Sec ion 3 we swi ch o he algeb aic iewpoin , explaining
he no ion o cyclic sliding and sliding ci cui s, and in oducing he se SC[m](x). Explo ing
he ela ion be ween sliding ci cui s and he powe s o a b aid, in Sec ion 4, we show how o
compu e one elemen in SC[m](x) o e e y xand m. We hen p oceed o s udy, in Sec ion 5, he
ela ion be ween he educ ion cu es, on he geome ic side, and he se s o sliding ci cui s, on
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he algeb aic side. A he end o his sec ion, we show ha Theo em 3.4 holds in gene al i i holds
o he special case o igid b aids. Sec ion 6 ea s he case o educible igid b aids, inishing he
p oo o Theo em 3.4 by showing ha i a igid, educible b aid has some in e io b aid which is
pseudo-Anoso , hen i s co esponding educ ion cu e is ound.
Acknowledgemen s: We wish o hank Volke Gebha d o many use ul discussions on his and
ela ed p oblems.
2 Round and almos ound educ ion cu es
2.1 De ini ions and no a ions
2.1.1 Canonical educ ion sys em and complexi y o cu es
Le Bnbe he b aid g oup on ns ands, whe e we ix as base poin s he se Pn={1,...,n} ∈ C.
E e y elemen x∈Bncan be seen as an au omo phism o Dn=D2 Pn, whe e D2deno es he disk
in Cwi h diame e [0, n + 1]. The e o e xinduces an ac ion on he iso opy classes o 1-mani olds
in Dn.
We will conside he ac ion o b aids on iso opy classes o simple cu es om he igh . Tha is,
we will deno e he iso opy class o a simple cu e Cby [C], and we will w i e [C]x, meaning he
iso opy class o he cu e ob ained om Ca e applying xconside ed as an au omo phism o he
n- imes punc u ed disk. By abuse o ocabula y, we shall o en say “cu es” when we eally mean
“iso opy classes o cu es”. Howe e , we shall ca e ully dis inguish he no a ions Cand [C].
A simple closed cu e Cin D2 Pnis said o be non-degene a e i i encloses mo e han one and
less han npoin s o Pn, and i is said o be ound i i is homo opic o a geome ic ci cle. I is
clea ha non-degene acy and oundness a e p ope ies which depend only on he iso opy class o
a cu e, so we can na u ally say ha some iso opy class [C] is non-degene a e, o is ound. A b aid
x∈Bnis said o be educible i [C](xm)= [C], o some posi i e in ege mand some non-degene a e
cu e C. Such a cu e Cis said o be a educ ion cu e o x. We say ha a educ ion cu e C
is essen ial i e e y o he educ ion cu e o xcan be iso oped o ha e emp y in e sec ion wi h
C[5].
The se o iso opy classes o essen ial educ ion cu es o a b aid xis called he canonical educ ion
sys em o x, and is deno ed CRS(x). I is well known ha CRS(x) = ∅i and only i xis ei he
pe iodic o pseudo-Anoso [5]. In o he wo ds, CRS(x)6=∅i and only i xis educible and non-
pe iodic. Since i is e y easy o de e mine whe he a gi en b aid x∈Bnis pe iodic (i su ices
o check i ei he xn−1o xnis equal o a powe o he hal wis ∆), he ques ion o de e mining
he geome ic ype o a b aid educes o he s udy o i s canonical educ ion sys em. We will hen
be in e es ed in educible, non-pe iodic b aids, and in hei essen ial educ ion cu es.
We will say ha a non-degene a e simple cu e Cin D2 Pnis almos ound i he e exis s a simple
elemen s(a pe mu a ion b aid) such ha [C]sis ound. This is equi alen o saying ha Ccan
be iso oped in D2 Pn o a cu e whose p ojec ion o he eal line has exac ly one local maximum
and one local minimum.
The e is an al e na i e cha ac e iza ion o almos ound cu es which will also allow us o in oduce
a no ion o complexi y o a simple closed cu e in he punc u ed disc. No ice ha a cu e [C] can
always be ans o med in o a ound cu e by a sui able au omo phism o he punc u ed disc, ha
is, by a sui able b aid y. Since he ull wis ∆2p ese es any gi en cu e, i ollows ha ∆2kβalso
ans o ms [C] in o a ound cu e, o e e y in ege k. Hence we can assume ha yis a posi i e
b aid, as e e y b aid becomes posi i e a e mul iplica ion by a su icien ly high powe o ∆2. I
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is shown in [20] ha gi en a amily Fo mu ually disjoin simple closed cu es in D2 Pn, he e
is a unique posi i e b aid y∈Bnsuch ha [F]yis a amily o ound cu es, and such ha y
has minimal leng h among all posi i e b aids sa is ying his p ope y (ac ually yis a p e ix o any
o he posi i e b aid sa is ying his p ope y). This b aid yis called he minimal s anda dize o F.
I Fconsis s o a single cu e C, we will call y he minimal s anda dize o C.
Now ecall ha he simple b aids (o pe mu a ion b aids) a e hose posi i e b aids o which e e y
pai o s ands c oss a mos once, and ha ℓ(x), he canonical leng h o a b aid x, is he minimal
numbe o simple ac o s in o which xcan be decomposed, no coun ing ac o s equal o he hal
wis ∆ – see also Sec ion 3. Al e na i ely, he canonical leng h ℓ(x) is he numbe o ac o s
di e en om ∆ in he le no mal o m o x.
De ini ion 2.1. Gi en a simple closed cu e Cin he punc u ed disc, we de ine he complexi y
o C o be he canonical leng h o he minimal s anda dize o C.
In o he wo ds, he complexi y o Cis he smalles possible canonical leng h o a posi i e b aid
sending [C] o a ound cu e. No ice ha his de ini ion could be equi alen ly exp essed he o he
way a ound: he complexi y o Cis he smalles possible canonical leng h o a posi i e b aid
sending a ound cu e o [C].
The cu es o complexi y 0 a e he ound cu es, and he cu es o complexi y 1 a e hose which
become ound by he ac ion o a simple elemen : hese a e p ecisely he almos ound, no ound
cu es.
2.1.2 Decomposi ion o a b aid along a amily o cu es
Reduc ion cu es allow us o decompose a b aid in o simple b aids. In ac , se e al p ocedu es
o speci ying such a decomposi ion a e concei able, bu we shall use he p ocedu e gi en in [18],
which we b ie ly explain now.
Le x∈Bn, and le Fbe a amily o disjoin simple closed cu es in D2 Pn. Le ybe he minimal
s anda dize o F, and le bx=y−1xy =: xyand b
F=Fy. No ice ha i xp ese es [F], hen bx
p ese es [ b
F] = [F]y, which is a amily o ound cu es. Howe e , e en i xdoes no p ese e [F],
i can s ill happen ha bxsends [ b
F] = [F]y o a amily o ound cu es (no necessa ily [ b
F] i sel ).
In his case we can de ine o e e y cu e C ∈ F ∪ {∂(D2)}, a b aid β[C∈F], called he componen
o xassocia ed o Cin F, as ollows.
Fo e e y subse I⊂ {1, . . . n}, we can de ine he subb aid (bx)I o be he b aid on #(I) s ands
ob ained om bxby keeping only hose s ands which s a a I. No ice ha his yields a well-
de ined elemen o B#(I), e en i he s ands s a ing a Ido no end a I– we jus equi e he
s ands o (bx)I o c oss in he same way as he s ands in bxs a ing a I, o de ails see [18].
Now gi en a cu e C ∈ F ∪ {∂(D2)}, le XCbe he only connec ed componen o D2 F which is
enclosed by C, and such ha C ⊂ XC. Then de ine DC=XC∪ C, which is homeomo phic o a
punc u ed disc. No ice ha DC DCis a amily o poin s and cu es, namely he ou e mos cu es
enclosed by C, and he poin s which a e enclosed by Cbu no enclosed by he men ioned cu es.
Simila ly, we le Xb
Cbe he only connec ed componen o D2 b
Fwhich is enclosed by b
C, and such
ha b
C ⊂ Xb
C. Then de ine Db
C=Xb
C∪b
C. This is a closed ound disk wi h some poin s and some
closed ound disks emo ed om i s in e io . Also, Db
C Db
Cis a amily o poin s and ound cu es.
De ini ion 2.2. [18] Le x∈Bnand le Fbe a amily o disjoin simple closed cu es in D2 Pn,
whose minimal s anda dize is y. Le bx=y−1xy, and suppose ha bxsends [b
F] = [F]y o a amily
o ound cu es. Le C ∈ F ∪ {∂(D2)}. Le I⊂ {1,...,n}consis o he indices o
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• hose punc u es ha appea in Db
C Db
C, and
• o each cu e in Db
C Db
C, exac ly one punc u e chosen a bi a ily among he punc u es en-
closed by ha cu e.
