Chasing the Rabbit

the rabbit conjecture
a brutal approach
a less brutal approach

Chasing the Rabbit

David Bessis

Les Houches, 28/01/2011

David Bessis Chasing the Rabbit

a brutal approach

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


a brutal approach


a brutal approach

Brou´-Malle-Rouquier’s setup:
e
V is an n-dimensional complex vector space.
W ⊆ GL(V ) is a finite group generated by complex
reflections.
A is the set of all reflecting hyperplanes.
V reg := V − w ∈W −{1} ker(w − 1) = V − H∈A H is the
hyperplane complement.
B(W ) := π1 (V reg /W ) is the braid group of W .


a brutal approach

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


a brutal approach

Additional requirements for good diagrams:
the generators of W should be reﬂections,
the braid generators should be braid reﬂections (aka
meridiens, aka generators-of-the-monodromy),
the number of generators should be minimal (when W is
irreducible, 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.)


a brutal approach

Brou´-Malle-Rouquier were able to solve this in all but six
e
exceptional cases:


a brutal approach

Various methods are used, applicable to particular families of
irreducible cases:
The inﬁnite family G (de, e, n) is monomial. Brou´-Malle-
e
Rouquier use ﬁbration arguments (` la Fadell-Neuwirth).
a
Real groups: the Artin presentation was obtained by Brieskorn.
A few exceptional non-real cases have a regular orbit space
isomorphic 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 – the
6 missing cases as of 1998.


a brutal approach

The rabbit conjecture (Brou´-Malle-Rouquier, 1998):
e


a brutal approach

Doesn’t look like a rabbit?


a brutal approach

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


a brutal approach

The discriminant of W is the image in V /W of the hyperplane
union H∈A H ⊆ V .
By Chevalley-Shephard-Todd theorem, V /W is an aﬃne space.
The equation of H in V /W can be writtten as an explicit
polynomial.


a brutal approach

In the 1930s, Zariski and Van Kampen proposed a “method” for
computing fundamental groups of complements of algebraic
hypersurfaces.
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 computing
the monodromy braids of the punctures in (V − H) ∩ L over
the space of generic lines.


a brutal approach

Diﬃculties.
what does “generic” mean?
can monodromy braids be computed with an exact software
algorithm?
can the computation be eﬃcient enough to address non-trivial
cases?
is there a good heuristic to simplify the (highly redundant)
presentations obtained this way?


a brutal approach

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


a brutal approach

Explicit Presentations for Exceptional Braid Groups
David Bessis and Jean Michel

CONTENTS
We give presentations for the braid groups associated with the
1. Introduction complex reflection groups G24 and G27 . For the cases of G29 ,
2. The Presentations G31 , G33 , and G34 , we give (strongly supported) conjectures.
3. Definitions and Preliminary Work These presentations were obtained with VKCURVE, a GAP pack-
4. Choosing the 2-Plane age implementing Van Kampen’s method.
5. The Package VKCURVE
6. Explicit Matrices of Basic Derivations
Acknowledgments
References

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 [Brou´ et al. 98].
e
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,
David Bessis including Coxeter himself, have described “Coxeter-like”
Chasing the Rabbit

Once a 2-plane P has been chosen, it is enough to feed
the rabbit conjecture ing ∆ by the resultant of ∆
a with the equation of the curve P ∩ H to ob-
VKCURVEbrutal approach
beautiful article [Hubbard
tain a apresentation of π1 (P − (P ∩ H)).
less brutal approach
can be made into a failsa
trarily good approximation
Example 5.1. For G31 , when computing the determinant
Since we will reuse them
of M31 and evaluating at z = y and t = 1 + x, we obtain
alities about complex poly
the following equation for P ∩ H:
α1 , . . . , αn be the complex
∆31 = 746496 + 3732480x − 3111936xy 2 P (z) = 0, we set NP (z)
93281756 4 58341596 6 the ﬁrst order approximat
− xy + xy + 7464960x2
27 27 P (NP (z)) to be close to 0
17556484 2 4 of starting with z0 ∈ C (
− 384y 2 − 9334272x2 y 2 + x y
27 as in [Hubbard et al. 01])
43196 2 6 3 756138248 3 2 zm+1 := NP (zm ), hoping th
+ x y + 7464576x − x y
27 81 a root of P —which indeed
192792964 3 4 16 3 6
+ x y + x y + 3730944x4 of z0 . How may we decid
81 81
139967996 4 84021416 4 2 82088 4 4 enough” approximation?
− y − x y + x y
81 27 27
43192 5 2 1720 5 4 Lemma 5.2. Assume P ha
+ 744192x5 + x y − x y Let z ∈ C, with P (z) =
27 27
124412 6 95896 6 2 {α1 , . . . , αn } such that |z −
− x + 777600800y 6 + x y
81 81
8 6 4 10364 7 4 7 2 4 8 Proof: If P (z) = 0, the re
− x y − x − x y + x
81 27 27 27 n
have P (z) =
(z) 1
i=1 z−αi .
8 8 4 8 2 4 P
− y − x y + x9 . j, |z − αi | ≤ |z − αj |. By t
81 27 81
P (z) 1 P
On a 3 GHz Pentium IV, VKCURVE needs about one P (z) − j=i |z−αj | ≥ P(
hour to deal with this example. follows.

