Garside Structures Research Articles

A new Garside structure on torus knot groups and some complex braid groups

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For [Formula: see text] two coprime integers, we introduce a new Garside monoid [Formula: see text] having as Garside group the [Formula: see text]-torus knot group, thereby generalizing to all torus knot groups a construction that we previously gave for the [Formula: see text]-torus knot group. As a byproduct, we obtain new Garside structures for the braid groups of a few exceptional complex reflection groups of rank two. Analogous Garside structures are also constructed for a few additional braid groups of exceptional complex reflection groups of rank two which are not isomorphic to torus knot groups, namely, for [Formula: see text] and for dihedral Artin groups of even type.

Lattices, Garside structures and weakly modular graphs

In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural A˜n shaped simplices, including Euclidean buildings of type A˜n and Cayley graphs of Garside groups and their quotients by the Garside elements. All these examples fit into the more general setting of lattices with order-increasing Z-actions and the associated lattice quotients proposed in a previous work by the first named author. We show that both the lattice quotients and the lattices themselves give rise to weakly modular graphs, which is a form of combinatorial non-positive curvature. We also show that several other objects fit into this setting of lattices/lattice quotients, including Artin complexes of Artin-Tits groups of type A˜n, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Hence our result also implies to these objects and shows that they give weakly modular graphs. Along the way, we also clarify the relationship between categorical Garside structures, lattices with Z action and different classes of complexes studied this article. We use this point of view to describe the first examples of Garside groups with exotic properties, like non-linearity or rigidity results.

Interval Garside structures related to the affine Artin groups of type [formula omitted

Garside theory emerged from the study of Artin groups and their generalizations. Finite-type Artin groups admit two types of interval Garside structures corresponding to their standard and dual presentations. Concerning affine Artin groups, Digne established interval Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to interval Garside structures. In this paper, shifting attention from dual presentations to other nice presentations for the affine Artin group of type A˜ discovered by Shi and Corran–Lee–Lee, I will construct interval Garside structures related to this group. This construction is the first successful attempt to establish interval Garside structures not related to the dual presentations in the case of affine Artin groups.

Dual Garside structures and Coxeter sortable elements

In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an $S$-reduced expression of its image in the Coxeter group, in a way which involves Reading's $c$-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahori-Hecke algebras have positivity properties. In the classical types, this requires to give an explicit description of the inverse of Reading's bijection from $c$-sortable elements to noncrossing partitions of a Coxeter element $c$, which might be of independent interest. The bijections are described in terms of the noncrossing partition models in these types. While the proof of the formula is case-by-case, it is entirely combinatorial and we develop an approach which reduces a uniform proof to uniformly proving a lemma about inversion sets of $c$-sortable elements.

Interval Garside structures for the complex braid groups 𝐵(𝑒,𝑒,𝑛)

We define geodesic normal forms for the general series of complex reflection groups G ( e , e , n ) G(e,e,n) . This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G ( e , e , n ) G(e,e,n) over the generating set of the presentation of Corran–Picantin. Using these geodesic normal forms, we construct intervals in G ( e , e , n ) G(e,e,n) that are proven to be lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group B ( e , e , n ) B(e,e,n) and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures.

Classifying spaces from Ore categories with Garside families

We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the Stein--Farley complexes used for various relatives of Thompson's groups. It recovers the fact that Garside groups have finite classifying spaces. We describe the categories and Garside structures underlying certain Thompson groups. The Zappa--Sz\'ep product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our methods we introduce the group Braided T and show that it is of type $F_\infty$.

A Garside-theoretic analysis of the Burau representations

We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the 4-strand braid group. In the dual context, it is shown that the Burau representation for arbitrary braid index is injective when restricted to the set of simply-nested braids.

Zappa–Szép products of Garside monoids

A monoid K is the internal Zappa–Szép product of two submonoids, if every element of K admits a unique factorisation as the product of one element of each of the submonoids in a given order. This definition yields actions of the submonoids on each other, which we show to be structure preserving. We prove that K is a Garside monoid if and only if both of the submonoids are Garside monoids. In this case, these factors are parabolic submonoids of K and the Garside structure of K can be described in terms of the Garside structures of the factors. We give explicit isomorphisms between the lattice structures of K and the product of the lattice structures on the factors that respect the Garside normal forms. In particular, we obtain explicit natural bijections between the normal form language of K and the product of the normal form languages of its factors.

Dual euclidean Artin groups and the failure of the lattice property

The irreducible euclidean Coxeter groups that naturally act geometrically on euclidean space are classified by the well-known extended Dynkin diagrams and these diagrams also encode the modified presentations that define the irreducible euclidean Artin groups. These Artin groups have remained mysterious with some exceptions until very recently. Craig Squier clarified the structure of the three examples with three generators more than twenty years ago and François Digne more recently proved that two of the infinite families can be understood by constructing a dual presentation for each of these groups and showing that it forms an infinite-type Garside structure. In this article I establish that none of the remaining dual presentations for irreducible euclidean Artin groups correspond to Garside structures because their factorization posets fail to be lattices. These are the first known examples of Artin groups where all of their dual presentations fail to form Garside structures. Nevertheless, the results presented here about the cause of this failure form the foundation for a subsequent article in which the structure of euclidean Artin groups is finally clarified.

Garside Structures on Monoids with Quadratic Square-Free Relations

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra \(A= \textbf{k}S\) has a Frobenius Koszul dualA! with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties.

Conjugacy in Garside groups III: Periodic braids

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.

Présentations duales des groupes de tresses de type affine $Ã$

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 \tilde A_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\in W such that c can be written c=pp' with l_R(c)=l_R(p)+l_R(p') . We define the monoid M(P_c) to be the monoid generated by a set \underline P_c in one-to-one correspondence, p\mapsto \underline p , with P_c with only relations \underline{pp'}=\underline p.\underline 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 \tilde A . 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 .

Explicit presentations for the dual braid monoids

Birman, Ko and Lee have introduced a new monoid B ∗ n —with an explicit presentation—whose group of fractions is the n-strand braid group B n . Building on a new approach by Digne, Michel and himself, Bessis has defined a dual braid monoid for every finite Coxeter type Artin–Tits group extending the type A case. Here, we give an explicit presentation for this dual braid monoid in the case of types B and D, and we study the combinatorics of the underlying Garside structures. To cite this article: M. Picantin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 843–848.