Then we de ine x[C∈F], he componen o xassocia ed o Cin F, as he subb aid (bx)I. I F=
CRS(x), he men ioned componen is jus deno ed xC.
We ema k ha x∂(D2)is usually called he ex e nal b aid associa ed o x, and is deno ed xex .
2.2 Canonical educ ion cu es o educible, posi i e b aids wi h i ial
in e io b aids a e ei he ound o almos ound
The aim o his sec ion is o p o e he ollowing esul :
P oposi ion 2.3. I Cis an essen ial educ ion cu e o a posi i e b aid x, wi h [C]x= [C], and
he s ands o xenclosed by Cdo no c oss each o he , hen Cis ei he ound o almos ound.
In o de o show his esul , i su ices o p o e ha such a cu e Ccanno be o complexi y wo,
i.e., i canno be he esul o a ound cu e a e he ac ion o a b aid o canonical leng h wo,
wi hou being ound o almos ound.
Ou i s aim is o unde s and wha a cu e o complexi y wo looks like ( o de ailed discussion
o mo e gene al ques ions see [24]). We shall i s s udy smoo h a cs α:I→D2in he disk D2
de ined on he uni in e al I= [0,1]; we shall es ic ou a en ion o smoo h a cs αwhich s a
and end in punc u e poin s, which may also a e se some punc u e poin s, bu whose angen
di ec ion is ho izon al and poin ing o he igh , a e e y punc u e poin . Fo b e i y, we shall
call hem a cs a e sing some punc u e poin s ho izon ally.
When s udying di eo opy classes o such a cs, we shall always mean di eo opies h ough amilies
o a cs which a e all supposed o a e se he punc u e poin s ho izon ally. We shall say ha a
simple closed cu e o an a c a e sing some punc u e poin s ho izon ally is educed i i has he
minimal possible numbe o in e sec ions wi h he ho izon al line, and also he minimal possible
numbe o e ical angencies in i s di eo opy class.
The ac ion o he b aid g oup on he se o di eo opy classes o a cs a e sing some punc u e
poin s ho izon ally, speci ically o a b aid xon an a c α, is de ined as ollows: xinduces a punc u e
dance, which in u n can be ex ended o a di eo opy o αin such a way ha a e e y momen
he in e sec ion o he a c wi h he punc u es is ho izon al. A he end o he dance we ob ain a
new a c a e sing some punc u e poin s ho izon ally, which is well-de ined up o di eo opy. This
is αx.
Fo an a c α a e sing some punc u e poin s ho izon ally, we de ine he angen di ec ion unc ion
α:I→R/2Zas he angle o he angen di ec ion o αagains he ho izon al, di ided by −π.
In pa icula , i he a c goes s aigh o he igh in α( ), hen α( ) = 0 + 2Z, i i goes s aigh
down hen α( ) = 1
2+ 2Z, and i i goes o he le hen α( ) = 1 + 2Z.
Fo e e y a c a e sing some punc u e poin s ho izon ally, we ha e a unique li ing o he unc-
ion α o a unc ion e
α:I→Rwi h e
α(0) = 0. Finally, i :R→Zdeno es he ounding
unc ion, which sends e e y eal numbe o he nea es in ege ( ounding down n+1
2), hen we
de ine he unc ion
τα:I→Z, 7→ ◦e
α( )
which one migh call he ounded li ed angen di ec ion unc ion.
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No ice ha , i αis an a c such ha τα akes he alue 0 in a neighbou hood o he poin s whe e
he a c a e ses a punc u e, hen he same is ue o i s image αxunde he ac ion o any b aid.
In o de o be able o cha ac e ize educ ion cu es o complexi y ze o, one, and wo, we gi e
now a de ailed desc ip ion o he punc u e dance associa ed o a posi i e pe mu a ion b aid. In
a i s s ep, he punc u es make a small e ical mo emen , wi h he punc u e in posi ion k∈Z
mo ing o posi ion k−k·ǫ·i∈C, o some small ǫ > 0. In a second s ep, he punc u es make a
ho izon al mo emen , pe mu ing hei R-coo dina es. In a hi d s ep, he punc u es make again
a small e ical mo emen , lining hem back up on he eal line.
Now, educ ion cu es Co complexi y ze o can be cha ac e ized as cu es enclosing an a c which
lies en i ely in he eal line, and which a e ses all he punc u es in he in e io o C. No ice ha
Ccan be seen as he bounda y o a egula neighbo hood o his a c.
Suppose now ha a cu e Chas complexi y one. Then i is ob ained om a ound cu e C0by
he ac ion o a simple b aid s. We can assume ha he punc u es enclosed by C0(which a e
consecu i e) do no c oss in s, as hose c ossings could be emo ed om swi hou modi ying
i s ac ion on C0. Hence, om he abo e desc ip ion o posi i e pe mu a ion b aids, we see ha
educ ion cu es Co complexi y one can be cha ac e ized as ollows: he e exis s a smoo h a c α
disjoin om C, a e sing all he punc u es in he in e io componen o D2 C ho izon ally such
ha ταis he cons an unc ion 0. (We a e going o say such an a c is almos ho izon al.)
The ac ion by a posi i e pe mu a ion b aid ans o ms an a c αwi h τα≡0 in o an a c α′which,
a e educ ion, has he ollowing p ope y: by an iso opy o D2 ha mo es he npunc u e poin s
only in he e ical di ec ion up o down, α′can be ans o med in o an a c whose imagina y
coo dina e is mono onically dec easing. The e o e, educ ion cu es Co complexi y wo can be
cha ac e ized as ollows: he e exis s a smoo h a c α′disjoin om Cbu a e sing ho izon ally
all he punc u es in he in e io componen o D2 C, such ha τα′only akes he alues 0 and 1
( o a mo e de ailed p oo see [24]).
One impo an p ope y is ha i a b aid xp ese es a cu e Co complexi y 2, and he s ands
inside Cdo no c oss in x, hen he men ioned a c is in a ian by x:
Lemma 2.4. Le x∈Bnand le Cbe a cu e such ha [C]x= [C]. Suppose ha he s ands
enclosed by Cdo no c oss in x. Le αby an a c a e sing ho izon ally some punc u es enclosed
by C. Then αx=α.
P oo . Le C0be a ound cu e and le y∈Bnbe such ha [C]y= [C0]. Conside he b aid
z=y−1xy, and he a c αy. No ice ha [C0]z= [C0]y−1xy = [C]xy = [C]y= [C0]. Hence z
p ese es he ound cu e C0. Mo eo e , as he punc u es enclosed by Cdo no c oss in y, we
can ind a ep esen a i e o zin which he punc u es enclosed by C0do no c oss. This implies
ha zcan be ep esen ed by a homeomo phism o he punc u ed disc whose es ic ion o he
componen enclosed by C0is i ial. As αyis a cu e enclosed by C0, one has (αy)z=αyand hen
αx= (αyz)y−1= (αy)y−1=α, as we wan ed o show.
We saw abo e ha a cu e Co complexi y 2 admi a smoo h a c α′disjoin om Cbu a e sing
ho izon ally all he punc u es in he in e io componen o D2 C, such ha τα′only akes he
alues 0 and 1. I xis a b aid p ese ing Cin which he s ands enclosed by Cdo no c oss,
he abo e lemma shows ha he smoo h a c α′is p ese ed by x. We shall call such an a c a
descending in a ian a c. No ice ha Cis he bounda y o a egula neighbo hood o α′.
Lemma 2.5. I xis a posi i e b aid, i αis an a c a e sing ho izon ally some punc u e poin s,
and i αxis i s educed image unde he ac ion o x, hen
max
∈Iταx( )⩾max
∈Iτα( )and min
∈Iταx( )⩾min
∈Iτα( )
7
P oo . I su ices o p o e his esul o x=σi, a single A in gene a o . I is an easy obse a ion
ha o e e y 0in I, we ha e ασi( 0) = α( 0) o ασi( 0) = α( 0) + 1. Some examples a e gi en
in Figu e 1.
0
2
3
33
3
3
43
0
1
0
−1 0
1 1
0 0
11
σiσiσi
σiσiσi
Figu e 1: The labels, which ep esen he alues o he unc ion α, can g ow unde he ac ion by
a gene a o σi, bu ne e go down.
Lemma 2.6. I [C]is an x-in a ian closed cu e o complexi y wo, whe e xis a posi i e b aid,
and he s ands o xenclosed by Cdo no c oss, hen o any p e ix x′o x he cu e [C]x′is o
complexi y wo.
