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Garside categories, periodic loops and cyclic sets

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Garside groupoids, as recently introduced by Krammer, generalise Garside groups. A weak Garside group is a group that is equivalent as a category to a Garside groupoid. We show that any periodic loop in a Garside groupoid \CG may be viewed as a Garside element for a certain Garside structure on another Garside groupoid \CG_m, which is equivalent as a category to \CG. As a consequence, the centraliser of a periodic element in a weak Garside group is a weak Garside group. Our main tool is the notion of divided Garside categories, an analog for Garside categories of B\"okstedt-Hsiang-Madsen's subdivisions of Connes' cyclic category. This tool is used in our separate proof of the K(π,1)K(\pi,1) property for complex reflection arrangements

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... Afterwards, near the end of the 2000s, Garside theory began to be further developed into the theory of Garside categories [6,30]. This culminated in the publication in 2015 of the reference book [20], which summarizes the state of the art and the adaptations of Garside theory to a categorical context. ...
... We then construct a categorical analogue of the order complex of Dehornoy and Lafont. To do this we introduce a notion of (locally) Gaussian category, which generalizes the notion of Garside category, as considered in [6]. The definitions are direct adaptations of [21,Section 4] to the categorical context. ...
... This category admits 88 objects, 660 atoms, and a total of 2603 simple morphisms (excluding the identities). The first possible approach used to study the homology of this category (see [16,Section 5.3]) was to construct the Charney-Meier-Whittlesey complex for this cat- egory (as defined in [6,Section 7]). Sadly this complex is too large to be dealt with without a strong computational power. ...
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The homology of a Garside monoid, thus of a Garside group, can be computed efficiently through the use of the order complex defined by Dehornoy and Lafont. We construct a categorical generalization of this complex and we give some computational techniques which are useful for reducing computing time. We then use this construction to complete results of Salvetti, Callegaro and Marin regarding the homology of exceptional complex braid groups. We most notably study the case of the Borchardt braid group B(G31)B(G31)B(G_{31}) through its associated Garside category.
... We follow the treatment of Garside categories in [13,68]. ...
... The simple connectedness of F(Y ) above be can alternatively deduced from [13,Corollary 7.6], where Bessis considers a graph on G x→ such that two vertices are adjacent if they differ by a simple morphism in C. It is proved that the flag complex F( ) is contractible. As is a proper subgraph of Y , it is not hard to deduce the simply-connectedness of F(Y ). ...
... Note that C 12 is a cell of type 1 ∩ 2 (Lemma 2.13), andC 1 ∩C 2 is a cell of type 1 ∩ 2 (Theorem 2.2), hence π(C 12 ) =C 1 ∩C 2 . Similarly, π(C 13 ...
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A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.
... This approach was pursued [1,19,25,20,8] and extended in several steps, first to Artin-Tits groups of spherical type [7,15], then to a larger family of groups now known as Garside groups [12,9,10]. More recently, it was realized that going to a categorical context allows for capturing further examples [18,3,11,23], and a coherent theory now emerges around a central unifying notion called a Garside family. The aim of this paper is to present the main basic results of this approach. ...
... Definition 4.20. If S is a germ, a map F from S [2] to S is said to obey the I-law if, for every g 1 |g 2 |g 3 in S [3] such that g 1 • g 2 is defined, we have ...
... (ii) Assume that I is an I-function obeying the I-law. Assume that g 1 |g 2 |g 3 lies in S [3] and g 1 • g 2 is defined. By assumption, we have I(g 1 , I(g 2 , g 3 ))= × I(g 1 • g 2 , g 3 ), which translates into ...
Article
Garside families have recently emerged as a relevant context for extending results involving Garside monoids and groups, which themselves extend the classical theory of (generalized) braid groups. Here we establish various characterizations of Garside families, that is, equivalently, various criteria for establishing the existence of normal decompositions of a certain type.
... Garside groups are equipped with a special element ∆, called the Garside element. An element g of a Garside group is said to be periodic if g k = ∆ for some integers k = 0 and [Bes06b,BGG08]. ...
... Roots of periodic elements. We begin with a definition of Bessis in [Bes06b]: for a Garside group G with Garside element ∆, an element g ∈ G is p/q-periodic if g q = ∆ p for p ∈ Z and q ∈ Z ≥1 . ...
... The following is a question of Bessis [Bes06b,Question 4]. ...
Article
Let G be a Garside group with Garside element Δ, and let Δm be the minimal positive central power of Δ. An element g∈G is said to be periodic if some power of it is a power of Δ. In this paper, we study periodic elements in Garside groups and their conjugacy classes.We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of G is cyclic; if gk=Δka for some nonzero integer k, then g is conjugate to Δa; every finite subgroup of the quotient group G/〈Δm〉 is cyclic.By a classical theorem of Brouwer, Kerékjártó and Eilenberg, an n-braid is periodic if and only if it is conjugate to a power of one of two specific roots of Δ2. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of Δm.We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type , , , and the braid group of the complex reflection group of type (e,e,n), endowed with the dual Garside structure, we may further assume the precentrality.
