chasing the rabbit

the rabbit conjecturea brutal approach

a less brutal approach

Chasing the Rabbit

David Bessis

Les Houches, 28/01/2011

David Bessis Chasing the Rabbit



This talk is a survey of the 10 years effort (from the mid-ninetiesto the mid-naughties) to find good diagrams for complex reflectiongroups and their braid groups.




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Broue-Malle-Rouquier’s setup:

V is an n-dimensional complex vector space.

W ⊆ GL(V ) is a finite group generated by complexreflections.

A is the set of all reflecting hyperplanes.

V reg := V −⋃w∈W−{1} ker(w − 1) = V −⋃H∈AH is thehyperplane complement.

B(W ) := π1(V reg/W ) is the braid group of W .




Problem. Find Coxeter-like diagrams for W , providing both aCoxeter-like presentation for W and an Artin-like presentation forB(W ).




Additional requirements for good diagrams:

the generators of W should be reflections,

the braid generators should be braid reflections (akameridiens, aka generators-of-the-monodromy),

the number of generators should be minimal (when W isirreducible, this could be either n or n + 1),

the product of the braid generators, raised to a certain power,should generate the center of B(W ),

the diagrams should be “pretty” – and, when possible, “cute”.

(Hope. Find a systematic replacement for Coxeter theory.)




Broue-Malle-Rouquier were able to solve this in all but sixexceptional cases:

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Various methods are used, applicable to particular families ofirreducible cases:

The infinite family G (de, e, n) is monomial. Broue-Malle-Rouquier use fibration arguments (a la Fadell-Neuwirth).

Real groups: the Artin presentation was obtained by Brieskorn.

A few exceptional non-real cases have a regular orbit spaceisomorphic to that of a real group (Orlik-Solomon).

2-dimensional exceptional groups: presentations are obtained“by hand” (Bannai).

There are 6 exceptional groups not covered by these methods – the6 missing cases as of 1998.




The rabbit conjecture (Broue-Malle-Rouquier, 1998):

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Doesn’t look like a rabbit?




A brutal approach to the rabbit conjectureD.B./J. Michel, 2001-2003




The discriminant of W is the image in V /W of the hyperplaneunion

⋃H∈AH ⊆ V .

By Chevalley-Shephard-Todd theorem, V /W is an affine space.

The equation of H in V /W can be writtten as an explicitpolynomial.




In the 1930s, Zariski and Van Kampen proposed a “method” forcomputing fundamental groups of complements of algebraichypersurfaces.Let H be an algebraic hypersurface in a complex affine space V .

for a “generic” complex 2-plane P, the map(V −H) ∩ P → V −H is a π1-isomorphism,

for a “generic” complex line L, the map(V −H) ∩ L→ V −H is a π1-epimorphism,

a presentation for π1(V −H) can be obtained by computingthe monodromy braids of the punctures in (V −H) ∩ L overthe space of generic lines.




Difficulties.

what does “generic” mean?

can monodromy braids be computed with an exact softwarealgorithm?

can the computation be efficient enough to address non-trivialcases?

is there a good heuristic to simplify the (highly redundant)presentations obtained this way?




VKCURVE (D.B., Jean Michel) is a software package thatimplements an efficient exact version of Van Kampen’s method.




Explicit Presentations for Exceptional Braid GroupsDavid Bessis and Jean Michel

CONTENTS

1. Introduction2. The Presentations3. Definitions and Preliminary Work4. Choosing the 2-Plane5. The Package VKCURVE6. Explicit Matrices of Basic DerivationsAcknowledgmentsReferences

2000 AMS Subject Classification: Primary 20F34;Secondary 20F36, 20F55

Keywords: Braid groups, complex reflection groups,fundamental groups, Van Kampen method

We give presentations for the braid groups associated with thecomplex reflection groups G24 and G27. For the cases of G29,

G31, G33, and G34, we give (strongly supported) conjectures.These presentations were obtained with VKCURVE, a GAP pack-age implementing Van Kampen’s method.

1. INTRODUCTION

To any complex reflection group W ! GL(V ), one may

attach a braid group B(W ), defined as the fundamental

group of the space of regular orbits for the action of W

on V [Broue et al. 98].

The “ordinary” braid group on n strings, introduced

by [Artin 47], corresponds to the case of the symmet-

ric group Sn, in its monomial reflection representation

in GLn(C). More generally, any Coxeter group can be

seen as a complex reflection group, by complexifying the

reflection representation. It is proved in [Brieskorn 71]

that the corresponding braid group can be described

by an Artin presentation, obtained by “forgetting” the

quadratic relations in the Coxeter presentation.

Many geometric properties of Coxeter groups still hold

for arbitrary complex reflection groups. Various authors,

including Coxeter himself, have described “Coxeter-like”

presentations for complex reflection groups. Of course,

one would like to have not just a “Coxeter-like” presen-

tation for W , but also an “Artin-like” presentation for

B(W ).

The problem can be reduced to the irreducible case.

Irreducible complex reflection groups have been classi-

fied by [Shephard et al. 54]: there is an infinite family

G(de, e, r) (which contains the infinite families of Coxeter

groups), plus 34 exceptional groups G4, . . . , G37 (among

them are the exceptional Coxeter groups).