P oo . Le αbe a descending in a ian a c associa ed o C. By Lemma 2.4 we know ha αx=α.
Now, he image o ταis equal o {0,1}, so he same holds o he image o ταx. Thus Lemma 2.5
implies ha o any p e ix x′o xone has
1 = max
∈Iταx( )⩾max
∈Iταx′( )⩾max
∈Iτα( ) = 1
and
0 = min
∈Iταx( )⩾min
∈Iταx′( )⩾min
∈Iτα( ) = 0.
Hence he image o ταx′is also equal o {0,1}. The e o e [C]x′has complexi y wo.
Le us in oduce some mo e no a ion. We shall suppose ha αis a descending in a ian a c o
some posi i e b aid x. We suppose also ha α′⊂αis a sub-a c whose wo ex emi ies lie in wo
in e io punc u es. We say an ex e io punc u e is le -blocked by α′i he e is no smoo h pa h
s a ing a his punc u e poin , e mina ing on he bounda y o he disk, disjoin om he a c α′,
and whose angen di ec ion has always a nega i e eal coo dina e. A igh -blocked punc u e is
de ined symme ically. We de ine in e io punc u es o be bo h le and igh blocked. We shall
call he wo in e io punc u es a he wo ends o he a c α′ he ex emal (in e io ) punc u es o α′.
The p oo o P oposi ion 2.3 will be comple ed by p o ing ha he e a e no blocked ex e io
punc u es a all, meaning ha he cu e Cis o complexi y 1. Fi s we ob ain wo pa ial esul s:
Lemma 2.7. Le Cbe an essen ial educ ion cu e o a posi i e b aid x. Suppose ha α′is a
sub-a c o a descending in a ian a c o x. Then he e canno be any ex e io punc u es which
a e le -blocked by α′and o he le o bo h ex emal punc u es o α′. Simila ly, he e canno be a
igh -blocked ex e io punc u e o he igh o bo h ex emal punc u es.
P oo . We shall p o e he i s sen ence, he p oo o he second one is e y simila . Mo eo e ,
we shall suppose ha he s a ing poin o he a c α′(which in he pic u e is “highe ” han he
end poin ) is o he le o he e minal poin , see Figu e 2(a). The p oo o he o he case (whe e
he s a ing poin o he a c α′is o he igh o he end poin , Figu e 2(b)) is simila , one simply
has o conside he posi i e b aid e (x), which is he image o xunde he an i-isomo phism
e : Bn→Bnwhich sends σi o i sel o e e y i= 1,...,n−1 ( ha is, e (x) is equal o x
w i en backwa ds).
We shall a gue by con adic ion: le us suppose ha he e is some le -blocked punc u e which is o
he le o he le ex emal in e io punc u e (see Figu e 2(a)). We obse e ha he co esponding
8
s ands canno c oss in he b aid x– indeed, i we hink o he b aid xas a dance o he punc u es,
hen du ing his dance he le -blocked punc u e canno mo e unde he le ex emal in e io
punc u e, o his would equi e a nega i e c ossing, and i canno mo e o e i , o his would
u n he cu e α′in o a cu e α′′ which possesses some poin s whe e he unc ion α′′ akes he
alue 2. Thus he se o punc u es which a e le -blocked by α′and which lie o he le o bo h
endpoin s o α′is s able du ing he whole dance.
01
1
0
0
1
igh ex emal punc u e o α′
le o bo h endpoin s o α′
(a)
α′
α
no le -blocked by α′
(b)
α
α′
igh -blocked punc u e o he
igh o bo h ex emal punc u es o α′
le -blocked punc u e o he
Figu e 2: (a) The s a ing poin o α′( he bold line segmen ) is o he le o he end poin . (b)
Vice e sa.
Now he e ical line h ough he le ex emal in e io punc u e, oge he wi h he a c α′, cu s
he disk in o a numbe o connec ed componen s, a leas one o which con ains some le -blocked
punc u es o he le o he le ex emal punc u e. Le Ξ be he union o all he componen s
con aining le -blocked punc u es. Le Ψ be he union o Ξ wi h an ini ial segmen o α′long
enough o ouch all he connec ed componen s o Ξ, bu no all o α′. (So Ψ looks in gene al
like some pea ls on a h ead, see Figu e 3(a).) Le N(Ψ) be a egula neighbou hood o Ψ. We
obse e ha N(Ψ) is p ese ed by he ac ion o x, and so is i s bounda y, which we shall call C′.
Mo eo e , C′in e sec s he canonical educ ion cu e C(which, we ecall, was he bounda y o
a egula neighbou hood o α) wice. This con adic s he de ini ion o a canonical educ ion
cu e.
(a)(a) (b)
α′
α′
Figu e 3: Cons uc ing in a ian cu es which in e sec he cu e c: (a) In he case whe e he e
is a le blocked punc u e o he le o bo h ex emal punc u es, and (b) in he o he case.
Lemma 2.8. Le Cbe an essen ial educ ion cu e o a posi i e b aid x. Suppose ha α′is a
sub-a c o a descending in a ian a c αo x. Also suppose ha α′does no a e se any in e io
punc u es (excep i s wo endpoin s). Then he e canno be any ex e io punc u es blocked by α′.
P oo . Again, we shall assume ha he s a ing poin o α′is o he le o he end poin , wi h
he o he case being simila . Lemma 2.7 oge he wi h he hypo hesis ha α′does no a e se
any in e io punc u e imply ha any blocked punc u es would ha e o lie be ween he le and he
igh ex emi y o α′. Supposing, o a con adic ion, ha such blocked punc u es exis , hen he e
mus be a pai o hem, wi h a igh -blocked punc u e abo e a le -blocked one (see Figu e 3(b)).
Le us now look a he b aid x, conside ed as a dance o he punc u es.
9
Co olla y 4.5. Gi en x∈Bnw i en as a p oduc o ℓsimple elemen s and i s in e ses, and
gi en m > 0, he e is an algo i hm ha compu es an elemen in SC[m](x)in ime O(Sℓn log n),
whe e S=
m
X
i=1
i Tn,iℓ.
P oo . The algo i hm compu es P(xi
[i−1]) and conjuga es x[i−1] by his elemen (ob aining x[i]),
o i= 1,...,m.
We s a wi h xw i en as a p oduc o ℓsimple elemen s and i s in e ses, and compu e i s
le no mal o m, which akes ime O(ℓ2nlog n) [10]. Now we apply i e a ed cyclic sliding o x
un il he i s epe i ion, which is x[1]. A each s ep, we ha e o compu e he p e e ed p e ix
o an elemen α, and conjuga e αby i . No ice ha a p e e ed p e ix is he g ea es common
di iso o wo pe mu a ion b aids: i α= ∆pα1···α is in le no mal o m and > 0, hen
p(α) = τ−p(α1)∧∂(α ). I he le no mal o m o αis known, he compu a ion o τ−p(α1)
and ∂(α ) akes ime O(n) [10], and compu ing hei gcd akes ime O(nlog n) [10]. Now α
is an i e a ed cyclic sliding o x, whe e cyclic sliding ne e inc eases he canonical leng h o
an elemen [16]. Hence he canonical leng h o αis a mos ℓ. The algo i hm akes αin le
no mal o m, and compu es he le no mal o m o i s conjuga e by p(α). As p(α) is a simple
elemen , and αhas canonical leng h a mos ℓ, his las s ep akes ime O(ℓn log n) [10]. Thus
compu ing p(α), conjuga ing αby i , and calcula ing he le no mal o m o he esul akes ime
O(ℓn log n). This is epea ed Tn,ℓ imes, so x[1] is compu ed in ime O(Tn,ℓ ℓn log n).
In he ollowing s eps o he algo i hm, one has x[i−1] and xi−1
[i−1] w i en in le no mal o m ( he
case o he p e ious pa ag aph is i= 1). No ice ha he canonical leng h o xi−1
[i−1] is a mos
(i−1)ℓ. The algo i hm hen compu es he le no mal o m o xi
[i−1]. This compu a ion, ob ained
om he p oduc o he le no mal o ms o x[i−1] and xi−1
[i−1], akes ime O((i−1)ℓ2nlog n). Now
he algo i hm compu es i e a ed cyclic slidings o xi
[i−1] un il he i s epe i ion. Mo e p ecisely,
he algo i hm s a s wi h α=x[i−1], and a each s ep i compu es he p e e ed p e ix p(αi),
and conjuga es bo h αand αiby his p e ix. The conjuga e o αis se as he new alue o α,
and he loop is epea ed. The loop ends a he i s epe i ion o αi. The complexi y o his
compu a ion is he same as ha o he p e ious pa ag aph, bu applied o a b aid o canonical
leng h iℓ, ins ead o ℓ. Hence, he compu a ion o x[i]and xi
[i] om x[i−1] and xi−1
[i−1] akes ime
O(Tn,iℓ iℓn log n). Adding up he complexi ies o each loop, we ob ain ha he whole algo i hm
akes ime O(ℓ2nlog n) + O(Sℓn log n). As ℓ < Tn,ℓ ⩽S, he esul ollows.