... Actually various groups can be equipped with a Garside group structure. Recently, Krammer [20], Bessis [3] and Digne-Michel [11] have extended the notion of a Garside group into the notion of a Garside groupoid, which turned out to be crucial in the proof of the long-standing question of the K(π, 1) property for complex reflection arrangements [3]. ...
... Actually various groups can be equipped with a Garside group structure. Recently, Krammer [20], Bessis [3] and Digne-Michel [11] have extended the notion of a Garside group into the notion of a Garside groupoid, which turned out to be crucial in the proof of the long-standing question of the K(π, 1) property for complex reflection arrangements [3]. ...
... For x and y in V (C), we denote by C x→y the set of morphisms from x to y. In order to be consistent with [8,3] and with the above naive definition of a small category, if v lies in C x→y and w lies in C y→z , then we denote by vw the morphism of C x→z obtained by composition. We denote by C x→· and by C ·→x the set of morphisms of the category C whose source and target, respectively, are x. ...
Article
We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to establish the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin-Tits groups of spherical type. For part I see J. Algebra 317, No. 1, 1-16 (2007; Zbl 1173.20027).
... The notion of a Garside monoid emerged at the end of the 1990's [24,19] as a development of Garside's theory of braids [32], and it led to many developments [2, 3, 5, 6, 7, 8, 13, 14, 15, 31, 33, 34, 41, 42, 45, 46, 47, ...]. More recently, Bessis [4], Digne-Michel [27], and Krammer [38] introduced the notion of a Garside category as a further extension, and they used it to capture new, nontrivial examples and improve our understanding of their algebraic structure. The concept of a Garside category is also implicit in [25] and [35], and maybe in the many diagrams of [18]. ...
... The seminal example of a Garside monoid is the braid monoid B + n equipped with Garside's fundamental braid ∆ n , see for instance [32,29]. Other classical examples are free abelian monoids and, more generally, all spherical Artin-Tits monoids [10], as well as the so-called dual Artin-Tits monoids [9,4]. Every Garside monoid embeds in a group of fractions, which is then called a Garside group. ...
... Recently, it appeared that a number of results involving Garside monoids still make sense in a wider context where categories replace monoids [4,27,38]. A category is similar to a monoid, but the product of two elements is defined only when the target of the first is the source of the second. ...
Article
In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This approach leads to a realistic program for establishing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhauser (2000), Chap. IX].
... In order to obtain shorter notation, we shall in the sequel use a, b, c... for σ 1 , σ 2 , σ 3 ..., and, symmetrically, A, B... for σ −1 1 , σ −1 2 ... (as in the caption of Figure 1). Then, a typical greedy normal form for a 4-braid is the sequence (−2; ac, abcb, bcba, a), i.e., equivalently, using (f (1), ..., f (n)) to specify a permutation f of {1, ..., n}, (−2; (2, 1, 4, 3), (2, 4, 3, 1), (4, 1, 3, 2), (2, 1, 3, 4)), consisting of an integer and four simple 4-braids, or, equivalently, four permutations of {1, ..., 4}: for instance, (2,1,4,3) is the permutation associated with ac, i.e., with σ 1 σ 3 . To check that we have a greedy normal form, we observe for instance that the descents of (2, 1, 4, 3) are 1 and 3, while the recoils of (2, 4, 3, 1), i.e., the descents of (4, 1, 2, 3), are 1 and 3 as well, so the normality condition is satisfied between (2,1,4,3) and (2,4,3,1). ...
... Then, a typical greedy normal form for a 4-braid is the sequence (−2; ac, abcb, bcba, a), i.e., equivalently, using (f (1), ..., f (n)) to specify a permutation f of {1, ..., n}, (−2; (2, 1, 4, 3), (2, 4, 3, 1), (4, 1, 3, 2), (2, 1, 3, 4)), consisting of an integer and four simple 4-braids, or, equivalently, four permutations of {1, ..., 4}: for instance, (2,1,4,3) is the permutation associated with ac, i.e., with σ 1 σ 3 . To check that we have a greedy normal form, we observe for instance that the descents of (2, 1, 4, 3) are 1 and 3, while the recoils of (2, 4, 3, 1), i.e., the descents of (4, 1, 2, 3), are 1 and 3 as well, so the normality condition is satisfied between (2,1,4,3) and (2,4,3,1). The other verifications are similar. ...
... Then, a typical greedy normal form for a 4-braid is the sequence (−2; ac, abcb, bcba, a), i.e., equivalently, using (f (1), ..., f (n)) to specify a permutation f of {1, ..., n}, (−2; (2, 1, 4, 3), (2, 4, 3, 1), (4, 1, 3, 2), (2, 1, 3, 4)), consisting of an integer and four simple 4-braids, or, equivalently, four permutations of {1, ..., 4}: for instance, (2,1,4,3) is the permutation associated with ac, i.e., with σ 1 σ 3 . To check that we have a greedy normal form, we observe for instance that the descents of (2, 1, 4, 3) are 1 and 3, while the recoils of (2, 4, 3, 1), i.e., the descents of (4, 1, 2, 3), are 1 and 3 as well, so the normality condition is satisfied between (2,1,4,3) and (2,4,3,1). The other verifications are similar. ...