Writing VKCURVE was of course the most diﬃcult Although elementary, th
part of our work. Bessis software accepts as input any
David This Chasing the Rabbit pensive (in terms of comput

a brutal approach

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


a brutal approach

Remarks
Our “Explicit Presentations” paper suffered from our attempt
to use Hamm-Le transversality conditions (based on Whitney
stratifications.)
There is a much simpler criterion to check that a 2-plane
section induces a π1 -isomorphism (see Section 4 of my
“K (π, 1)” paper.)
Zariski-Van Kampen method is indeed a fully implementable
algorithm.
In particular, the task of finding good diagrams for all
complex reflection groups is now complete.


a brutal approach

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


a brutal approach

Strategy of proof
View B(G31 ) as the centralizer of a periodic element in the Artin
group of type E8 (braid version of Springer’s theory of regular
elements).
Ingredients:
Conjectures on periodic elements in braid groups
(Brou´-Michel)
e
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 structure
Lyashko-Looijenga morphisms


a brutal approach

Remark. The construction for G31 is just one particular case in a
general theory applicable to “almost all” complex reﬂection groups
and providing good diagrams, Garside structures, natural geometric
objects, 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).


a brutal approach

Idea 1 (Brou´-Michel)
e

Springer theory of regular elements in W should have an analog in
terms of periodic elements (roots of central elements) in B(W ).
As G31 is the centralizer of a 4-regular element in E8 , one could
expect B(G31 ) to be the centralizer of a 4-periodic element in
B(E8 ).


H
r´flexion H de W the arabbit conjecture. La proposition 3.2 r´sulte alors du
e contenant H
brutal approach e
fait que, par hypoth`se, V approach
a less brutal (w) n’est contenu dans aucun hyperplan de
e
r´flexion de W .
e
Sur certains ´l´ments r´guliers
ee e
des groupes de Weyl et les vari´t´s e e
Remarque. Comme not´ dans [DeLo] et [Le], la proposition prć´dente
e e e
et sa d´monstration s’´tendent au cas plus associés W est un groupe
e de Deligne–Lusztig gń´raleo`
e e e u
engendr´ par des pseudo-r´flexions (et mˆme au cas o` w est un ´l´ment
e e e u ee
r´gulier qui normalise W sans nćessairement lui appartenir), grˆce au
e e a
thór`me fondamental de Steinberg ([St1],Michel affirme que le fix-
e e Michel Brou´ et Jean 1.5) qui
e
ateur d’un sous-espace (“sous-groupe parabolique”) est aussi engendré
par des pseudo-r´flexions.
e [. . . ]
Sommaire
On 1.eVari´t´s de Deligne–Lusztig M(w) le lacet d’origine x0 d´fini par
d´signee par π : [0, 1] →
e e
π(θ) = e2πiθ x0 , et et notations par w : [0, 1] → M(w) le chemin de x0
A. Contexte on d´signe
e
eB. Les vari´t´s de Deligne–Lusztig
ee
` w.x0 d´fini par w(θ) = e2πiθ/d x0 . On note encore π et w respective-
a
2. La vari´t´ Xπ
ee
ment les ´lÓp´ration de B+ sur XB(w) ainsi d´finis. On voit que, dans le
A.ements de P(w) et π
e e e
groupe B(w), on a caract`res – Conjectures
B. Valeurs de e
3. Bons ´l´ments r´guliers
ee e
é wd = π
A. El´ments r´guliers et groupes des .tresses associ´s
e e
B. Racines de π et ´l´ments r´guliers
ee e
4. Groupes de r´flexions complexes et alg`bres de Hecke associés
e e e
Comme nousralit´fait remarquer R. Rouquier, il est facile de v´rifier
A. Gń´ l’a es
e e e
que B. Caract`res et degr´s fantˆmes
e e o
C. Groupes de tresse
D. Alg`bres de Hecke
e
3.4. l’´lÉ. Valeurs estcaract`res dans B(w). de π
e ement w de central sur les racines
e
5. Vari´t´s associés aux racines de π
ee e
3.5. Question. L’injectionBessis
A. Quelques propri´t´s naturelle de Vthe Rabbit
ee
David Chasing (w) dans V d´finit-elle un
e

a brutal approach

Idea 2 (Jean Michel)

Let φ be a diagram automorphism of a (ﬁnite) Coxeter group W .
The Deligne/Brieskorn-Saito normal form allows an easy
computation 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 ), where
W = CW (w0 ), as predicted by the braid version of Springer theory.