We ema k ha i Conjec u e 3.5 holds, ha is, i Tn,ℓ is a polynomial in nand ℓ, hen he
complexi y o he algo i hm in Co olla y 4.5 is polynomial in n,ℓand m. As we shall only need o
compu e one elemen in SC[m](x) o m⩽||∆||3=n3(n−1)3/8 (see Theo em 3.4), he complexi y
in his case will be polynomial in nand ℓ, always p o ided Conjec u e 3.5 holds.
5 Sliding ci cui s and educ ion cu es
5.1 Sliding ci cui s and ound cu es
In his sec ion we shall in es iga e he p ope ies o he elemen s belonging o SC[m](x), wi h
espec o hei canonical educ ion sys ems. The simples case occu s when his educ ion sys em
is made o ound cu es. The ollowing esul assu es he exis ence o hese examples
Theo em 5.1. [1] (see also [20]) Le x∈Bnbe a posi i e b aid whose le no mal o m is x1···x .
I [C]is a ound cu e such ha [C]xis also ound, hen [C]x1···xiis ound o i= 1,..., .
16

In o he wo ds, i he oundness o a cu e is p ese ed by a b aid x, hen i is p ese ed by each
ac o in he le no mal o m o x. Since ∆±1p ese es he oundness o e e y cu e, he abo e
esul can be applied o e e y b aid, no necessa ily posi i e. This is used in [1] o show ha ,
i a b aid p ese es a ound cu e, i s cycling and i s decycling also p ese e ound cu es. This
immedia ely implies ha o e e y educible b aid x, he e is some elemen in i s supe summi
se SSS(x) which p ese es a ound cu e [1]. Clea ly, one can eplace SSS(x) by USS(x) in he
p e ious s a emen . E en be e , one can eplace i by SC(x), as we will now see, bu he p oo
o his ac is sligh ly di e en : we need o show he ollowing esul , conce ning in a ian amilies
o ound cu es.
P oposi ion 5.2. Le x∈Bn, and le Fbe a amily ound cu es such ha [F]x= [F]. Then
[F]p(x)is also a amily o ound cu es. Hence, i xp ese es a amily o ound cu es, hen so
does s(x).
P oo . We can assume > 0. Le ∆px1···x be he le no mal o m o x. By Theo em 5.1
applied o each pa icula cu e o F, one has ha [F]∆px1is a amily o ound cu es, and since
∆px1=τ−p(x1)∆p, i ollows ha he cu es o [F]τ−p(x1)a e ound. Le F2be a amily o cu es
such ha [F2] = [F]τ−p(x1). In he same way, Theo em 5.1 ells us ha he cu es o [F]∆px1···x −1
a e ound. Le F1be such ha [F1] = [F]∆px1···x −1. No ice ha [F1]x = [F](∆px1···x −1)x =
[F]x= [F].
We hen ha e [F1]x τ−p(x1)= [F2], whe e F1and F2a e amilies o ound cu es. Now, by
de ini ion, he le no mal o m o x τ−p(x1) is equal o y1y2, whe e y1=x p(x). By Theo em 5.1
again, we ob ain ha he cu es o [F1]y1a e ound. Bu [F1]y1= [F1]x p(x)= [F]p(x), hence
[F]p(x)is a amily o ound cu es, as we wan ed o show.
Co olla y 5.3. Fo e e y educible b aid x∈Bnand e e y m > 0, he e is some y∈SC[m](x)
such ha CRS(y)consis s o ound cu es. Mo eo e , all elemen s in he sliding ci cui o y
sa is y he same p ope y.
P oo . The canonical educ ion sys em CRS(x) is a amily o disjoin simple cu es on he punc-
u ed disc. Hence some o ien able au omo phism o he punc u ed disc ela i e o he bounda y,
will send i o a collec ion o (possibly nes ed) ound cu es. This au omo phism co esponds o a
b aid γ∈Bn. In o he wo ds, he e is some γ∈Bnsuch ha [CRS(x)]γconsis s o ound cu es.
I is well known ha [CRS(x)]γ= [CRS(xγ)], hence z=xγis a conjuga e o xwhose canonical
educ ion sys em consis s o ound cu es.
Now ecall ha z[m], which is he conjuga e o zby P(z)P((z[1])2)P((z[2])3)···P((z[m−1])m),
belongs o SC[m](z) = SC[m](x). We will show ha all he cu es in CRS(z[m]) a e ound ci cles
by induc ion on m. We know ha his is ue o m= 0 since z[0] =z, so we assume CRS(z[m−1])
consis s o ound cu es o some m > 0.
In o de o compu e z[m], we conjuga e z[m−1] by P((z[m−1])m). Recall ha he canonical educ ion
sys em o an elemen coincides wi h he canonical educ ion sys em o each nonze o powe , hence
CRS((z[m−1])m) consis s o ound cu es. Applying i e a ed cyclic sliding o (z[m−1])mun il he
i s epe i ion, ha is, conjuga ing i by P((z[m−1])m), one ob ains (z[m])m. By P oposi ion 5.2,
all cu es in CRS((z[m−1])m) keep hei oundness a e each applica ion o s. Hence all cu es in
CRS((z[m])m) = CRS(z[m]) a e ound, as we wan ed o show.
We ha e hen shown ha he e is some y∈SC[m](x) all o whose educ ion cu es a e ound. By
P oposi ion 5.2 again, he same happens o e e y elemen ob ained by applying i e a ed cyclic
sliding o y, ha is, o e e y elemen in he sliding ci cui o y.
No ice ha he abo e p oo does no p o ide an algo i hm o ind y, since we do no know a
17
p io i which is he b aid γ ha conjuga es x o z. Ne e heless, since SC[m](x) is a ini e se , one
can compu e he whole SC[m](x) and check o each elemen whe he i p ese es some amily o
ound cu es. In his way one can ind a educ ion cu e o y, and hen o x.
The compu a ion o he whole se SC[m](x), s a ing om a single elemen , pa allels he usual
cons uc ions gi en in [9, 3, 16], so we will skip i he e. Fo ou pu poses, i su ices o know ha
he e is one elemen yin SC[m](x) all o whose essen ial cu es a e ound. Such elemen s ha e a
pa icula ly nice beha io wi h espec o no mal o ms, as i is shown in [20] and [18].
Lemma 5.4. (see o ins ance [20]) Le y∈Bn, and le Fbe a amily o ound cu es such
ha Fyis also ound. Suppose ha yis a posi i e b aid, and le y1···y be i s le no mal
o m, whe e some o he ini ial ac o s may be equal o ∆. Le C ∈ F ∪ ∂(D). Fo i= 1,..., ,
deno e [Ci] = [C]y1···yi−1and [Fi] = [F]y1···yi−1. Then he le no mal o m o y[C∈F ]is p ecisely
y1[C1∈F1]y2[C2∈F2]···y [C ∈F ]. In his no mal o m, some o he ini ial ac o s could be hal wis s,
and some o he inal ac o s could be i ial.
Lemma 5.5. Le x, y ∈Bnbe b aids, le Fbe a amily o ound cu es, and le [C]∈[F]∪∂(D).
Suppose ha Fxand Fya e ound. Then Fx∧yis also ound, and (x∧y)[C∈F ]=x[C∈F]∧y[C∈F ].
P oo . The i s sen ence is shown be Lee and Lee [20], and he second one in [18].
Lemma 5.6. [18] Le y∈Bn, and le Fbe a amily o ound cu es such ha Fyis also ound.
Le C ∈ F ∪ ∂(D). Then ι(y)p ese es he oundness o [F], and ι(y)[C∈F]is ei he a hal wis
o equal o ι(y[C∈F ]).
P oposi ion 5.7. [18] Le y∈Bn, and le Fbe a amily o ound cu es such ha [F]y= [F].
Conside he p e e ed p e ix p(y), and le C ∈ F ∪ ∂(D). Then p(y)[C∈F]is ei he a hal wis , o
equal o p(y[C∈F ]), o o ι(y[C∈F]), o o ι(y−1
[C∈F]).