Article
We describe the most efficient solutions to the word problem of Artin's braid group known so far, i.e., in other words, the most efficient solutions to the braid isotopy problem, including the Dynnikov method, which could be especially suitable for cryptographical applications. Most results appear in literature; however, some results about the greedy normal form and the symmetric normal form and their connection with grid diagrams may have never been stated explicitly.
... The things we added since 2004 are that we noticed that it makes sense to consider categories which are only left or right locally Garside, and that a sufficient condition to make things work is a Noetherianness property (before that, we imposed the homogeneity which comes from an additive length). We also added a discussion of the relation between our definitions and the notion of Garside categories, for which we use the definition introduced by Bessis [Be1]. We define what we call left Garside categories in this context; part of this reflects inspiring discussions we had with Bessis and with Krammer in april 2006. ...
... The three following definitions are adaptations to one-sided Garside categories of the definitions of [Be1]. Definition 2.12 is almost equivalent to [Be1,2.5]. ...
... The three following definitions are adaptations to one-sided Garside categories of the definitions of [Be1]. Definition 2.12 is almost equivalent to [Be1,2.5]. ...
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We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.
... However, it is possible to replace B(W ) by a sort of categorical barycentric subdivision, its d-divided Garside category M d , on which µ d acts diagram automorphisms. This construction is explained in my separate article [5]. The fixed subcategory M µ d d is again a Garside category. ...
... This article focuses the geometric aspects of the proof of Theorem 0.3 – it is probably fair to say that the true explanation lies in the properties of M d and in the general theorems about periodic elements in Garside groupoids that are explained in [5]. By-products. ...
... We are able to solve these problems when W is well-generated: Theorem 12.5 contains a complete description of the roots of the generator of the center of the pure braid group P (W ) and of their centralisers. As for Theorem 0.3, the main conceptual ingredient towards the proof of Theorem 12.5 is a general property of Garside categories, explained in our separate paper [5]. What is done here is the minor step consisting of re-interpreting the general Kerékjártó theorem for Garside categories from [5] in terms of the S 1 -structure on the regular orbit space W \V reg . ...
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Let V be a finite dimensional complex vector space and W\subseteq \GL(V) be a finite complex reflection group. Let V^{\reg} be the complement in V of the reflecting hyperplanes. We prove that V^{\reg} is a K(π,1)K(\pi,1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and Orlik-Solomon, only six exceptional cases remained open. In addition to solving this six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about \pi_1(W\cq V^{\reg}), the braid group of W. This includes a description of periodic elements in terms of a braid analog of Springer's theory of regular elements.
... Garside categories were originally introduced near the end of the 2000s as a natural generalization of Garside monoids (see for instance [Kra08] or [Bes07]). A comprehensive survey of the general theory of Garside categories is made in [DDGKM]. ...
... This proposition allows us to restrict our attention to (p, q)-periodic elements with p and q coprime. The study of periodic elements in a general Garside context was introduced in [Bes07], in which the following notion of divided category is introduced. We follow the exposition given in [DDGKM, Section XIV.1.1]. ...
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In his proof of the K(pi,1) conjecture for complex reflection arrangements, Bessis defined Garside categories suitable for studying braid groups of centralizers of Springer regular elements in well-generated complex reflection groups. We provide a detailed study of these categories, which we call Springer categories. We describe in particular the conjugacy of braided reflections of regular centralizer in terms of the Garside structure of the associated Springer category. In so doing we obtain a pure Garside theoretic proof of a theorem of Digne, Marin and Michel on the center of finite index subgroups in complex braid groups in the case of a regular centralizer in a well-generated group. We also provide a "Hurwitz-like" presentation of Springer categories. To this aim we provide additional insights on noncrossing partitions in the infinite series. Lastly, we use this "Hurwitz-like" presentation, along with a generalized Reidemeister-Schreier method we introduce for groupoids, to deduce nice presentations of the complex braid group B(G31).
... Moreover, as with Springer-regular elements, Bessis has shown [ibid] that all d th roots of π m are conjugate. Such results essentially "lift" Springer theory to braid groups; they rely on Garside-like structures in [4]. However, we should warn the reader that this does not imply the existence of nice sections from W to B(W ). ...
... In fact Opdam's elements g C,0 of this Galois group correspond precisely to our Σ C of Defn. 4 As far as the BMR-freeness theorem goes, and again because we are really interested in the "geometric" Galois group Gal C(v)/C(u) , it is possible that we could replace it by Losev's weaker but uniform theorem [37]. We hope to be able to clarify this in the future. ...
... In [4, Theorem 2.24], it is shown that the center of the braid group B of an irreducible reflection group W is an infinite cyclic group generated by β : t → exp(2iπt/|Z(W )|)x 0 (where x 0 ∈ V reg is a base point) for all but six exceptional reflection groups. In his articles [1][2], Bessis proves that the result holds for all reflection groups but the exceptional one G 31 . This remark is a first step toward the case of G 31 : we show that if ZB is an infinite cyclic group, it is generated by β. ...
