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# Chasing the Rabbit

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Slides from a (very specialized) math talk on finding good diagrams for braid groups of exceptional complex reflection groups.

Slides from a (very specialized) math talk on finding good diagrams for braid groups of exceptional complex reflection groups.

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## Chasing the RabbitPresentation Transcript

• the rabbit conjecture a brutal approacha less brutal approach Chasing the Rabbit David Bessis Les Houches, 28/01/2011 David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachThis 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approacha less brutal approach David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachBrou´-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 . David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachProblem. Find Coxeter-like diagrams for W , providing both aCoxeter-like presentation for W and an Artin-like presentation forB(W ). David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachAdditional 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.) David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachBrou´-Malle-Rouquier were able to solve this in all but six eexceptional cases: David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachVarious methods are used, applicable to particular families ofirreducible 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 – the6 missing cases as of 1998. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachThe rabbit conjecture (Brou´-Malle-Rouquier, 1998): e David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachDoesn’t look like a rabbit? David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approacha less brutal approach David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachA brutal approach to the rabbit conjecture D.B./J. Michel, 2001-2003 David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachThe discriminant of W is the image in V /W of the hyperplaneunion 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 explicitpolynomial. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachIn 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 computing the monodromy braids of the punctures in (V − H) ∩ L over the space of generic lines. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachDiﬃculties. 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? David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachVKCURVE (D.B., Jean Michel) is a software package thatimplements an eﬃcient exact version of Van Kampen’s method. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachExplicit Presentations for Exceptional Braid GroupsDavid Bessis and Jean MichelCONTENTS We give presentations for the braid groups associated with the1. 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 VKCURVE6. Explicit Matrices of Basic DerivationsAcknowledgmentsReferences 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 [Hubbardtain a apresentation of π1 (P − (P ∩ H)). less brutal approach can be made into a failsa trarily good approximationExample 5.1. For G31 , when computing the determinant Since we will reuse themof M31 and evaluating at z = y and t = 1 + x, we obtain alities about complex polythe 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 POn 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, thpart of our work. Bessis software accepts as input any David This Chasing the Rabbit pensive (in terms of comput
• the rabbit conjecture a brutal approach a less brutal approachThe presentation-shrinking heuristics implemented in VKCURVEallowed us to recover the Rabbit diagram for G31 . David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachRemarks 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachA less brutal approach to the rabbit conjecture D.B., 2006 (unpublished) David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachStrategy 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 David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachRemark. 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). David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachIdea 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 ). David Bessis Chasing the Rabbit
• Hr´ﬂexion H de W the arabbit conjecture. La proposition 3.2 r´sulte alors du e contenant H brutal approach efait que, par hypoth`se, V approach a less brutal (w) n’est contenu dans aucun hyperplan de er´ﬂexion de W . e Sur certains ´l´ments r´guliers ee e des groupes de Weyl et les vari´t´s e eRemarque. Comme not´ dans [DeLo] et [Le], la proposition pr´c´dente e e eet sa d´monstration s’´tendent au cas plus associ´es W est un groupe e de Deligne–Lusztig g´n´raleo` e e e uengendr´ par des pseudo-r´ﬂexions (et mˆme au cas o` w est un ´l´ment e e e u eer´gulier qui normalise W sans n´cessairement lui appartenir), grˆce au e e ath´or`me fondamental de Steinberg ([St1],Michel aﬃrme que le ﬁx- e e Michel Brou´ et Jean 1.5) qui eateur d’un sous-espace (“sous-groupe parabolique”) est aussi engendr´epar 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 eπ(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 eement les ´l´Op´ration de B+ sur XB(w) ainsi d´ﬁnis. On voit que, dans le A.ements de P(w) et π e e egroupe B(w), on a caract`res – Conjectures B. Valeurs de e 3. Bons ´l´ments r´guliers ee e ´e 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 associ´es e e e Comme nousralit´fait remarquer R. Rouquier, il est facile de v´riﬁer A. G´n´ l’a es e e eque B. Caract`res et degr´s fantˆmes e e o C. Groupes de tresse D. Alg`bres de Hecke e3.4. l’´l´E. Valeurs estcaract`res dans B(w). de π e ement w de central sur les racines e 5. Vari´t´s associ´es aux racines de π ee e3.5. Question. L’injectionBessis A. Quelques propri´t´s naturelle de Vthe Rabbit ee David Chasing (w) dans V d´ﬁnit-elle un e
• the rabbit conjecture a brutal approach a less brutal approachIdea 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. David Bessis Chasing the Rabbit