Before this note, presentations were known for all but

6 exceptional groups (see the tables of [Broue et al. 98]):

c! A K Peters, Ltd.1058-6458/2004$ 0.50 per page

Experimental Mathematics 13:3, page 257

-

87




Bessis and Michel: Explicit Presentations for Exceptional Braid Groups 263

is good for our purposes. In our examples, it is easy to

construct such planes, since Disc(!X) is monic in one

of the remaining variables. This is how we obtained, for

G24 and G27, theorems rather than conjectures.

Note that, for other groups, all assumptions used here

(including the monicity of Disc(!X)) remain valid, ex-

cept that we do not know whether !2(E, y0) = 0. Instead

of answering Question 4.2, checking that !2(E, y0) = 0

would turn our conjectures into theorems.

5. THE PACKAGE VKCURVE

Once a 2-plane P has been chosen, it is enough to feed

VKCURVE with the equation of the curve P ! H to ob-

tain a presentation of !1(P " (P ! H)).

Example 5.1. For G31, when computing the determinant

of M31 and evaluating at z = y and t = 1 + x, we obtain

the following equation for P ! H:

!!31 = 746496 + 3732480x " 3111936xy2

" 93281756

27xy4 +

58341596

27xy6 + 7464960x2

" 384y2 " 9334272x2y2 +17556484

27x2y4

+43196

27x2y6 + 7464576x3 " 756138248

81x3y2

+192792964

81x3y4 +

16

81x3y6 + 3730944x4

" 139967996

81y4 " 84021416

27x4y2 +

82088

27x4y4

+ 744192x5 +43192

27x5y2 " 1720

27x5y4

" 124412

81x6 + 777600800y6 +

95896

81x6y2

" 8

81x6y4 " 10364

27x7 " 4

27x7y2 +

4

27x8

" 8

81y8 " 4

27x8y2 +

4

81x9.

On a 3 GHz Pentium IV, VKCURVE needs about one

hour to deal with this example.

Writing VKCURVE was of course the most di"cult

part of our work. This software accepts as input any

square-free polynomial in Q[i][X,Y ] and computes a pre-

sentation for the fundamental group of the complement of

the corresponding complex algebraic curve. The program

does not use floating point computations (even when

computing monodromy braids); therefore, there is no is-

sue of numerical accuracy, and the result is “certified” to

be correct (provided that our implementation does not

contain mathematical errors).

The remainder of this section is an overview of the

algorithms used in VKCURVE. We rely on the version of

Van Kampen’s method exposed in [Bessis 03, Procedure

4], where it is decomposed into four steps.

5.1 Implementing Steps 1 and 2

Starting with our polynomial P # Q[i][X,Y ], we view it

as a one-variable polynomial in Q[i][Y ][X] and compute

its discriminant ! # Q[i][Y ]. The discriminant ! may

not be reduced; to compute approximations y1, . . . , yr #Q[i] of its complex roots y1, . . . , yr, we apply Newton’s

method to the reduced polynomial !0 obtained by divid-

ing ! by the resultant of ! and !!. As is proved in the

beautiful article [Hubbard et al. 01], Newton’s method

can be made into a failsafe algorithm producing arbi-

trarily good approximations of y1, . . . , yr.

Since we will reuse them later, we recall a few trivi-

alities about complex polynomials. Let P # C[Z]. Let

"1, . . . ,"n be the complex roots of P . Let z # C. If

P !(z) $= 0, we set NP (z) := z " P (z)P !(z) . Considering

the first order approximation of P around z, we expect

P (NP (z)) to be close to 0. Newton’s method consists

of starting with z0 # C (chosen randomly, or smartly

as in [Hubbard et al. 01]) and constructing iteratively

zm+1 := NP (zm), hoping that (zm) will converge towards

a root of P—which indeed happens for “many” choices

of z0. How may we decide that a given zn is a “good

enough” approximation?

Lemma 5.2. Assume P has n distinct roots "1, . . . ,"n.

Let z # C, with P !(z) $= 0. Then, there exists " #{"1, . . . ,"n} such that |z " "| % n

!!! P (z)P !(z)

!!!.

Proof: If P (z) = 0, the result is trivial. Otherwise, we

have P !(z)P (z) =

"ni=1

1z"!i

. Choose i such that, for all

j, |z " "i| % |z " "j |. By triangular inequality, 1|z"!i| &!!!P !(z)

P (z)

!!!""

j #=i1

|z"!j | &!!!P !(z)

P (z)

!!!"(n"1) 1|z"!i| . The result

follows.

Although elementary, this lemma provides a very inex-

pensive (in terms of computational time) test for deciding

whether a tentative list "1, . . . , "n of complex numbers

“separates” the roots (i.e., whether there exists #1, . . . , #n

such that the disks D("i, #i) do not overlap and each of

them contains a root of P ).

Instead of working with the exact Newton’s method,

we use a truncated version, where NP (z) is replaced by

an approximate (a+ib)10k, where a, b # Z and k is an in-

teger slightly smaller than log10

!!!P !(z)P (z)

!!!. This is to avoid

-

93




The presentation-shrinking heuristics implemented in VKCURVEallowed us to recover the Rabbit diagram for G31.




Remarks

Our “Explicit Presentations” paper suffered from our attemptto use Hamm-Le transversality conditions (based on Whitneystratifications.)