... In each orbit C ∈ H /W , we choose a hyperplane H C ∈ C and write ZC = Ind W NC (Z) where N C ⊂ W is the stabilizer of H C . The Shapiro's isomorphism lemma [5, Proposition III.6.2] shows ...
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The final result of this article gives the order of the extension 1P/[P,P]jB/[P,P]pW1 1 \longrightarrow P/[P,P]{\mathop{\longrightarrow} \limits^j} B/[P,P] {\mathop{\longrightarrow} \limits^p} W \longrightarrow 1as an element of the cohomology group H 2(W, P/[P, P]) (where B and P stands for the braid group and the pure braid group associated to the complex reflection group W). To obtain this result, we first refine Stanley-Springer’s theorem on the abelianization of a reflection group to describe the abelianization of the stabilizer N H of a hyperplane H. The second step is to describe the abelianization of big subgroups of the braid group B of W. More precisely, we just need a group homomorphism from the inverse image of N H by p (where p: B → W is the canonical morphism) but a slight enhancement gives a complete description of the abelianization of p −1(W′) where W′ is a reflection subgroup of W or the stabilizer of a hyperplane. We also suggest a lifting construction for every element of the centralizer of a reflection in W.
... The weak order of Coxeter groups is an example of a combinatorial Garside structure. The construction of the complex associated in [1,7] to any Garside structure can be generalized to complexified hyperplane arrangements and leads to the construction of 'Garside-type' combinatorial models for the covers of complexified arrangements (see [11,Chapter 6]). These models are tiled by copies of the order complexes of the posets of regions. ...
... Indeed, let γ represent a vertex of U + m \ U + m−1 : it is a positive path of length m that ends, say, in the chamber C. Its link in U + m is spanned by all vertices indexed by positive paths γ ′ such that (1) γ ∼ γ ′ (C ′ → C) and C ′ = C, ...
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We examine Deligne's classical proof of the asphericity of simplicial arrangements from the viewpoint of the combinatorics of the poset of regions of the arrangement. This turns out to be very natural. In particular, we show that an arrangement is simplicial only if it satisfies Deligne's property on positive paths, thus answering a question posed by Luis Paris.
... p/q-periodicity, uniqueness of roots and a question of Bessis. First, we introduce a definition of Bessis in [Bes06b]: an element g ∈ G is p/q-periodic if g q = ∆ p for p ∈ Z and q ∈ Z ≥1 . ...
... The above question is answered almost positively in the case of the braid group B n : each periodic element in B n is conjugate to a power of one of the particular braids δ and ε which are the Garside elements in the dual Garside structures of B n and A(B n−1 ), respectively, where A(B n−1 ) denotes the Artin group of type B n−1 viewed as a subgroup of B n . In [Bes06b], Bessis showed that the above question is answered almost positively in Garside groupoid setting. ...
Article
Full-text available
Let G be a Garside group with Garside element Δ\Delta. An element g in G is said to be \emph{periodic} if some power of g lies in the cyclic group generated by Δ\Delta. This paper shows the following. (i) The periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of G is cyclic. (ii) If gk=Δkag^k=\Delta^{ka} for some nonzero integer k, then g is conjugate to Δa\Delta^a. (iii) Every finite subgroup of the quotient group G/<Δm>G/<\Delta^m> is cyclic, where Δm\Delta^m is the minimal positive central power of Δ\Delta.
... We refer the reader to [Deh15], [Bes03], [Bes06b] and [HH22] for references concerning Garside and weak Garside groups. ...
Preprint
We motivate the study of metric spaces with a unique convex geodesic bicombing, which we call CUB spaces. These encompass many classical notions of nonpositive curvature, such as CAT(0) spaces and Busemann-convex spaces. Groups having a geometric action on a CUB space enjoy numerous properties. We want to know when a simplicial complex, endowed with a natural polyhedral metric, is CUB. We establish a link condition, stating essentially that the complex is locally a lattice. This generalizes Gromov's link condition for cube complexes, for the \ell^\infty metric. The link condition applies to numerous examples, including Euclidean buildings, simplices of groups, Artin complexes of Euclidean Artin groups, (weak) Garside groups, some arcs and curve complexes, and minimal spanning surfaces of knots.
... Moreover, as with Springer-regular elements, Bessis has shown [ibid] that all d th roots of π m are conjugate. Such results essentially "lift" Springer theory to braid groups; they rely on garside-like structures in [Bes06]. However, we should warn the reader that this does not imply the existence of nice sections from W to B(W ). ...
Preprint
We describe an approach, via Malle's permutation Ψ\Psi on the set of irreducible characters Irr(W)\text{Irr}(W), that gives a uniform derivation of the Chapuy-Stump formula for the enumeration of reflection factorizations of the Coxeter element. It also recovers its weighted generalization by delMas, Reiner, and Hameister, and further produces structural results for factorization formulas of arbitrary regular elements.
... In fact B n admits two distinct Garside structures for all n ≥ 3, but we will not prove this here. See [4] for example. ...