• 4 DAVID BESSIS the rabbit conjecture a brutal approachw0 on W is a diagram automorphism (i.e., it is induced a less brutal approach 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. 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 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 Garside 1element is a rotation δ of angle 2π n . In [5], we used this Garside structure to computethe centralisers of powers of δ, 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 aDavid Bessis group, andthe Rabbit to compute. Note that the Garside Chasing is easy
• the rabbit conjecture a brutal approach a less brutal approachIdea 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less 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. IntroductionThe positive braid monoid (on n  1 strings) is the monoid B that admitsthe presentation hx1 ; . . . ; xn jxi xj  xj xi if ji À jj > 2; xi xi  1 xi  xi  1 xi xi  1 if i  1; . . . ; n À 1i: ÃIt was considered by Garside in [18] and plays a prominent role in the theory ofbraid groups. In particular, several properties of the braid groups are derived fromextensive 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 thelength of any expression of a. For a, b in B , we say that a is a left divisor of bor, equivalently, that b is a right multiple of a if there exists c in B such that bis ac. The existence of the length function guarantees that left divisibility is apartial order on B . It was actually proved in [18] that any two elements ofB have a lowest common right multiple. Moreover, B has left and rightcancellation properties, namely, ab  ac implies b  c, and ba  ca impliesb  c. Ores criterion says: if a monoid M has left and right cancellationproperties, and if any two elements of M have a common right multiple, then Membeds 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 congruencerelation generated by the pairs xxÀ1 ; 1 ChasingÀ1 x; 1, with x in M. By the David Bessis and x the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachIdea 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. David Bessis Chasing the Rabbit
• N the rabbit conjecture N we have w0 = m=1 sm = a brutal approach. m=1 tN −m+1 2 a less brutal approach 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 productThe 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 allreal 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. 31In 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 2One SÉRIE – TOMEvery powerful algebraic analogue of Coxeter theory. 4e gets a 36 – 2003 – N 5 ◦ David Bessis 80 Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachQuestion. 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 σ1 δ is periodic: (σ1 δ)n−1 = δ n = ∆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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachIdea 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approachISSN 1364-0380 269 a less brutal approachGeometry & Topology G T G G TT T T T GVolume 3 (1999) 269–302 G G T G TPublished: 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.eduAbstractArtin 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 Teichm¨ ller space that usatisﬁes 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approach280 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 = baba single ∆, and in the latter case we agree to push this ∆ to the last slot. There David Bessis Chasing the Rabbit
• and the homomorphism is the length modulo 2δ . (Note that the argument of the rabbit conjectureTheorem 4.1 gives another proof of this fact.) a brutal approach a less brutal approachIn 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 σ ⊂ 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 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 anatom B , and the vertices ∗, B, BB, BBB, · · · , (BB)(m−1)/2 B (all atoms) arepermuted 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 orderthat is preserved by the stabilizer Stab(σ) < G .Proof We can translate σ 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 σ (equivalently, by the lengths of the atoms −1Bi ). We need to argue that the left translationRabbit Bi is going to pro- David Bessis Chasing the by
• the rabbit conjecture a brutal approach a less brutal approachDiﬃculty. The ﬁxed point may not be a vertex (conjugacy isn’talways a diagram automorphism.) David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachIdea 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.) David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachIdea 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.” David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachTheoremLet 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.) David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachThe proof is constructive. The divided category Cd is designed to √contain a formal root d φ of the Garside automorphism. Its objectsare 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?) David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachThe 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! David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachDivided 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachBestvina’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
• the rabbit conjecture a brutal approach a less brutal approachDivided 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 = ∆. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachDivided 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachRemark. This is the “gar” version of a key construction in: David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachDivided categories is the Garside version of Springertheory 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachIdeas 8, 9, ... : connect this to the geometry ofV /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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachOn 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. David Bessis Chasing the Rabbit
• the rabbit conjecture a brutal approach a less brutal approachHomework: 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. David Bessis Chasing the Rabbit