There is a much simpler criterion to check that a 2-planesection induces a π1-isomorphism (see Section 4 of my“K (π, 1)” paper.)

Zariski-Van Kampen method is indeed a fully implementablealgorithm.

In particular, the task of finding good diagrams for allcomplex reflection groups is now complete.




A less brutal approach to the rabbit conjectureD.B., 2006 (unpublished)




Strategy of proof

View B(G31) as the centralizer of a periodic element in the Artingroup of type E8 (braid version of Springer’s theory of regularelements).

Ingredients:

Conjectures on periodic elements in braid groups(Broue-Michel)

The dual braid monoid (Birman-Ko-Lee, D.B.)

Non-positively curved aspects of Artin groups (Bestvina)

Garside categories (Krammer; see also Digne-Michel)

“Tits-like” geometric objects in V /W and a “chamber-like”decomposition. Sub-ingredients:

Kyoji Saito’s flat structureLyashko-Looijenga morphisms




Remark. The construction for G31 is just one particular case in ageneral theory applicable to “almost all” complex reflection groupsand providing good diagrams, Garside structures, natural geometricobjects, and much more.

We emphasize G31 because it is the most pathological example:

non-monomial,

high-dimensional (dimension 4),

regular orbit space doesn’t coincide with that of a real group,

not well-generated (it needs 5 reflections).




Idea 1 (Broue-Michel)

Springer theory of regular elements in W should have an analog interms of periodic elements (roots of central elements) in B(W ).

As G31 is the centralizer of a 4-regular element in E8, one couldexpect B(G31) to be the centralizer of a 4-periodic element inB(E8).




Sur certains elements reguliersdes groupes de Weyl et les varietes

de Deligne–Lusztig associees

Michel Broue et Jean Michel

Sommaire

1. Varietes de Deligne–LusztigA. Contexte et notationsB. Les varietes de Deligne–Lusztig

2. La variete X!

A. Operation de B+ sur X!

B. Valeurs de caracteres – Conjectures3. Bons elements reguliers

A. Elements reguliers et groupes des tresses associesB. Racines de ! et elements reguliers

4. Groupes de reflexions complexes et algebres de Hecke associeesA. GeneralitesB. Caracteres et degres fantomesC. Groupes de tresseD. Algebres de HeckeE. Valeurs de caracteres sur les racines de !

5. Varietes associees aux racines de !A. Quelques proprietesB. ConjecturesC. Valeurs de caracteres et applications

6. Le cas non-deployeA. GeneralitesB. Elements reguliers et !-racines de !C. Algebres de Hecke “avec automorphisme”D. Varietes associees aux !-racines de !Annexe 1 : Les bonnes racines de !Annexe 2 : Les groupes de Coxeter tordusAnnexe 3 : Les groupes de Coxeter cyclotomiques

Nous remercions Francois Digne, Gunter Malle et Raphael Rouquier pour de

nombreuses conversations utiles a la mise au point de cet article, et Pierre Deligne

pour nous avoir communique les resultats contenus dans son manuscrit [De2].

[. . . ]

92 Michel Broue et Jean Michel

des reflexions et donc n’est pas contenu dans SL(V ). Ce groupe WH!

est normalise par w, donc on deduit de 3.3 que

(WH! != {1}) "# (CWH! (w) != {1}) .

Mais CWH! (w) = W (w)H! , et par consequent ce groupe n’est pas trivialsi et seulement si H ! $ A(w). On voit donc que A(w) est l’ensemble deshyperplans H ! de V (w) tels que WH! ne soit pas trivial, donc (puisqueWH! est engendre par des reflexions) tels qu’il existe un hyperplan dereflexion H de W contenant H !. La proposition 3.2 resulte alors dufait que, par hypothese, V (w) n’est contenu dans aucun hyperplan dereflexion de W . !

Remarque. Comme note dans [DeLo] et [Le], la proposition precedenteet sa demonstration s’etendent au cas plus general ou W est un groupeengendre par des pseudo-reflexions (et meme au cas ou w est un elementregulier qui normalise W sans necessairement lui appartenir), grace autheoreme fondamental de Steinberg ([St1], 1.5) qui a!rme que le fix-ateur d’un sous-espace (“sous-groupe parabolique”) est aussi engendrepar des pseudo-reflexions.

On designe par ! : [0, 1] % M(w) le lacet d’origine x0 defini par!(!) = e2!i"x0 , et on designe par w : [0, 1] % M(w) le chemin de x0

a w.x0 defini par w(!) = e2!i"/dx0 . On note encore ! et w respective-ment les elements de P(w) et B(w) ainsi definis. On voit que, dans legroupe B(w), on a

wd = ! .

Comme nous l’a fait remarquer R. Rouquier, il est facile de verifierque

3.4. l’element w est central dans B(w).

3.5. Question. L’injection naturelle de V (w) dans V definit-elle unmorphisme injectif P(w) "% P ?

Remarquons que si tel est le cas, alors

3.6. les fleches naturelles induisent le diagramme commutatif suivant,ou les fleches horizontales sont injectives, et les suites verticales sont




Idea 2 (Jean Michel)

Let φ be a diagram automorphism of a (finite) Coxeter group W .The Deligne/Brieskorn-Saito normal form allows an easycomputation of the centralizer of φ in A(W ).