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The notion of a Garside group was first introduced in a paper of Dehornoy and Paris (14). Over the past decade they have been used as a tool to better understand the structure of Artin's braid groups (2) and their generalizations. In general, one can use the Garside structure associated with a Garside group to solve the word and conjugacy problems, as well as create a finite dimensional Eilenberg-MacLane space for the group. In general it may take some eort construct a Garside structure for a given group and many groups do not have Garside structures. Recently McCammond (19) has approached this problem from a dierent angle by combinatorially creating groups with built-in Garside structures. This paper hopes to serve as an expose on recent developments in the theory of Garside structures, especially McCammond's combinatorial Garside struc- tures, filling in non-trivial details that are often omitted in other expositions.
... This implies that the properties of braid groups that one is able to show using the techniques introduced by Garside (and developed by several other authors), will hold in every Garside group. Some authors like David Bessis [6], François Digne and Jean Michel [27], Daan Krammer [53] and Patrick Dehornoy [22], are extending the scope by introducing Garside categories, for which Garside groups are a particular case. The main ideas in Garside's work are the following. ...
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These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin's presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. We also describe the centralizer of a given braid. Most proofs are classical ones, using modern terminology. I have chosen those which I find simpler or more beautiful. Comment: To appear in Annales Math\'ematiques Blaise Pascal. 45 pages, 11 figures
... the recent years, there has been an increasing interest in a particular class of algebraic structures generically called Garside structures, see for in- stance [6, 9, 24, 22, 27, 38, 40, 42]. Several versions exist, but, in this survey, we shall only mention Garside monoids and Garside groups. ...
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We summarize the main known results involving subword reversing, a method of semigroup theory for constructing van Kampen diagrams by referring to a preferred direction. In good cases, the method provides a powerful tool for investigating presented (semi)groups. In particular, it leads to cancellativity and embeddability criteria for monoids and to efficient solutions for the word problem of monoids and groups of fractions. The text includes some new results about mixed reversing (combination of left- and right-reversings) and about the combinatorial distance of braids.
... , w m ) then the cyclic action on N C m+1 (W ) defined by (6.4) restricts to the subset N C m (W ), and coincides with the cyclic action defined in (6.5). The latter cyclic action on N C m (W ) may also be interpreted in terms of the Garside automorphism of the (m + 1)-divided category associated with the dual braid monoid of W (see [10]). Even the special case where m = 1 in Conjecture 6.5 is interesting, and an open question, which strictly generalizes Theorem 1.1. ...
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We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for well-generated complex reflection groups.
... Quasi-Garside monoids have the same definition as the Garside monoids except they are not required to be finitely generated. Recently, this notion was extended to the notion of Garside categories [107], [108], [76], [14], which, in some sense, has to be considered as a geometric object more than as an algebraic one. Garside categories are a central concept in Bessis' solution to the K(π, 1) problem for complex reflection arrangements (see [15]). ...
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This article is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of the theory: the faithful linear representations, the cohomology, and the geometrical representations.
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We show that for a large class of Artin groups with Dynkin diagrams being a tree, the K(π,1)K(\pi ,1)-conjecture holds. We also establish the K(π,1)K(\pi ,1)-conjecture for another class of Artin groups whose Dynkin diagrams contain a cycle, which applies to some Artin groups whose Dynkin diagrams are of hyperbolic type. This is based on a new approach to the K(π,1)K(\pi ,1)-conjecture for Artin groups.
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We show that for a large class of Artin groups with Dynkin diagrams being a tree, the K(π,1)K(\pi,1)-conjecture holds. We also establish the K(π,1)K(\pi,1)-conjecture for another class of Artin groups whose Dynkin diagrams contain a cycle, which applies to some hyperbolic type Artin groups. This is based on a new approach to the K(π,1)K(\pi,1)-conjecture for Artin groups.
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The homology of a Garside monoid, thus of a Garside group, can be computed efficiently through the use of the order complex defined by Dehornoy and Lafont. We construct a categorical generalization of this complex and we give some computational techniques which are useful for reducing computing time. We then use this construction to complete results of Salvetti, Callegaro and Marin regarding the homology of exceptional complex braid groups. We most notably study the case of the Borchardt braid group B(G31)B(G_{31}) through its associated Garside category.
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The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced by the third author by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrist's theorem on the homotopy dimension of the configuration space of graphs. We also make sample calculations of the fundamental group of ordered and unordered configuration spaces of two points for small graphs.
Book
This volume comprises the Lecture Notes of the CIMPA/TUBITAK Summer School Arrangements, Local systems and Singularities held at Galatasaray University, Istanbul during June 2007. The volume is intended for a large audience in pure mathematics, including researchers and graduate students working in algebraic geometry, singularity theory, topology and related fields. The reader will find a variety of open problems involving arrangements, local systems and singularities proposed by the lecturers at the end of the school.