5.2 Rigidi y, sliding ci cui s and p e e ed conjuga o s
The key ing edien o showing he main heo em will be he p ope ies o he p e e ed conjuga-
o P(y) o a b aid ywhich p ese es a amily o ound cu es. In ac , we won’ be able o gain
su icien con ol o e P(y), and we ha e o s udy he p e e ed conjuga o P(yk) o some sui able
powe yko yins ead. The need o aking powe s o ob ain a be e beha io o he p e e ed
conjuga o is he eason why we ha e o wo k wi h he se SC[m](x), a he han simply he se
o sliding ci cui s SC(x).
The p ope y we will equi e o a powe o y∈SC(x) in ol es he no ion o igidi y in oduced
in [3], which measu es how he le no mal o m o an elemen a ies when aking i s squa e. Mo e
p ecisely, i x= ∆px1···x is in le no mal o m wi h > 0, one could expec ha he le
no mal o m o x2is ∆2pτp(x1)···τp(x )x1···x , bu in gene al his is no he case. We say
ha he igidi y o xis R(x) = k/ i kis he bigges in ege in {0,1,..., }such ha he i s
2|p|+k ac o s in he le no mal o m o x2a e ∆2pτp(x1)···τp(xk). The wo ex eme cases a e
R(x) = 0, in which all ac o s in he le no mal o m o xa e modi ied when conside ing x2, and
R(x) = 1, in which no ac o is modi ied, and he le no mal o m o x2is he expec ed one we
saw abo e. In his la e case we say ha xis igid.
We will be in e es ed in he case in which R(x)>0 and R(x−1)>0. This kind o elemen s a e
cha ac e ized by he ollowing esul .
Lemma 5.8. [3, Lemmas 3.4, 3.5 and Co olla y 3.6] Le x∈Bnwi h ℓ(x)>0. The ollowing
condi ions a e equi alen :
1. R(x)>0.
18
2. in (x2) = 2 in (x)and ι(x2) = ι(x).
3. in (xm) = min (x)and ι(xm) = ι(x) o e e y m > 0.
The ollowing condi ions a e also equi alen :
1. R(x−1)>0.
2. sup(x2) = 2 sup(x)and ϕ(x2) = ϕ(x).
3. sup(xm) = msup(x)and ϕ(xm) = ϕ(x) o e e y m > 0.
These equali ies o in ima, sup ema, ini ial and inal ac o s yield a good beha io o he p e e ed
conjuga o s, as we shall see. Mo eo e , his condi ion is p ese ed by cyclic sliding, i he elemen
is in i s supe summi se :
Lemma 5.9. Le x∈Bnand y∈SSS(x)wi h ℓ(y)>0. Then R(s(y)) ⩾R(y)and R(s(y)−1)⩾
R(y−1).
P oo . Le =ℓ(y)>0. Since y∈SSS(x) one has s(y)∈SSS(x), hence ℓ(s(y)) = . No ice ha
he p ope y R(y)⩾k/ can be ew i en as y2∧∆2p+k= (y∧∆p+k)∆p. One can apply o his
equali y he anspo map based a y[16]. This map sends y o s(y), ∆ o i sel , and p ese es
p oduc s and g ea es common di iso s. Hence one ob ains s(y)2∧∆2p+k= (s(y)∧∆p+k)∆p, which
is equi alen o R(s(y)) ⩾k/ . Hence R(y)⩾k/ implies R(s(y)) ⩾k/ o e e y k∈ {0,..., },
so one has R(s(y)) ⩾R(y).
Replacing yby y−1, which is also in i s supe summi se , one has R(s(y−1)) ⩾R(y−1). The
esul ollows as s(y−1) = s(y)−1(see he a gumen ha ollows De ini ion 3.1).
The elemen s in a sliding ci cui ha ul ill he equi ed igidi y condi ions also sa is y he ollowing
impo an p ope y: hei p e e ed conjuga o is igid.
P oposi ion 5.10. Le x∈Bnand y∈SC(x)wi h ℓ(y)>0. I R(y)>0and R(y−1)>0, hen
he p oduc p(y)p(s(y)) is le -weigh ed, and P(y)is igid.
P oo . Le us i s p o e ha p(y)p(s(y)) is le -weigh ed. Conside he bigges elemen α4p(s(y))
such ha p(y)αis simple. Le ∆py1···y be he le no mal o m o y. No ice ha p(s(y)) 4
ι(s(y)) 4s(y)∆−p=p(y)−1yp(y)∆−p. Hence p(y)p(s(y)) 4yp(y)∆−p4y2∆−2p. Since ysa is ies
he equi ed igidi y condi ions, Lemma 5.8 ells us ha in (y2) = 2p, hence he ini ial ac o o
y2∆−2pis p ecisely ι(y2), which is equal o ι(y), again by Lemma 5.8. Since we a e assuming ha
p(y)αis a simple p e ix o p(y)p(s(y)), i ollows ha p(y)α4ι(y2) = ι(y). In he same way, as
p(y−1) = ι(y−1)∧ι(y) = p(y) one has p(s(y−1)) = p(s(y)−1) = p(s(y)), we can apply he abo e
a gumen o y−1and i ollows ha p(y)α4ι(y−1). The e o e p(y)α4ι(y)∧ι(y−1) = p(y), so
α= 1, and he i s hal o he p oposi ion is p o en.
Now, i he hypo heses o P oposi ion 5.10 a e sa is ied by y, hen by Lemma 5.9 hey a e also
sa is ied by sk(y) o e e y k > 0. So no only he p oduc p(y)p(s(y)) is le -weigh ed as w i en,
bu also p(si(y)) p(si+1(y)) is le -weigh ed o e e y i > 0. Thus he le no mal o m o P(y) is
p ecisely p(y)p(y(1))···p(y(N−1)), whe e Nis he leng h o he sliding ci cui o y. Mo eo e , as
y(N)=y, he p oduc p(y(N−1))p(y) is also le -weigh ed, hence he le no mal o m o P(y)2is
p(y)p(y(1))···p(y(N−1))p(y)p(y(1))···p(y(N−1)), which means ha P(y) is igid.
Once we ha e seen ha i R(y)>0 and R(y−1)>0 hen P(y) is igid, we a e in e es ed in
inding elemen s which sa is y hese igidi y condi ions, so we can gain su icien con ol o e hei
p e e ed conjuga o . In he nex esul ,we will see ha i N=||∆||3− ||∆||2, e e y elemen in
SC[N](x) has a powe which sa is ies he equi ed igidi y condi ions.
19
P oposi ion 5.11. Le x∈Bn, and le N=||∆||3− ||∆||2. Gi en y∈SC[N](x), he e is an
in ege mwi h 0< m < N such ha R(ym)>0and R(y−m)>0.
P oo . In [21] i is shown ha o e e y x∈Bn he e exis s some k⩽||∆||2such ha e e y elemen
in SSS(xk) is pe iodically geodesic. Tha is, o e e y z∈SSS(xk) one has in (z ) = ·in (z)
and sup(z ) = ·sup(z) o all > 0. In pa icula , since y∈SC[N](x) and k < N, one has
yk∈SC(xk)⊂SSS(xk), so ykis pe iodically geodesic. This means ha in (yk ) = ·in (yk) and
sup(yk ) = ·sup(yk) o all > 0.
Once ykis known o be pe iodically geodesic, one has a chain ι(yk)4ι(y2k)4ι(y3k)4··· ( he
ini ial ac o o yik is a p e ix o he ini ial ac o o y(i+1)k). No ice ha his chain s abilizes a
he i s epe i ion, hence i mus s abilize in less han ||∆|| s eps. In he same way, since ykis
pe iodically geodesic one has a chain ··· <ϕ(y3k)<ϕ(y2k)<ϕ(yk) ( he inal ac o o yik is a
su ix o he inal ac o o y(i+1)k), which mus also s abilize in less han ||∆|| s eps. The e o e,
o some ⩽||∆|| − 1 one has ι(y k) = ι(y2 k) and ϕ(y k) = ϕ(y2 k). We can ake m=k ⩽
||∆||3− ||∆||2and we will ha e, on he one hand, in (y2m) = 2 in (ym) and ι(y2m) = ι(ym)
( hus R(ym)>0), and on he o he hand sup(y2m) = 2 sup(ym) and ϕ(y2m) = ϕ(ym) ( hus
R(y−m)) >0), so he esul ollows.
Now we will place ou sel es in he case in which a b aid y∈SC(x) sa is ies he abo e igidi y
condi ions, ha is, R(y)>0 and R(y−1)>0 (by P oposi ion 5.11 we know how o ind a b aid
which ul ill hese equi emen s). We saw in P oposi ion 5.10 ha in his case P(y) is igid. We
will now see ha , i o some eason we need o conside some powe o y, his makes no ha m, as
e e y powe o ysa is ies he same p ope ies (e en he p ope y o belonging o a sliding ci cui ).