In particular, if ∆ ∈ B(W ) is the Tits lift of w0, the centralizer of∆ in B(W ) is isomorphic to the braid group B(W ′), whereW ′ = CW (w0), as predicted by the braid version of Springer theory.




Example. How to view F4 in E6:

4 DAVID BESSIS

w0 on W is a diagram automorphism (i.e., it is induced by a permutation of S) and thecentraliser W ! := CW (w0) is a Coxeter group with Coxeter generating set S ! indexed byw0-conjugacy orbits on S. At the level of Artin groups, one shows (see for example [28])that

A(W !, S !) ! CA(W,S)(!),

which is an algebraic reformulation of the case p = 1, q = 2 of Question 3 (applied toW\V reg, as in Example 0.2).

Example 0.4. When A(W, S) is a Artin group of type E6, the !-conjugacy action is thenon-trivial diagram automorphism and the centraliser is an Artin group of type F4.

• •• •

• •s1 s2

s3 s4

s!3 s!

4

!!!!!!!!!!

""""""""""

!!

!

""

,

!!

!

""

• • • •s1 s2 s3s

!3 s4s

!4

The main strategy throughout this article is to construct Garside structures with suf-ficient symmetries, so that centralisers of periodic elements can be computed as easily asin Example 0.4.

Birman-Ko-Lee showed that the classical braid group Bn admits, in addition to thetype An"1 Artin group structure, another Garside group structure where the Garsideelement is a rotation ! of angle 2" 1

n. In [5], we used this Garside structure to compute

the centralisers of powers of !, which solves Question 3 for Bn and q|n. Thanks to somerather miraculous diagram chasing, we were also able to obtain the remaining case q|n"1.

Whenever G is a group and ! # G is the Garside element of a certain Garside structure,the centraliser CG(!) is again a Garside group, and is easy to compute. Note that thenotion of periodic element may be extended to this setting: we say that # # G is p

q-periodic

if#q = !p.

This of course is relative to the choice of a particular Garside structure. However, for Bn,the Artin Garside element ! and the Birman-Ko-Lee Garside element ! are commensu-rable

!2 = !n.

In particular, whether a given element # is periodic or not does not depend on the choicebetween the two standard Garside structures (although the actual p and q may vary).

As the above particular case illustrates, it is very easy to compute centraliser of Garsideelements in Garside groups. In particular, when trying to answer Questions 3 and 3’ in aspace whose fundamental group is a Garside group, it is very tempting to expect to builda proof on a positive answer to:

Question 4. Let G be a Garside group with Garside element !. Let # # Gbe a periodic element with respect to !. Does G admit a Garside structurewith Garside element #?

Note that a positive answer to Question 4 would imply that the following question alsoadmits a positive answer:




Idea 3 (Dehornoy-Paris)

What really matters in Deligne/Brieskorn-Saito normal form canbe axiomatized.

A Garside monoid is a monoid with properties emulating those ofspherical type Artin monoids:

finite positive homogeneous presentation,

lattice for the divisibility order,

a Garside element ∆, common multiple to all generators, suchthat conjugating by ∆ is “diagram automorphism” (apermutation of the generators).

(Some of these properties can be relaxed.)

Some (not all) of Broue-Malle-Rouquier diagrams provideexamples.




!"#$$%"& !'(#)$ "&* !"'$%*+ !'(#)$,-.( !+&+'"/%$"-%(&$ (0 "'-%& !'(#)$

)"-'%12 *+3('&(4 !"# /#%$ )"'%$

5'676896: ; (7<=>6? @AA;B'698C6: @D $6E<6F>6? @AADG

@H $"%&'#()%*'"

-I6 +',*%*-. /&!*# 0'"'*# J=K "! @ C<?8KLCM 8C <I6 F=K=8: 1! <IN< N:F8<C<I6 E?6C6K<N<8=K

"2@! H H H ! 2" #2* 23 $ 23 2* 8O # *% 3 #! P! 2* 2*!@ 2* $ 2*!@ 2* 2*!@ 8O * $ @! H H H ! "% @&"

%< QNC 7=KC8:6?6: >R !N?C8:6 8K 5!"G NK: ESNRC N E?=F8K6K< ?=TS6 8K <I6 <I6=?R =O>?N8: L?=UECH %K EN?<87USN?, C696?NS E?=E6?<86C =O <I6 >?N8: L?=UEC N?6 :6?896: O?=F6V<6KC896 8K96C<8LN<8=KC =O <I6 E=C8<896 >?N8: F=K=8:C JC66, O=? 6VNFES6, 5#, !$, !%GMH" W?C< =>C6?9N<8=K 8C <IN< <I6 :6WK8KL ?6SN<8=KC =O 1! N?6 I=F=L6K6=UCH -IUC,