Article
Lorsque W est un groupe de réflexion complexe bien engendré, le treillis NCP_W des partitions non-croisées de type W est un objet combinatoire très riche, généralisant la notion de partitions non-croisées d'un n-gone, et intervenant dans divers contextes algébriques (monoïde de tresses dual, algèbres amassées...). De nombreuses propriétés combinatoires de NCP_W sont démontrées au cas par cas, à partir de la classification des groupes de réflexion. C'est le cas de la formule de Chapoton, qui exprime le nombre de chaînes de longueur donnée dans le treillis NCP_W en fonction des degrés invariants de W. Les travaux de cette thèse sont motivés par la recherche d'une explication géométrique de cette formule, qui permettrait une compréhension uniforme des liens entre la combinatoire de NCP_W et la théorie des invariants de W. Le point de départ est l'utilisation du revêtement de Lyashko-Looijenga (LL), défini à partir de la géométrie du discriminant de W. Dans le chapitre 1, on raffine des constructions topologiques de Bessis, permettant de relier les fibres de LL aux factorisations d'un élément de Coxeter. On établit ensuite une propriété de transitivité de l'action d'Hurwitz du groupe de tresses B_n sur certaines factorisations. Le chapitre 2 porte sur certaines extensions finies d'anneaux de polynômes, et sur des propriétés concernant leurs jacobiens et leurs discriminants. Dans le chapitre 3, on applique ces résultats au cas des extensions définies par un revêtement LL. On en déduit — sans utiliser la classification — des formules donnant le nombre de factorisations sous-maximales d'un élément de Coxeter de W en fonction des degrés homogènes des composantes irréductibles du discriminant et du jacobien de LL.
Article
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Man erh~ilt also die Coxeter-Gruppen in nattirlicher Weise als gewisse Restklassengruppen der Artin-Gruppen. Ftir den Fall der Z6pfegruppen ergeben sich so gerade die symmetrischen Gruppen, was nattirlich schon lange bekannt ist. Die Coxeter-Gruppen sind seit ihrer Einftihrung durch Coxeter im Jahre 1935 eingehend studiert worden, und eine sch6ne Darstellung der dabei erhaltenen Resultate findet man bei Bourbaki [1]. Ftir die Artin- Gruppen gab es, von den freien Gruppen einmal abgesehen, nur ftir die Z6pfegruppen eine Reihe yon Untersuchungen, yon denen die wichtigste in letzter Zeit die L~Ssung des Konjugationsproblems durch Garside war. Ftir die anderen Artin-Gruppen gibt es einige vereinzelte Resultate in [2, 3 und 5]. Diese beziehen sich ebenso wie der gr6gte Teil der vorliegen- den Arbeit auf den Fall, dab die zu der Artin-Gruppe G geh~Srige Coxeter- Gruppe G endlich ist. Die endlichen Coxeter-Gruppen sind schon yon Coxeter selbstWoestimmt worden: Es sind die endlichen Spiegelungs- gruppen, also - im irreduziblen Fall - die Gruppen vom Typ A,, B,,
Article
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We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conjugacy problem given by the authors in [9], are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups [2]. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.
Article
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We give an easily handled algorithm for the word problem in each of Artin's braid groups, Bn, based on Garside's methods, but framed more directly in terms of the set of positive braids in which each pair of strings crosses at most once. We develop a natural partial order on each braid group defined in terms of positive braids, and apply this to compare braids with different powers ¢r of the fundamental half-twist braid ¢. This leads to an improvement of Garside's conjugacy algorithm, using a much smaller finite subset of each conjugacy class, which we term the super summit set, to represent the class, in place of Garside's summit set.
Article
Full-text available
We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for Artin braid groups and some other cases. Our proof makes use of results by Birman-Ko-Lee. We give a new intrinsic account of these results. Notations If G is a group acting on a set X, we denote by X G the subset of X of elements fixed by all elements of G. If (X, x) is a pointed topological space, we denote by Ω(X, x) the corresponding loop space, by ∼ the homotopy relation on Ω(X, x) and by π1(X, x) the fundamental group. For all n ∈ N, we denote by µn the set of n-th roots of unity in C.
Article
Finite subgroups of GLn(Q) generated by reflections, known as Weyl groups, classify simple complex Lie Groups as well as simple algebraic groups. They are also building stones for many other significant mathematical objects like braid groups and Hecke algebras. Through recent work on representations of reductive groups over fi- nite fields based upon George Lusztig's fundamental work, and motivated by conjectures about modular representations of general finite groups, it has become clearer and clearer that finite subgroups of GLn(C) generated by pseudo-reflections ("complex reflection groups") behave very much like Weyl groups, and might even be as important. We present here a concatenation of some recent work (mainly by D. Bessis, G. Malle, J. Michel, R. Rouquier and the author) on complex reflection groups, their braid groups and Hecke algebras, emphasizing the general properties which generalize basic properties of Weyl groups. By many aspects, the family of finite groups G(q )o ver finite fields with q elements behave as if they were the specialisations at x = q of an object depending on an indeterminate x. Convincing indices tend to show that, although complex reflection groups which are not Weyl groups do not define finite groups over finite fields , they might be associated to similar mysterious objects. We present here some aspects of the ma- chinery allowing to emphasize this point of view. We use this machinery to state the general conjectures about representations of finite reductive groups over � -adic rings which, ten years ago, originated this work .