P oposi ion 5.12. Le x∈Bnand y∈SC(x)wi h ℓ(y)>0. I R(y)>0and R(y−1)>0, hen
o e e y m≥1one has y∈SC[m](x)( ha is, ym∈SC(xm)), R(ym)>0,R(y−m)>0, and
P(y)is a posi i e powe o P(ym).
P oo . We ecall om [3, P oposi ion 3.9] ha i y∈USS(x) and ℓ(y)>0, hen R(y)⩽R(ym)
o all m⩾1. Hence, i y∈SC(x) is such ha R(y)>0 and R(y−1)>0, he same happens o
e e y powe o y.
By Lemma 5.8, ι(ym) = ι(y) and ϕ(ym) = ϕ(y). Hence p(ym) = ι(ym)∧∂(ϕ(ym)) = ι(y)∧
∂(ϕ(y)) = p(y). The e o e s(ym) = (s(y))m. By Lemma 5.9, s(y) also sa is ies he equi ed
igidi y condi ions, ha is, R(s(y)) >0 and R(s(y)−1)>0. Hence p(s(y)m) = p(s(y)) and hen
s2(ym) = s(s(ym)) = s(s(y)m) = (s2(y))m o e e y m > 0. I e a ing his a gumen , one ob ains
p((s (y))m) = p(s (y)) and s (ym) = (s (y))m o e e y , m > 0. In o he wo ds, applying i e a ed
cyclic sliding o ymis he same hing as applying i e a ed cyclic sliding o yand hen aking
he m h powe , since he conjuga ing elemen s coincide. As yis in a sliding ci cui , applying
i e a ed cyclic sliding leads back o y, and he same happens o ym. Tha is, ymis also in a
sliding ci cui , as we wan ed o show. Mo eo e , some posi i e powe o P(ym) equals P(y) as he
p e e ed p e ixes along he ci cui s o yand ymcoincide. Ac ually, we will ha e P(ym) = P(y),
unless he e is some zin he sliding ci cui o ysuch ha zm=ym, in which case P(ym) will be
sho e han P(y), bu con inuing along he sliding ci cui o ymone will ob ain se e al epe i ions
o P(ym) being equal o P(y).
We end his sec ion wi h a esul abou p e e ed conjuga o s which we shall need soon. I says
ha he p e e ed conjuga o s o any wo elemen s in he same se o sliding ci cui s a e conjuga e,
up o aising hose p e e ed conjuga o s o some sui able powe s. This will allow us o ob ain
in o ma ion conce ning P(y), o some y∈SC(x), jus by compa ing P(y) wi h P(z), o some
o he z∈SC(x). This ime we do no equi e any igidi y condi ion.
20
Lemma 5.13. Le x∈Bnand y, z ∈SC(x). Then P(y)sis conjuga e o P(z) o some s, > 0,
and one can ake as conjuga ing elemen any b aid αconjuga ing y o z.
P oo . Le Nand Mbe he leng hs o he sliding ci cui s o yand z, espec i ely. Tha is,
sN(y) = yand sM(z) = z. Le αbe such ha α−1yα =z. We can apply o α he anspo map
de ined in [16]. I one applies his anspo map k imes o α, we ob ain an elemen deno ed α(k),
which is a conjuga ing elemen om sk(y) o sk(z). Namely,
α(k)=p(y)p(s(y)) ···p(sk−1(y))−1αp(z)p(s(z)) ···p(sk−1(z)).(1)
In [16, Lemma 8] i is shown ha , in his si ua ion, z∈SC(x) i and only i α(sN)=α o
some s > 0. This means ha αconjuga es ssN (y) = y o ssN (z), bu since he conjuga e o yby α
is p ecisely z, i ollows ha ssN (z) = zhence sN = M o some > 0. Bu hen Equali y (1),
eplacing kby sN, eads α= (P(y)s)−1α P(z) o , in o he wo ds, α−1P(y)sα=P(z) .
5.3 Sliding ci cui s and canonical educ ion sys ems
P oposi ion 5.14. Le x∈Bnand y∈SC(x). I R(y)>0and R(y−1)>0, hen CRS(P(y)) ⊂
CRS(y).
P oo . No ice ha he esul holds i yis pe iodic, since he only pe iodic elemen s sa is ying he
igidi y hypo hesis a e powe s o ∆, and hen P(y) = 1, so bo h canonical educ ion sys ems a e
emp y. I yis pseudo-Anoso he esul also holds, since P(y) is in he cen alize o yso i mus
be ei he pseudo-Anoso o pe iodic [19], and in ei he case CRS(P(y)) = ∅=CRS(y). We can
hen assume ha yis non-pe iodic and educible, ha is, CRS(y)6=∅. And o cou se we can
assume ha CRS(P(y)) 6=∅, o he wise he esul is i ially ue.
By P oposi ion 5.12, we can make he u he assump ion ha yis pu e, since ymwill sa is y he
same hypo hesis as y, and he canonical educ ion sys ems o yand o i s p e e ed conjuga o
a e p ese ed by aking powe s o y. Replacing P(y) by a powe i necessa y in he ollowing
discussion, we will also assume ha P(y) is pu e.
Le F=CRS(y)∪ {∂(D2)}, and le us assume o a momen ha all cu es in Fa e ound.
Since P(y) is pu e and commu es wi h y,P(y) sends F o i sel , cu e-wise. This implies ha
an essen ial educ ion cu e o P(y) ei he belongs o F(as we wan o show) o can be iso oped
o be disjoin om F. In he la e case, i would co espond o an essen ial educ ion cu e
o P(y)[C∈F] o some [C]∈ F. Thus we mus show ha P(y)[C∈F ]does no admi an essen ial
educ ion cu e, o e e y [C]∈ F.
Le hen [C]∈ F. We know ha y[C∈F]is ei he pe iodic o pseudo-Anoso , and ha he b aid
P(y)[C∈F]commu es wi h y[C∈F ]. I y[C∈F ]is pseudo-Anoso , hen P(y)[C∈F ]mus be ei he
pseudo-Anoso o pe iodic, hence i admi s no essen ial cu es. I y[C∈F]is pe iodic, i has o be
a powe o he ull wis , since yis pu e. Bu in his case P oposi ion 5.7 ells us ha p(y)[C∈F]is
ei he i ial o a hal wis (he e we use ha Fconsis s o ound cu es). Hence, applying cyclic
sliding o y, we ob ain a b aid whose componen associa ed o Cis also a powe o he hal wis ,
and we can epea he a gumen un il one ge s back o y, o conclude ha P(y)[C∈F]is a (possibly
i ial) powe o ∆. Hence P(y)[C∈F ]does no admi an essen ial cu e, also in his case. The e o e,
all essen ial educ ion cu es o P(y) a e essen ial cu es o y, ha is, CRS(P(y)) ⊂CRS(y) i
CRS(y) is a amily o ound cu es.
Now we show he gene al case, in which he cu es in CRS(y) a e no necessa ily ound. We canno
apply he abo e a gumen as we do no know, a p io i, ha he componen s o P(y) co esponding
21

o he pe iodic componen s o ya e powe s o ∆. Ne e heless, we will be able o show his by
compa ing p e e ed p e ixes wi h he aid o Lemma 5.13. We jus need o ind a sui able b aid
whose educ ion cu es a e ound and which sa is ies he hypo hesis o P oposi ion 5.14, ha is,
i belongs o a sliding ci cui , and bo h he b aid and i s in e se ha e nonze o igidi y.
By Co olla y 5.3, o e e y N > 0 he e is some elemen z∈SC[N](y) whose essen ial cu es
a e all ound. We can hen ake N=||∆||3− ||∆||2and use P oposi ion 5.11 o conclude ha
o some mwi h 0 < m ⩽Nwe ha e R(zm)>0 and R(z−m)>0. As m≤N, we also ha e
zm∈SC(ym). No ice ha he canonical educ ion sys ems o zand zmcoincide, so zmis a b aid
whose canonical educ ion sys em is made o ound cu es, which belongs o a sliding ci cui ,
and such ha R(zm)>0 and R(z−m)>0, so zmis he b aid we we e looking o . To simpli y
no a ion, we ecall om P oposi ion 5.12 ha he esul will be shown o yi i is shown o ym,
so we can eplace yby ym, and his will eplace zby zm. We can hen assume ha zis a b aid
whose canonical educ ion sys em is made o ound cu es, which belongs o a sliding ci cui , and
such ha R(z)>0 and R(z−1)>0.