=K6 FNR :6NS Q8<I N S6KL<I OUK7<8=K !X 1! ' ! QI87I NCC=78N<6C <= ! 8K 1! <I6S6KL<I =O NKR 6VE?6CC8=K =O !H 0=? !, / 8K 1!, Q6 CNR <IN< ! 8C N 4.5% #*-*,'& =O /=?, 6YU89NS6K<SR, <IN< / 8C N &*67% 0(4%*+4. =O ! 8O <I6?6 6V8C<C ) 8K 1! CU7I <IN< /8C !)H -I6 6V8C<6K76 =O <I6 S6KL<I OUK7<8=K LUN?NK<66C <IN< S6O< :898C8>8S8<R 8C NEN?<8NS =?:6? =K 1!H %< QNC N7<UNSSR E?=96: 8K 5!"G <IN< NKR <Q= 6S6F6K<C =O1! IN96 N S=Q6C< 7=FF=K ?8LI< FUS<8ES6H Z=?6=96?, 1! INC 4.5% !"# &*67%)!").44!%*'" +&'+.&%*.,, KNF6SR, !/ $ !) 8FES86C / $ ), NK: /! $ )! 8FES86C/ $ )H (?6[C 7?8<6?8=K CNRCX 8O N F=K=8: 8 INC S6O< NK: ?8LI< 7NK76SSN<8=KE?=E6?<86C, NK: 8O NKR <Q= 6S6F6K<C =O 8 IN96 N 7=FF=K ?8LI< FUS<8ES6, <I6K 86F>6:C 8K 8<C 6&'(+ '5 J&*67%M 5&!)%*'", JC66 5!&, -I6=?6F @HP\GMH -I8C L?=UE 8C(8 )8%@*!+, QI6?6 8%@ 8C <I6 :UNS F=K=8: =O 8, NK: + 8C <I6 7=KL?U6K76?6SN<8=K L6K6?N<6: >R <I6 EN8?C (22%@! @* NK: (2%@2! @*, Q8<I 2 8K 8H ]R <I6E?698=UC 7=KC8:6?N<8=KC, 1! CN<8CW6C (?6[C 7=K:8<8=KC, NK:, <I6?6O=?6, 6F>6:C 8K 8<CL?=UE =O O?N7<8=KCH -I8C 8C <I6 >?N8: L?=UE =K "! @ C<?8KLCH-I6 5("#!0."%!4 .4.0."% =O 1!, UCUNSSR :6K=<6: >R ", 8C <I6 S=Q6C< 7=FF=K

?8LI< FUS<8ES6 =O 2@! H H H ! 2"H %< 8C NSC= <I6 S=Q6C< 7=FF=K S6O< FUS<8ES6 =O2@! H H H ! 2", NK: "P L6K6?N<6C <I6 76K<?6 =O <I6 >?N8: L?=UEH 0U?<I6?F=?6, <I6 C6< =OS6O< :898C=?C =O " 8C 6YUNS <= <I6 C6< =O ?8LI< :898C=?C =O "H-I8C C8<UN<8=K QNC C8FUS<NK6=UCSR L6K6?NS8C6: >R ]?86C^=?K NK: $N8<= 5'G, NK:

>R *6S8LK6 5!'G, <= N ONF8SR =O F=K=8:C NK: L?=UEC 7NSS6: WK8<6 1=V6<6? <RE6"?<8K F=K=8:C NK: L?=UECH /8^6 <I6 >?N8: L?=UEC, <I6C6 L?=UEC IN96 K876 K=?FNSO=?FC JC66 5'G NK: 5!'GM, IN96 ONC< Q=?: E?=>S6F C=SU<8=KC JC66 5#"GM, NK: N?6>8NU<=FN<87 JC66 5"G NK: 5(GM, NSS <I6C6 E?=E6?<86C >68KL E?=96: <I?=ULI N :66EC<U:R =O <I6 "?<8K F=K=8:CH%K <I8C ENE6?, Q6 CINSS 6V<6K: <I6 E?698=UC ?6CUS<C <= N SN?L6? 7SNCC =O F=K=8:C

NK: L?=UEC, QI87I Q6 KN<U?NSSR E?=E=C6 <= <6?F 9!&,*#.H -I6C6 L?=UEC N?6

@AA@ 8!%7.0!%*), :(/3.)% ;4!,,*<)!%*'"X E?8FN?R P_0_`, P_0\ab C67=K:N?R P_]c_, P_Z_`H

=&')> ?'"#'" 8!%7> :')> J\M ;A J@AAAM `aAda_cH




Idea 4 (Birman-Ko-Lee)

In the Artin group of type An−1, there is an alternate Garsidestructure whose Garside element δ = σ1 . . . σn−1 is related to theclassical Garside element by the relation:

δn = ∆2.

In a joint work with Francois Digne and Jean Michel, we used thisto compute centralizers of periodic elements in type A braidgroups.




The dual braid monoid generalizes Birman-Ko-Lee’s constructionto all well-generated complex reflection groups (this includes allreal types, as well as all high-dimensional exceptions except G31.)

In the real case, the new generating set consists of all reflections:

678 D. BESSIS

LEMMA 5.1.1. – For any positive integer m, set

tm :=

!m"

i=1

si

#!m!1"

i=1

si

#!1

.

We have T = {t1, . . . , tN}, and

w0 =N"

m=1

tN!m+1.

Proof. – Set tm := p(tm). By [4], Ch. V, §6 Ex. 2, pp. 139–140, we have T = {t1, . . . , tN}.Using the commutation relations within L, we see that, when 1 ! m ! k, tm = sm. When

k + 1 ! m ! n, the commutation relations within R yield

tm = s1 . . .sm!1sms!1m!1 . . .s!1

1 = s1 . . .snsms!1n . . .s!1

1 = csmc!1.

We have proved$s1, . . . , sk,csk+1c

!1, . . . ,csnc!1%

= {t1, . . . , tn}.