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Article
Artin-Tits groups of spherical type have two well-known Garside structures, coming respectively from the divisibility properties of the classical Artin monoid and of the dual monoid. For general Artin-Tits groups, the classical monoids have no such Garside property. In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type à n we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter element in the Artin-Tits group. More precisely, if W is a Coxeter group, one can consider the length l R on W with respect to the generating set R consisting of all reflections. Let c be a Coxeter element in W and let P c be the set of elements p ∈ W such that c can be written c = pp′ with l R (c) = I R (p) + l R (p′). We define the monoid M(P c) to be the monoid generated by a set P c in one-to-one correspondence, p → p, with P c with only relations pp′ = p·p′ whenever p, p′ and pp′ are in P c and l R (pp′) = l R(p) + l R(p′). We conjecture that the group of quotients of M (P c) is the Artin-Tits group associated to W and that it has a simple presentation (see 1.1 (ii)). These conjectures are known to be true for spherical type Artin-Tits groups. Here we prove them for Artin-Tits groups of type Ã. Moreover, we show that for exactly one choice of the Coxeter element (up to diagram automorphism) we obtain a (quasi-) Garside monoid. The proof makes use of non-crossing paths in an annulus which are the counterpart in this context of the non-crossing partitions used for type A.
Article
Small Gaussian groups are a natural generalization of spherical Artin groups in which the existence of least common multiples is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be of Coxeter type. We show here how to extend the Elrifai-Morton solution for the conjugacy problem in braid groups to every small Gaussian group.
Article
The purpose of this note is to give a self-contained (apart from simple facts about Coxeter groups) and we hope a bit shorter and more understandable account of some results of [C1,C2] on normal forms of braids which are themselves based on the papers [D1,T]. In particular a motivation was to give a proof of Proposition 5.1 that we use in [B-M]. Some proofs and results from Section 2 onwards seem to be new. I thank several people for improvements from earlier versions of the manuscript: M. Geck for pointing out some errors, F. Digne for pointing out that some results don't need the braid group to be of finite type, and J.-Y. Hée for suggesting (and providing) further improvements in that direction.
Article
It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lower common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right l.c.m.'s satisfy an additional symmetry condition. A Gaussian group and a Garside group are respectively the group of fractions of a Gaussian monoid and of a Garside monoid. Braid groups and, more generally, finite Coxeter type Artin groups are Garside groups. We determine algorith-mic criterions in terms of presentations for recognizing Gaussian and Garside monoids and groups, and exhibit infinite families of such groups. We describe simple algorithms that solve the word problem in a Gaussian group, show that theses algorithms have a quadratic complexity if the group is a Garside group, and prove that Garside groups have quadratic isoperimetric inequalities. We construct normal forms for Gaussian groups, and prove that, in the case of a Garside group, the language of normal forms is regular, symmetric, and geodesic, has the 5-fellow traveller property, and has the uniqueness property. This shows in particular that Garside groups are geodesically fully biautomatic. Finally, we consider an automorphism of a finite Coxeter type Artin group derived from an automorphism of its defining Coxeter graph, and prove that the subgroup of elements fixed by this automorphism is also a finite Coxeter type Artin group that can be explicitely determined.
Article
We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair (W,S) by (W,T), with T the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type A case, we recover the monoid constructed by Birman, Ko and Lee.RésuméNous étudions une nouvelle structure de monoı̈de pour les groupes d'Artin associés aux systèmes de Coxeter finis. Ce nouveau monoı̈de est, tout comme le classique monoı̈de des tresses positives, un monoı̈de de Garside. Nous en donnons différentes constructions : algébriquement, le nouveau monoı̈de apparaı̂t quand on étudie les systèmes de Coxeter avec un point de vue “dual”, qui consiste à remplacer la paire (W,S) par (W,T), où T est l'ensemble de toutes les réflexions ; géométriquement, il apparaı̂t quand on observe l'arrangement de réflexions depuis un point-base particulier. Pour les systèmes de type A, nous retrouvons le monoı̈de construit par Birman, Ko et Lee.
Article
Define a Garside monoid to be a cancellative monoid where right and left lcm's exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garside monoid. The family of Garside groups contains the braid groups, all spherical Artin-Tits groups, and various generalizations previously considered.2 Here we prove that Garside groups are biautomatic, and that being a Garside group is a recursively enumerable property, i.e., there exists an algorithm constructing the (infinite) list of all small Gaussian groups. The latter result relies on an effective, tractable method for recognizing those presentations that define a Garside monoid.
Article
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 Teichmueller space that satisfies a weak nonpositive curvature condition and also a space "at infinity" analogous to the space of projective measured laminations. Using these constructs, we deduce several group-theoretic properties of Artin groups of finite type that are well-known in the case of braid groups.