4 DAVID BESSIS
a brutal approach
w0 on W is a diagram automorphism (i.e., it is induced
a less brutal approach by a permutation of S) and the
centraliser W := CW (w0 ) is a Coxeter group with Coxeter generating set S indexed by
w0 -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 to
W V reg , as in Example 0.2).
Example 0.4. How to viewis a Artin group of type E6 , the ∆-conjugacy action is the
Example. When A(W, S) F4 in E6 :
non-trivial diagram automorphism and the centraliser is an Artin group of type F4 .
s3 s4
•
ppp ` •`
pp
, ppp
• • pNN
N
pN ∆ ∆ • • • •
s1 s2 NNNNN ~ ~ s1 s2 s3 s3 s4 s4
N
• •
s3 s4
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 as
in Example 0.4.
Birman-Ko-Lee showed that the classical braid group Bn admits, in addition to the
type An−1 Artin group structure, another Garside group structure where the Garside
1
element is a rotation δ of angle 2π n . In [5], we used this Garside structure to compute
the centralisers of powers of δ, which solves Question 3 for Bn and q|n. Thanks to some
rather 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 aDavid Bessis group, andthe Rabbit to compute. Note that the
Garside Chasing is easy

a brutal approach

Idea 3 (Dehornoy-Paris)

What really matters in Deligne/Brieskorn-Saito normal form can
be axiomatized.
A Garside monoid is a monoid with properties emulating those of
spherical type Artin monoids:
ﬁnite positive homogeneous presentation,
lattice for the divisibility order,
a Garside element ∆, common multiple to all generators, such
that conjugating by ∆ is “diagram automorphism” (a
permutation of the generators).
(Some of these properties can be relaxed.)
Some (not all) of Brou´-Malle-Rouquier diagrams provide
e
examples.


a brutal approach

GAUSSIAN GROUPS AND GARSIDE GROUPS,
TWO GENERALISATIONS OF ARTIN GROUPS

PATRICK DEHORNOY and LUIS PARIS

[Received 7 October 1997ÐRevised 18 September 1998]

1. Introduction
The positive braid monoid (on n ‡ 1 strings) is the monoid B‡ that admits
the presentation
hx1 ; . . . ; xn jxi xj ˆ xj xi if ji À jj > 2; xi xi ‡ 1 xi ˆ xi ‡ 1 xi xi ‡ 1 if i ˆ 1; . . . ; n À 1i:
Ã
It was considered by Garside in [18] and plays a prominent role in the theory of
braid groups. In particular, several properties of the braid groups are derived from
extensive investigations of the positive braid monoids (see, for example, [2, 16, 17]).
A ®rst observation is that the de®ning relations of B‡ are homogeneous. Thus,
one may deal with a length function n: B‡ 3 N which associates to a in B‡ the
length of any expression of a. For a, b in B‡ , we say that a is a left divisor of b
or, equivalently, that b is a right multiple of a if there exists c in B‡ such that b
is ac. The existence of the length function guarantees that left divisibility is a
partial order on B‡ . It was actually proved in [18] that any two elements of
B‡ have a lowest common right multiple. Moreover, B‡ has left and right
cancellation properties, namely, ab ˆ ac implies b ˆ c, and ba ˆ ca implies
b ˆ c. Ore's criterion says: if a monoid M has left and right cancellation
properties, and if any two elements of M have a common right multiple, then M
embeds in its group of (right) fractions (see [10, Theorem 1.23]). This group is
…M Ã M À1 †=, where M À1 is the dual monoid of M, and is the congruence
relation generated by the pairs …xxÀ1 ; 1† ChasingÀ1 x; 1†, with x in M. By the
David Bessis and …x the Rabbit

a brutal approach

Idea 4 (Birman-Ko-Lee)

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

δ n = ∆2 .

In a joint work with Fran¸ois Digne and Jean Michel, we used this
c
to compute centralizers of periodic elements in type A braid
groups.