For all m, we have tm+n = ctmc!1. From this and the above description of {t1, . . . , tn}, wededuceT = {t1, . . . , tN}.From the Bourbaki exercice quoted above, we also get w0 =

&Nm=1 sm. Since

(s1, . . . , sN ) ! RedS(w0),

we have w0 =&N

m=1 sm =&N

m=1 tN!m+1. !These facts are summarized in Table 1.The final line has the following explanation: in [1], a certain class of presentations of braid

groups is constructed. Each of these presentations corresponds to a regular degree d. The productof the generators, raised to the power d (which is the order of the image of this product in thereflection group), is always central.For an irreducible Coxeter group, 2 and h are the respectively smallest and largest degrees;

they are always regular; it is possible to choose intermediate regular degrees but they do not seemto yield Garside monoids.

Table 1

Classical monoid Dual monoid

Set of atoms S T

Number of atoms n N

! w0 c

Length of! N n

Order of p(!) 2 h

Product of the atoms c w0

Regular degree h 2

4e SÉRIE – TOME 36 – 2003 – N! 5

-

80

One gets a very powerful algebraic analogue of Coxeter theory.




Question. Do all periodic elements in B(W ) correspond toGarside structures?

That would be very natural and very beautiful.

That would make computing centralizers a trivial task.

Bad news. It doesn’t seem to work. In Birman-Ko-Lee’s setup,the element σ1δ is periodic:

(σ1δ)n−1 = δn = ∆2,

yet no-one could find a Garside monoid structure with symmetry oforder n − 1 on the braid group with n strings. (See Ko-Han forexplicit obstructions).

Between 2001 and 2005, I became convinced that this approachwould never work. That was very frustrating, very discouraging.




Idea 5 (Bestvina)

Let A be a spherical type Artin group, with Garside element ∆.

The group A/∆2 exhibits non-positive curvature features.

In particular, torsion elements in A/∆2 (and, correspondingly,periodic elements in A) can be classified thanks to a Cartan fixedpoint theorem.




Geometry & Topology GGGGGGGG G GGGG

GG

T TTTTTT

TTTTTTTT

Volume 3 (1999) 269–302

Published: 11 September 1999

Non-positively curved aspects of Artin groupsof finite type

Mladen Bestvina

Department of Mathematics, University of UtahSalt Lake City, UT 84112, USA

Email: [email protected]

Abstract

Artin groups of finite type are not as well understood as braid groups. This isdue to the additional geometric properties of braid groups coming from theirclose connection to mapping class groups. For each Artin group of finite type,we construct a space (simplicial complex) analogous to Teichmuller space thatsatisfies a weak nonpositive curvature condition and also a space “at infinity”analogous to the space of projective measured laminations. Using these con-structs, we deduce several group-theoretic properties of Artin groups of finitetype that are well-known in the case of braid groups.

AMS Classification numbers Primary: 20F32, 20F36

Secondary: 55P20

Keywords: Artin groups, nonpositive curvature

Proposed: Walter Neumann Received: 27 November 1998

Seconded: Wolfgang Metzler, Joan Birman Revised: 5 August 1999

ISSN 1364-0380

Copyright Geometry and Topology

269




a

a.a

a.a.aa.a.ab

a.ab a.ab.b

a.ab.ba

a.ab.ba.ab

a.ab.b.b

a.ab.b.ba

ab

ab.bab.ba

!

b

ba

b.b

ba.ab

b.ba

Figure 1: X(G) for G = A/!2 , A = !a, b | aba = bab"

a single !, and in the latter case we agree to push this ! to the last slot. Thereis an oriented edge from a vertex v to a vertex w if the special representativeof w is obtained from the special representative of v by rightmultiplying byan atom. The simplicial complex X !(G) is defined to be the flag complexdetermined by the resulting graph, ie, a collection of vertices spans a simplexprovided all pairs span an edge. The natural quotient map G # G/!!" extendsto a simplicial map p : X !(G) # X(G). The preimage of the basepoint $ %X(G) is the edge [$,!] & X !(G). More generally, the preimage of a simplex! is the (triangulated) prism ! ' [0, 1]. There is a global homeomorphismX !(G) (= X(G) ' [0, 1]. In the first coordinate this homeomorphism is given byp and in the second it is the simplicial map that sends the vertices whose specialrepresentative has no !’s to 0 and the vertices whose special representative hasone ! to 1.

Analogously, there is a complex X(A) with vertex set A and similarly definedsimplicial structure: edges are drawn from $ to the atoms and extended equiv-ariantly, and then the higher-dimensional simplices are filled in. The quotientmap extends to the natural simplicial map q : X(A) # X(G) and there is aglobal homeomorphism X(A) (= X(G) ' R in the first coordinate given by qand in the second it is the map that is linear on each simplex and sends a vertex

Mladen Bestvina

Geometry and Topology, Volume 3 (1999)

280




4.2 Finite subgroups of G

It is well-known to the experts that all finite subgroups of G are cyclic, and infact the kernel of the homomorphism G ! Z/2! is torsion-free, where ! = ||!||and the homomorphism is the length modulo 2! . (Note that the argument ofTheorem 4.1 gives another proof of this fact.)

In this section we will use the geometric structure of X(G) to give a classificationof finite subgroups of G up to conjugacy.