Article
This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset NC(k)(W)NC^{(k)}(W) for each finite Coxeter group W and for each positive integer k. When k=1, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When W is the symmetric group, we obtain the poset of classical k-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to related background material. Before defining our generalization NC(k)(W)NC^{(k)}(W), we develop from scratch the theory of algebraic noncrossing partitions NC(W). This involves studying a finite Coxeter group W with respect to its generating set T of {\em all} reflections, instead of the usual Coxeter generating set S. This is the first time that this material has appeared in one place. Finally, it turns out that our poset NC(k)(W)NC^{(k)}(W) shares many enumerative features in common with the ``generalized nonnesting partitions'' of Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In particular, there is a generalized ``Fuss-Catalan number'', with a nice closed formula in terms of the invariant degrees of W, that plays an important role in each case. We give a basic introduction to these topics, and we describe several conjectures relating these three families of ``Fuss-Catalan objects''.
Article
Let V be a finite dimensional complex vector space and W\subseteq \GL(V) be a finite complex reflection group. Let V^{\reg} be the complement in V of the reflecting hyperplanes. We prove that V^{\reg} is a K(π,1)K(\pi,1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and Orlik-Solomon, only six exceptional cases remained open. In addition to solving this six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about \pi_1(W\cq V^{\reg}), the braid group of W. This includes a description of periodic elements in terms of a braid analog of Springer's theory of regular elements.
Article
An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B_n, which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.
Article
We construct a class of Garside groupoid structures on the pure braid groups, one for each function (called labelling) from the punctures to the integers greater than 1. The object set of the groupoid is the set of ball decompositions of the punctured disk; the labels are the perimeters of the regions. Our construction generalises Garside's original Garside structure, but not the one by Birman-Ko-Lee. As a consequence, we generalise the Tamari lattice ordering on the set of vertices of the associahedron.
Article
We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman-Ko-Lee and by the author of new Garside monoids for Artin groups of spherical type. Conjecturally, an analog construction should be available for arbitrary Artin groups and for braid groups of well-generated complex reflection groups. This article continues the exploration of the theory of Artin groups and generalised braid groups from the new point of view introduced by Birman-Ko-Lee in [BKL] for the classical braid group on n strings. In [B1], we generalised their construction to Artin groups of spherical type. In the current article, we study the case of the free group, which is the Artin group associated with the universal Coxeter group. The formal analogs of the main statements in [B1] turn out to be elementary consequences of classical material (some of which was known to Hurwitz and Artin). In an attempt to interpolate some recent generalisations of the dual monoid construction (by Digne for the Artin group of type Ãn, [D]; by Corran and the author for the braid group of the complex reflection group G(e, e, n), [BC]), we propose two conjectures describing properties of a generalised
Article
A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(\pi,1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(\pi,1)s enjoy Bestvina's weak non-positive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually abelian.
  • P Dehornoy
  • Groupes De Garside
P. Dehornoy, Groupes de Garside, Ann. Sci. Ecole Norm. Sup. (4) 35 (2002) 267–306.
Sur certainsélémentscertainséléments réguliers des groupes de Weyl et les variétés de Deligne- Lusztig associées, Proceedings de la Semaine de Luminy " Représentations des groupes réductifs finis
  • M Broué
  • J Michel
M. Broué, J. Michel, Sur certainsélémentscertainséléments réguliers des groupes de Weyl et les variétés de Deligne- Lusztig associées, Proceedings de la Semaine de Luminy " Représentations des groupes réductifs finis ", 73-139, Birkhaüser, 1996. [14] L. E. J. Brouwer, ¨ Uber die periodischen Transformationen der Kugel, Math. Ann. 80 (1919), 39–41.
Catégories de Garside, personal communication
  • F Digne
  • J Michel
F. Digne, J. Michel, Catégories de Garside, personal communication, fall 2004.
Notes on the cyclic sieving phenomenon for non-crossing partitions
  • D Bessis
  • V Reiner
D. Bessis, V. Reiner, Notes on the cyclic sieving phenomenon for non-crossing partitions, manuscript in preparation.
  • J Birman
  • V Gebhardt
  • J González-Meneses
J. Birman, V. Gebhardt, J. González-Meneses, Conjugacy in Garside groups III: periodic braids, arXiv:math.GT/0609616.
  • E Brieskorn
  • K Saito
  • Artin-Gruppen Und Coxeter-Gruppen
E. Brieskorn, K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245-271.
Sur certainséléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées
  • M Broué
  • J Michel
M. Broué, J. Michel, Sur certainséléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Proceedings de la Semaine de Luminy "Représentations des groupes réductifs finis", 73-139, Birkhaüser, 1996.
  • N Franco
  • J González-Meneses
N. Franco, J. González-Meneses, Computation of centralizers in braid groups and Garside groups, Re. Mat. Iberoamericana 19 2003, 367-384.
  • D Krammer
D. Krammer, A class of Garside groupoid structures on the pure braid group, arXiv:math.GR/0509165.
45 rue d'Ulm, 75230 Paris cedex 05, France E-mail address: david dot bessis at ens dot fr
  • École Dma
  • Normale Supérieure
DMA,École normale supérieure, 45 rue d'Ulm, 75230 Paris cedex 05, France E-mail address: david dot bessis at ens dot fr