As he esul is shown o elemen s whose canonical educ ion sys em is made o ound cu es,
CRS(P(z)) ⊂CRS(z). Bu ecall om Lemma 5.13 ha P(z)sis conjuga e o P(y) o
some s, > 0, and ha a conjuga ing elemen αis p ecisely a conjuga ing elemen om z o y.
Since he essen ial cu es o P(z) and P(z)scoincide, we ha e CRS(P(z)s) = CRS(P(z)) ⊂
CRS(z). Conjuga ing bo h P(z)sand zby α, co esponds o applying α o hei essen ial cu es,
hence i ollows ha CRS(P(y) )⊂CRS(y). As he essen ial cu es o P(y) and P(y) coincide,
his means CRS(P(y)) ⊂CRS(y), as we wan ed o show.
We ha e now assembled mos o he ing edien s o showing ha ou main esul , Theo em 3.4,
ollows om he igid case. The key lemma o his educ ion o he igid case is as ollows.
Lemma 5.15. Le x∈Bnbe a non-pe iodic, educible b aid. Le N=||∆||3− ||∆||2. Fo e e y
elemen y∈SC[N](x) he e is some m⩽Nsuch ha ei he ymis igid, o P(ym)is igid, admi s
essen ial educ ion cu es, and all i s essen ial educ ion cu es a e essen ial educ ion cu es o y.
P oo . Le x∈Bnbe a non-pe iodic, educible b aid, N=||∆||3− ||∆||2and y∈SC[N](x). By
P oposi ion 5.11 he e is some powe ymwi h m⩽Nsuch ha R(ym)>0 and R(y−m)>0.
No ice also ha ym∈SC(xm). Hence ymsa is ies he hypo hesis o P oposi ions 5.10 and 5.14,
so P(ym) is igid and CRS(P(ym)) ⊂CRS(ym). I CRS(P(ym)) 6=∅, he esul ollows.
Suppose on he con a y ha CRS(P(ym)) = ∅. This means ha P(ym) mus be ei he pe iodic
o pseudo-Anoso . I canno be pseudo-Anoso , as i commu es wi h he non-pe iodic, educible
b aid ym, while pseudo-Anoso elemen s can only commu e wi h pseudo-Anoso o pe iodic ones.
Hence P(ym) is pe iodic. No ice ha P(ym) canno be a non i ial powe o ∆, since by P oposi-
ion 5.10 he le no mal o m o P(ym) is a p oduc o p e e ed p e ixes, each o hem no equal
o ∆ by de ini ion. As he only igid, pe iodic b aids a e he powe s o ∆, i ollows ha P(ym)
mus be i ial. This is equi alen o saying ha ymis igid.
The ollowing esul ells us how o deal wi h he igid case. We will assume o he momen ; i
will be shown in he nex sec ion:
Theo em 5.16. Le β∈Bnbe a non-pe iodic, educible b aid which is igid. Then he e is some
posi i e in ege k⩽nsuch ha one o he ollowing condi ions holds:
1. βkp ese es a ound essen ial cu e, o
2. in (βk)and sup(βk)a e e en, and ei he ∆−in (βk)βko β−k∆sup(βk)is a posi i e b aid which
p ese es an almos ound essen ial educ ion cu e whose co esponding in e io s ands do
no c oss.
22
In pa icula , some essen ial educ ion cu e o βis ei he ound o almos ound.
We can inally show ou main esul , assuming ha Theo em 5.16 holds.
P oo o Theo em 3.4. Le x∈Bnbe a non-pe iodic, educible b aid, N=||∆||3− ||∆||2and
y∈SC[N](x). Le m⩽Nbe he in ege gi en by Lemma 5.15. I ymis igid, hen by Theo em 5.16
CRS(ym) con ains a cu e which is ei he ound o almos ound. As CRS(ym) = CRS(y), he
esul ollows in his case.
I ymis no igid, hen by Lemma 5.15, P(ym) is igid and ∅ 6=CRS(P(ym)) ⊂CRS(ym) =
CRS(y). By Theo em 5.16 again, some cu e in CRS(P(ym)), and hus in CRS(y), is ei he
ound o almos ound.
This shows ha e e y elemen in SC[N](x) admi s an essen ial educ ion cu e which is ei he
ound o almos ound. This implies he esul .
6 Reducible igid b aids
This sec ion is de o ed o he p oo o Theo em 5.16.
Le β∈Bnbe a non-pe iodic, educible b aid which is igid. Then βbelongs o a sliding ci cui
(as s(β) = β), also ℓ(β)>0 and CRS(β)6=∅. Also, any powe βko βis also non-pe iodic,
educible and igid, and has he same canonical educ ion sys em as β. No ice ha o e e y
cu e C ∈ CRS(β), he e is some ⩽n/2 such ha [C]β = [C]. Replacing β by i s squa e i
necessa y, i ollows ha o e e y C ∈ CRS(x) he e is some e en k⩽nsuch ha βkp ese es [C],
and bo h in (βk) and sup(βk) a e e en.
Fix an inne mos cu e C ∈ CRS(β) and conside βk o some e en k⩽nsuch ha [C]βk= [C].
Le ∆2px1···x be he le no mal o m o βk, and deno e x=x1···x = ∆−in (βk)βk. No ice
ha CRS(β) = CRS(βk) = CRS(∆2px) = CRS(x), as ∆2p ese es e e y simple closed cu e o
he punc u ed disc. Mo eo e x=x1···x is non-pe iodic, educible and igid.
Deno e F=CRS(x) = CRS(β)6=∅. As Cis an inne mos cu e o F, he componen x[C∈F ]
mus be ei he pe iodic o pseudo-Anoso . Recall ha in o de o de ine x[C∈F ]one conjuga es x
by he minimal s anda dize o F o ob ain y=bx, and he cu e co esponding o C, namely b
C,
is an inne mos essen ial cu e o ywhich is ound. By [20, Theo em 4.9] ybelongs o i s Ul a
Summi Se p o ided xdoes. I is no di icul o modi y he p oo in [20] o show ha ybelongs
o SC(x) p o ided xdoes. This is he case, as xis igid. Bu i is shown in [16] ha , i xis igid,
SC(x) consis s p ecisely o he igid conjuga es o x. Hence yis igid. Mo eo e , as xp ese es
[C], y[
b
C∈
b
F]is a conjuga e o x[C∈F], which is ei he pe iodic o pseudo-Anoso .
Suppose y[
b
C∈
b
F]is pe iodic. As yis a igid, posi i e b aid, whose le no mal o m has he o m
y1···y , one has ha y y1is le weigh ed as w i en. Bu he le no mal o m o y[
b
C∈
b
F]is
de e mined by he le no mal o m o y, in he sense explained in Lemma 5.4. Hence y[
b
C∈
b
F]
mus be a igid, posi i e b aid whose le no mal o m is he p oduc o (possibly i ial) simple
elemen s. Since he only pe iodic igid elemen s a e powe s o ∆, i ollows ha ei he y[
b
C∈
b
F]is
i ial, o y[
b
C∈
b
F]= ∆
k(whe e kis he numbe o s ands inside b
C).
I y[
b
C∈
b
F]is i ial, he in e io b aid o x= ∆−in (βm)βmassocia ed o Cmus also be i ial, as i
is a conjuga e o y[
b
C∈
b
F]. By P oposi ion 2.3, Cis ei he ound o almos ound, so Theo em 5.16
holds in his case.
23
Suppose ha y[
b
C∈
b
F]= ∆
k, and no ice ha = sup(βm)−in (βm) is e en. Le us conside he
n-s and b aids x′and y′such ha xx′= ∆ and yy′= ∆ . We ema k ha x′and y′a e basically
he in e ses o xand y, mul iplied by some e en powe o ∆ so ha hei in imum becomes 0.
Hence x′and y′a e posi i e, igid b aids o in imum 0 and canonical leng h , whose canonical
educ ion sys ems coincide wi h hose o xand y, espec i ely. Le αbe such ha α−1xα =y.
Since α−1∆ α= ∆ as is e en, we ob ain ha α−1x′α=y′. Mo eo e , y′
[
b
C∈
b
F]is i ial. Hence,
he s ands o x′in e io o Cdo no c oss. By P oposi ion 2.3, Cis ei he ound o almos ound.
Now no ice ha x′=x−1∆ =β−m∆in (βm)+ =β−m∆sup(βm). Hence Theo em 5.16 also holds
in his case.
I only emains o p o e Theo em 5.16 in he case in which x[C∈F ]is pseudo-Anoso .