N the rabbit conjecture
N
we have w0 = m=1 sm = a brutal approach.
m=1 tN −m+1 2
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 product
The dual braid raised to the power d (which isBirman-Ko-Lee’s construction
of the generators, monoid generalizes the order of the image of this product in the
reflection group), is always central.
to all well-generated complex reflection groupssmallest and largest degrees;
For an irreducible Coxeter group, 2 and h are the respectively (this includes all
real types, asregular; as possible to choose intermediate regular degrees but they do not seem
they are always well it is all high-dimensional exceptions except G .)
to yield Garside monoids. 31

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

One SÉRIE – TOMEvery powerful algebraic analogue of Coxeter theory.
4e gets a 36 – 2003 – N 5 ◦

David Bessis 80 Chasing the Rabbit

a brutal approach

Question. Do all periodic elements in B(W ) correspond to
Garside 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 σ1 δ is periodic:

(σ1 δ)n−1 = δ n = ∆2 ,

yet no-one could ﬁnd a Garside monoid structure with symmetry of
order n − 1 on the braid group with n strings. (See Ko-Han for
explicit obstructions).
Between 2001 and 2005, I became convinced that this approach
would never work. That was very frustrating, very discouraging.


a brutal approach

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 ﬁxed
point theorem.


a brutal approach
ISSN 1364-0380 269

Geometry Topology G T
G G TT T T T
G
Volume 3 (1999) 269–302 G
G T G T
Published: 11 September 1999 G T G G T
G T G T
GG G T TT

Non-positively curved aspects of Artin groups
of finite type
Mladen Bestvina

Department of Mathematics, University of Utah
Salt Lake City, UT 84112, USA
Email: bestvina@math.utah.edu

Abstract

Artin groups of finite type are not as well understood as braid groups. This is
due to the additional geometric properties of braid groups coming from their
close connection to mapping class groups. For each Artin group of finite type,
we construct a space (simplicial complex) analogous to Teichm¨ ller space that
u
satisfies 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 finite
type that are well-known in the case of braid groups.


a brutal approach

280 Mladen Bestvina

a.a.ab
a.a.a
a.ab.b.b

a.a a.ab.b.ba
ba.ab

a.ab a.ab.b
ba a

∗ a.ab.ba

ab
b
b.ba
a.ab.ba.ab
ab.b
b.b ab.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. There

and the homomorphism is the length modulo 2δ . (Note that the argument of
Theorem 4.1 gives another proof of this fact.)
a brutal approach
In this section we will use the geometric structure of X(G) to give a classification
of 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 σ ⊂ X(G) that
contains ∗ 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 . The
vertices of σ are ∗, B, B 2 , · · · , B m−1 (all atoms) and B m = ∆. Necessarily,
B = B (since ∆ fixes the whole simplex).
Type 2 The order m of H is odd, the group is generated by B∆ for an
atom B , and the vertices ∗, B, BB, BBB, · · · , (BB)(m−1)/2 B (all atoms) are
permuted cyclically and faithfully by the group (so the dimension of σ 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 = σ1 σ3 σ2 in the braid group B4 / ∆2
(of order 4). An example of a type 2 group is σ1 ∆ in B3 / ∆2 (of order 3).
The key to this is:

Lemma 4.6 The set of vertices of any simplex σ in X admits a cyclic order
that is preserved by the stabilizer Stab(σ) G .

Proof We can translate σ so that ∗ is one of its vertices. Let the cyclic
order be induced from the linear order ∗ B1 B2 · · · Bk given by
the orientations of the edges of σ (equivalently, by the lengths of the atoms
−1
Bi ). We need to argue that the left translationRabbit Bi is going to pro-
David Bessis Chasing the by

a brutal approach

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


a brutal approach

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 ) = ∆−1 g ∆

by taking ∆ to be a natural transformation from the identity
functor to φ.
This is how Garside theory was meant to be! (See the diagram
chasing in Dehornoy’s papers.)


a brutal approach

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.”


a brutal approach

Theorem
Let C be a Garside category with groupoid of fractions G . Let d
be a positive integer. There exists a Garside category Cd with
groupoid 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) is
conjugate to a Garside element (= a morphism in the natural
family ∆d ).
The categories C and Cd are not equivalent (the whole point is to
construct Garside structures with new symmetries.)

a brutal approach

The proof is constructive. The divided category Cd is designed to
√
contain 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 exotic
Garside structures such as the Birman-Ko-Lee monoid (the only
point we miss is whether these structures can be constructed with
a single object, but should this really be an issue?)