Theorem 4.5 Every finite subgroup H < G is cyclic. Moreover, after con-jugation, H transitively permutes the vertices of a simplex " " X(G) thatcontains # and H has one of the following two forms:

Type 1 The order of H is even, say 2m. It is generated by an atom B . Thevertices of " are #, B,B2, · · · , Bm!1 (all atoms) and Bm = !. Necessarily,B = B (since ! fixes the whole simplex).

Type 2 The order m of H is odd, the group is generated by B! for anatom B , and the vertices #, B,BB,BBB, · · · , (BB)(m!1)/2B (all atoms) arepermuted cyclically and faithfully by the group (so the dimension of " is m$1).Since m is odd, the square BB of the generator also generates H .

An example of a type 1 group is %B& for B = "1"3"2 in the braid group B4/%!2&(of order 4). An example of a type 2 group is %"1!& in B3/%!2& (of order 3).

The key to this is:

Lemma 4.6 The set of vertices of any simplex " in X admits a cyclic orderthat is preserved by the stabilizer Stab(") < G .

Proof We can translate " so that # is one of its vertices. Let the cyclicorder be induced from the linear order # < B1 < B2 < · · · < Bk given bythe orientations of the edges of " (equivalently, by the lengths of the atomsBi ). We need to argue that the left translation by B!1

i is going to pro-duce the same cyclic order. We can write Bi+1 = BiCi for atoms Ci , sothat the vertices of " are #, C1, C1C2, · · · , C1 · · · Ck . After translation by B!1

1

the vertices are B!11 , #, C2, C2C3, · · · , C2 · · · Ck , where B!1

1 should be replacedby the special representative of the coset B!1

1 %!&. Let Y be the atom withC1C2 · · · CkY = !. Then B!1

1 ! = C!11 ! = C2C3 · · · CkY and C2C3 · · · CkY

is this canonical representative (it is a subword of !, so it is an atom). SinceC2 · · · Ck < C2 · · · CkY , it follows that the induced ordering on the vertices is acyclic permutation of the old one. Repeating this i times gives the claim.

Non-positively curved aspects of Artin groups of finite type


293




Difficulty. The fixed point may not be a vertex (conjugacy isn’talways a diagram automorphism.)




Idea 6 (Krammer – see also Digne-Michel)

Garside theory works equally well in a categorical setup:

Replace “monoid” by “category” and “group” by “groupoid.”

Lattice property: existence of limits and colimits.

Category automorphism φ (= diagram automorphism.)

Categorify the conjugacy relation

φ(g) = ∆−1g∆

by taking ∆ to be a natural transformation from the identityfunctor to φ.

This is how Garside theory was meant to be! (See the diagramchasing in Dehornoy’s papers.)




Idea 7: stop kidding yourself.

Be serious about the categorical viewpoint.

Remember all this is homotopy theory.

Learn from your teachers.

Think “up to equivalence of categories.”




Theorem

Let C be a Garside category with groupoid of fractions G . Let dbe a positive integer. There exists a Garside category Cd withgroupoid of fractions Gd , together with a functor

θd : C → Cd ,

inducing an equivalence of categories

G → Gd ,

and such that, for any d-periodic loop γ ∈ C , the image θd(γ) isconjugate to a Garside element (= a morphism in the naturalfamily ∆d).

The categories C and Cd are not equivalent (the whole point is toconstruct Garside structures with new symmetries.)




The proof is constructive. The divided category Cd is designed tocontain a formal root d

√φ of the Garside automorphism. Its objects

are factorisations in d terms of elements in the natural family ∆.

We get a general argument explaining the existence of exoticGarside structures such as the Birman-Ko-Lee monoid (the onlypoint we miss is whether these structures can be constructed witha single object, but should this really be an issue?)




The Rabbit conjecture (sans monodromy)

Applying the theorem to the dual braid monoid of type E8 andd = 4, we get a fixed subcategory R31 with 88 objects.

If we believe Broue-Michel’s approach, the groupoid of fractions ofR31 should be equivalent (as a category) to the braid group of G31.(This is actually shown in Sections 11-12 of my K (π, 1) paper.)

Exercise.

Implement the construction.

Write down (in RAM, not on paper) a presentation for R31.

Using heuristics to simplify presentations, show that theautomorphism group of an object in the groupoid of fractionsis presented by the Rabbit diagram.

Disclaimer: I haven’t tried the exercise.Reclaimer: Jean just told me he worked out the same exercise, anddid obtain the Rabbit diagram!




Divided Garside categories: geometric viewpoint

Bestvina’s Cartan fixed point theorem: any periodic elementpreserves a simplex of the “almost non-positively curved”classifying complex for C/∆.

The category Cd is essentially a barycentric subdivision of C .

In other words, Cd/∆d is homotopy equivalent to C/∆. It onlyhas a bigger 0-skeleton.

The equivalence of categories is just a fancy way of saying that Gand Gd are fundamental groupoids of the “same” space, but withrespect to a different set of basepoints.