Lemma 6.1. Le x∈Bn. Gi en wo elemen s y, z ∈SC(x), he e is a sequence o conjuga ions
y=α1
s1
−→ α2
s2
−→ α3··· s
−→ α +1 =z
such ha o e e y i= 1, . . . , one has αi+1 =αsi
i∈SC(x), and ei he si4ι(αi)o si4ι(α−1
i).
P oo . This p oo ollows he ideas in [9, 13, 4]. Fi s , we can assume ha ℓ(y)>0, o he wise
SC(x) = {∆p} o some p, and he esul becomes i ial as y=z. Now yand za e conjuga e
since hey belong o SC(x). Mul iplying any conjuga ing elemen by a su icien ly la ge powe
o ∆, i ollows ha z=yα o some posi i e elemen α. This conjuga ing elemen αcan ob iously
be decomposed in o a p oduc o indecomposable conjuga ing elemen s, ha is, α=s1···s , whe e
αi+1 =ys1···si∈SC(x) o i= 1,..., , and siis posi i e and canno be decomposed as a p oduc
o wo non i ial posi i e elemen s si=ab such ha αa
i∈SC(x). No ice ha simus be simple,
o he wise we could ake a=si∧∆ (which by Theo em 4.1 sa is ies αa
i∈SC(x)) o decompose si.
We mus show ha such an indecomposable elemen simus be a p e ix o ei he ι(αi) o ι(α−1
i).
Deno e =si∧ι(α−1
i). We claim ha (αi) ∈SC(x). Indeed, by de ini ion, one has ι(α−1
i) =
∆∧(α−1
i∆−in (α−1
i)). Since αi∈SC(x), i is clea ha α∆
i∈SC(x) and ha α(α−1
i∆−in (α−1
i))
i∈
SC(x). By Theo em 4.1, αι(α−1
i)
i=α∆∧(α−1
i∆−in (α−1
i))
i∈SC(x). Bu αsi
i=αi+1 ∈SC(x), so
applying Theo em 4.1 again one has αι(α−1
i)∧si
i= (αi) ∈SC(x), as we wan ed o show.
We hen ha e a posi i e p e ix 4sisuch ha (αi) ∈SC(x). Since siis an indecomposable
conjuga o , i ollows ha ei he =sio = 1. In he o me case si= =si∧ι(α−1
i), which
implies si4ι(α−1
i), hence he esul holds in his case.
Suppose hen ha = 1. This means ι(α−1
i)∧si=∂(ϕ(αi)) ∧si= 1, which is equi alen o say
ha ϕ(αi)siis le weigh ed as w i en. Le ∆pa1···a be he le no mal o m o αi. We ha e hen
shown ha a siis le weigh ed as w i en, so ∆pa1···a siis he le no mal o m o αisi. Bu
we know ha αi+1 =s−1
iαisi∈SC(x). In pa icula ℓ(αi+1) = , whe e αi+1 =s−1
i∆pa1···a si.
This implies ha τp(si)4a1···a si, whe e he igh hand side is in le no mal o m and he le
hand side is a simple elemen , hence τp(si)4a1, ha is, si4τ−p(a1) = ι(αi), so he esul also
holds in his case.
Finally, he e is he esul ha comple es he p oo o Theo em 5.16:
P oposi ion 6.2. Le xbe a educible igid b aid, and le Cbe an in a ian cu e o xwhose
co esponding in e io b aid is pseudo-Anoso . Then Cis ound.
P oo . We know om [16] ha SC(x) is he se o igid conjuga es o x, hence x∈SC(x), and
we know om Co olla y 5.3 ha he e is an elemen ˜x∈SC(x) whose educ ion cu es a e all
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ound. By Lemma 6.1 he e is a chain o conjuga ions
˜x=α1
1
−→ α2
2
−→ α3···
−→ α +1 =x
such ha o e e y i= 1, . . . , one has αi+1 =α i
i∈SC(x), and ei he i4ι(αi) o i4ι(α−1
i).
Suppose ha Cis no ound. This means ha he cu e C˜xo ˜xco esponding o Cis a ound
cu e which loses i s oundness a e he applica ion o 1··· . This implies ha he e mus be
wo igid b aids y, z ∈SC(x) (p ecisely αiand αi+1 o some i), conjuga e by a simple elemen s
(p ecisely i), a ound in a ian cu e Cyo ywhose co esponding in e io b aid is pseudo-Anoso ,
and he co esponding in a ian cu e o z, [Cz] = [Cy]s, which is no ound. Mo eo e sis ei he
a p e ix o ι(y) o a p e ix o ι(y−1) (as s= i). Since he in e se o a pseudo-Anoso b aid is also
pseudo-Anoso , and he igidi y and educ ion cu es o a b aid a e p ese ed by aking in e ses,
we can eplace yand zby y−1and z−1i necessa y, so we can assume ha sis a p e ix o ι(y−1).
Since aking powe s and mul iplying igid b aids by ∆2ka e ope a ions which do no a ec hei
igidi y, hei ini ial ac o s, hei in a ian cu es o he geome ic ype o hei co esponding
in e io b aids, we can u he assume ha yand za e pu e b aids, and ha in (y) = in (z) = 0.
Suppose ha some non i ial posi i e p e ix s′4sis such ha [Cy]s′is ound, and deno e by ρ he
minimal posi i e elemen such ha s′4ρand yρis igid (equi alen ly, yρ∈SC(x)). Since sis an
indecomposable conjuga o , we mus ha e ρ=s. Bu we will now see ha ρsends [Cy] o a ound
cu e, while [Cy]ρ= [Cy]s= [Cz] is no ound. A con adic ion ha will imply ha s′= 1. Indeed,
by [17, Algo i hm 2, s ep 3(b)], ρcan be compu ed in he ollowing way: i s , while ys′/∈SSS(x),
eplace s′by
s′·1∨(ys′)−1∆in y∨ys′∆−sup y.
No ice ha he h ee elemen s 1, (ys′)−1∆in yand ys′∆−sup ysend [Cy]s′ o a ound cu e. In
he e minology o [20], he h ee elemen s belong o he s anda dize o [Cy]s′. Since i is shown
in [20] ha he s anda dize o a cu e is closed unde ∨, i ollows ha each s ep o his p ocedu e
eplaces s′by a bigge elemen , which belongs o he s anda dize o [Cy]. Hence we can assume
ha ys′∈SSS(x). The second s ep o compu e ρ, explained in [16, Theo em 2], consis s o
applying i e a ed sliding o ys′un il one eaches a igid elemen . Mul iplying s′on he igh by
all conjuga ing elemen s, one ob ains ρ. Bu each conjuga ing elemen o sliding main ains he
oundness o ou dis inguished cu e, om P oposi ion 5.2. The e o e, ρsends Cy o a ound
cu e, bu [Cy]ρ= [Cz] is no ound. A con adic ion. I ollows ha s′= 1, o in o he wo ds,
he e is no non i ial p e ix s′4sis such ha [Cy]s′is ound.
Le p, p + 1,...,q be he punc u es inside Cy. We will collec he s ands o sin o h ee se s,
L={1,...,p−1},I={p, p + 1,...,q}and R={q+ 1, q + 2,...,n}, depending whe he hey
s a o he le , inside o o he igh o Cy. Since e e y p e ix o smus de o m he ound cu e Cy,
and he b aid sis simple, i ollows ha he s ands in L( esp. in Iand in R) do no c oss each
o he in s, since his would imply ha wo consecu i e s ands in L( esp. in Iand in R) would
c oss in s, and he co esponding c ossing would be a p e ix o sp ese ing he oundness o Cy, a
con adic ion. Also, no s and o sin Lcan c oss all he s ands in I, since his would imply ha
he s and p−1 would c oss all he s ands in I, and hen σp−1σp···σq−1would be a p e ix o s
p ese ing he oundness o Cy, a con adic ion. In he same way, no s and o sin Rcan c oss
all he s ands in I. In summa y, sis a simple b aid o a e y pa icula o m: some s ands o L
may c oss some (bu no all) s ands o I, some s ands o Rmay c oss some (bu no all) s ands
o I, and any wo s ands belonging o he same g oup (L,I, o R) ne e c oss.
Recall ha yand za e igid, and le y1···y and z1···z be hei le no mal o ms. Fo
i= 0 ..., , we deno e [Cy,i] = [Cy]y1···yiand [Cz,i] = [Cz]z1···zi. By Theo em 5.1, Cy,i is ound o
e e y i, and by he igidi y o zi ollows ha Cz,i is no ound o any i. Now, o i= 0,..., ,
conside he b aid si= (y−1
i···y−1
1)s(z1···zi), which is he i h anspo o sunde cycling
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