a brutal approach

The Rabbit conjecture (sans monodromy)
Applying the theorem to the dual braid monoid of type E8 and
d = 4, we get a ﬁxed subcategory R31 with 88 objects.
If we believe Brou´-Michel’s approach, the groupoid of fractions of
e
R31 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 the
automorphism group of an object in the groupoid of fractions
is presented by the Rabbit diagram.
Disclaimer: I haven’t tried the exercise.
Reclaimer: Jean just told me he worked out the same exercise, and
did obtain the Rabbit diagram!

a brutal approach

Divided Garside categories: geometric viewpoint

Bestvina’s Cartan ﬁxed point theorem: any periodic element
preserves 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 only
has a bigger 0-skeleton.
The equivalence of categories is just a fancy way of saying that G
and Gd are fundamental groupoids of the “same” space, but with
respect to a diﬀerent set of basepoints.


a brutal approach

Bestvina’s complex, redux
280 Mladen Bestvina
With enough basepoints, any ﬁnite subgroup of C /∆ ﬁxes a vertex:

a.a.ab
a.a.a
a.ab.b.b

a.a a.ab.b.ba
ba.ab

a.ab a.ab.b
ba a

∗ a.ab.ba

ab
b
b.ba
a.ab.ba.ab
ab.b
b.b ab.ba

2
Figure 1: X(G) for Bessis A/∆Chasing the Rabbit = bab
David G = , A = a, b | aba

a brutal approach

Divided Garside categories: cyclic structure 1

Bestvina/Charney-Meier-Whittlesey’s viewpoint: rather than
looking at the “bar” resolution, the cohomology of a Garside
group(oid) G can be understood on a smaller, ﬁnite dimensional
resolution of G .
The “bar” resolution is the nerve (in the categorical sense) of the
universal 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 classifying
space for G . One only considers sequences whose product is a
preﬁx of ∆. Its k-skeleton consists of sequences (g0 , . . . , gk ) such
that g0 . . . gk = ∆.


a brutal approach

Divided Garside categories: cyclic structure 2

In addition to the simplicial structure, the operator

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

turns the “gar” resolution into (a mild variant of) a cyclic set, in
the 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 cyclic
group of order d.
The divided groupoid Gd is designed such that its “gar” resolution
is the “0 modulo d”-skeleton of G ’s “gar” resolution.


a brutal approach

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


a brutal approach

Divided categories is the Garside version of Springer
theory

The regular orbit space comes equipped with a natural S 1 -action
(and an action of each ﬁnite µd ⊆ S 1 ).
The topological realization of a cyclic set comes equipped with a
natural S 1 -action. (Actually, there is more. Dwyer-Hopkins-Kan:
the homotopy category of cyclic sets is equivalent to that of
S 1 -spaces).
Springer theory is about µd -action. Divided categories are also
about µd -action.
They are meant to get along.


a brutal approach

Ideas 8, 9, ... : connect this to the geometry of
V /W
So far, we have defined a Rabbit category R31 . How does it
compare 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-like
decomposition of V /W .
the observation that Springer theory is compatible with all
these structures.
Moral. The dual braid monoid, and its divided categories, can be
explicitly interpreted in terms on natural geometric constructions.
It works. No joke. It might be as good as Coxeter theory.

a brutal approach

On the center of B(G31 )
Brou´-Malle-Rouquier also conjectured that the centers of braids
e
group of irreducible complex reﬂection groups are cyclic. As of
2006, the only case left was G31 (it isn’t addressed in my preprints).
√
24
Let ∆2 ∈ B(E8 ) be a periodic element associated with the
regular degree 24. Its centralizer is a cyclic group (rank 1 braid
group).
Using Springer theory for braids, we get:
√
24
√
4
Z B(Z/24Z) = CB(E8 ) ∆2 ⊆ CB(E8 ) ∆2 B(G31 )
√
24
√
4
As ∆2 ∈ CB(E8 ) ∆2 , this seems to indicate that ZB(G31 ) is
cyclic.
Disclaimer. Last minute early morning slide, I haven’t checked
whether this really works.

a brutal approach

Homework: Lehrer-Springer theory in braid groups
Disclaimer. 1. I haven’t done it. 2. It is up for grabs. 3. It might
be a good subject for a PhD student.
the numerology of divided categories and their ﬁxed
subcategories is controlled by an instance of the cyclic sieving
phenomenon (see my joint paper with Vic Reiner.)
in the dual braid monoid setup, Drew Armstrong studied a
cyclic structure closely related, yet not identical, to divided
categories.
Hurwitz action explains the nuance: my φd corresponds to the
braid δ = σ1 . . . σd−1 , while Armstrong’s comes from σ1 δ.
Using Armstrong’s action and my geometric tools, get
Lehrer-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.

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