Bestvina’s complex, redux

With enough basepoints, any finite subgroup of C/∆ fixes a vertex:

a

a.a

a.a.aa.a.ab

a.ab a.ab.b

a.ab.ba

a.ab.ba.ab

a.ab.b.b

a.ab.b.ba

ab

ab.bab.ba

!

b

ba

b.b

ba.ab

b.ba

Figure 1: X(G) for G = A/!2 , A = !a, b | aba = bab"

a single !, and in the latter case we agree to push this ! to the last slot. Thereis an oriented edge from a vertex v to a vertex w if the special representativeof w is obtained from the special representative of v by rightmultiplying byan atom. The simplicial complex X !(G) is defined to be the flag complexdetermined by the resulting graph, ie, a collection of vertices spans a simplexprovided all pairs span an edge. The natural quotient map G # G/!!" extendsto a simplicial map p : X !(G) # X(G). The preimage of the basepoint $ %X(G) is the edge [$,!] & X !(G). More generally, the preimage of a simplex! is the (triangulated) prism ! ' [0, 1]. There is a global homeomorphismX !(G) (= X(G) ' [0, 1]. In the first coordinate this homeomorphism is given byp and in the second it is the simplicial map that sends the vertices whose specialrepresentative has no !’s to 0 and the vertices whose special representative hasone ! to 1.

Analogously, there is a complex X(A) with vertex set A and similarly definedsimplicial structure: edges are drawn from $ to the atoms and extended equiv-ariantly, and then the higher-dimensional simplices are filled in. The quotientmap extends to the natural simplicial map q : X(A) # X(G) and there is aglobal homeomorphism X(A) (= X(G) ' R in the first coordinate given by qand in the second it is the map that is linear on each simplex and sends a vertex

Mladen Bestvina


280




Divided Garside categories: cyclic structure 1

Bestvina/Charney-Meier-Whittlesey’s viewpoint: rather thanlooking at the “bar” resolution, the cohomology of a Garsidegroup(oid) G can be understood on a smaller, finite dimensionalresolution of G .

The “bar” resolution is the nerve (in the categorical sense) of theuniversal cover of G : its k-skeleton consists of sequences(g0, . . . , gk−1) of composable arrows. It has a simplicial structure.

The “gar” resolution is another way to construct a classifyingspace for G . One only considers sequences whose product is aprefix of ∆. Its k-skeleton consists of sequences (g0, . . . , gk) suchthat g0 . . . gk = ∆.




Divided Garside categories: cyclic structure 2

In addition to the simplicial structure, the operator

(g0, . . . , gk) 7→ (g1, . . . , gk , φ(g0))

turns the “gar” resolution into (a mild variant of) a cyclic set, inthe sense of Connes.

The “0 modulo d”-skeleton of “gar” (i.e. the collection of the(dk)k∈Z≥0

-skeletons) comes equipped with an action of the cyclicgroup of order d .

The divided groupoid Gd is designed such that its “gar” resolutionis the “0 modulo d”-skeleton of G ’s “gar” resolution.




Remark. This is the “gar” version of a key construction in:




Divided categories is the Garside version of Springertheory

The regular orbit space comes equipped with a natural S1-action(and an action of each finite µd ⊆ S1).

The topological realization of a cyclic set comes equipped with anatural S1-action. (Actually, there is more. Dwyer-Hopkins-Kan:the homotopy category of cyclic sets is equivalent to that ofS1-spaces).

Springer theory is about µd -action. Divided categories are alsoabout µd -action.

They are meant to get along.




Ideas 8, 9, ... : connect this to the geometry ofV /W

So far, we have defined a Rabbit category R31. How does itcompare with the actual topologically defined braid group B(G31)?Long story, involving:

Saito’s flat structure of V /W .

Lyashko-Looijenga morphisms.

analogs of chambers and galleries, and a cell-likedecomposition of V /W .

the observation that Springer theory is compatible with allthese structures.

Moral. The dual braid monoid, and its divided categories, can beexplicitly interpreted in terms on natural geometric constructions.It works. No joke. It might be as good as Coxeter theory.




On the center of B(G31)

Broue-Malle-Rouquier also conjectured that the centers of braidsgroup of irreducible complex reflection groups are cyclic. As of2006, the only case left was G31 (it isn’t addressed in my preprints).

Let24√

∆2 ∈ B(E8) be a periodic element associated with theregular degree 24. Its centralizer is a cyclic group (rank 1 braidgroup).Using Springer theory for braids, we get:

Z ' B(Z/24Z) = CB(E8)

(24√

∆2)⊆ CB(E8)

(4√

∆2)' B(G31)

As24√

∆2 ∈ CB(E8)

(4√

∆2)

, this seems to indicate that ZB(G31) is

cyclic.

Disclaimer. Last minute early morning slide, I haven’t checkedwhether this really works.




Homework: Lehrer-Springer theory in braid groups

Disclaimer. 1. I haven’t done it. 2. It is up for grabs. 3. It mightbe a good subject for a PhD student.

the numerology of divided categories and their fixedsubcategories is controlled by an instance of the cyclic sievingphenomenon (see my joint paper with Vic Reiner.)

in the dual braid monoid setup, Drew Armstrong studied acyclic structure closely related, yet not identical, to dividedcategories.

Hurwitz action explains the nuance: my φd corresponds to thebraid δ = σ1 . . . σd−1, while Armstrong’s comes from σ1δ.

Using Armstrong’s action and my geometric tools, getLehrer-Springer theory in braid groups. Hint: imitate Section 11

of my K (π, 1) paper and simply remove the constant ramification

stratum to get a Lehrer-Springer version of Lemma 11.4.


chasing the rabbit

Education

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