arXiv research

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 groupo…

The Garside group, as a generalization of braid groups and Artin groups of finite types, is defined as the group of fractions of a Garside monoid. We show that the semidirect product of Garside monoids is a Garside monoid. We use the semidirect product $\mathbb Z\ltimes G^n$ of the infinite cyclic group $\mathbb Z$ and…

New Garside structures found for torus knot groups and related braid groups.

Invites geometers to Garside theory for mapping class groups.

In this paper, we show that for every abelian subgroup $H$ of a Garside group, some conjugate $g^{-1}Hg$ consists of ultra summit elements and the centralizer of $H$ is a finite index subgroup of the normalizer of $H$. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the a…

Completes results on complex braid group parabolic subgroups.

New Garside structures derived from groups, leading to new group properties.

In Garside groups, axes of Morse elements are strongly contracting.

It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.

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(π,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 suff…

withdrawn and included in our new manuscript "Abelian subgroups of Garside groups", math.GT/0609683

The study connects lattices, Garside structures, and weakly modular graphs.

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 th…

Let $G$ be a Garside group with Garside element $Δ$. An element $g$ in $G$ is said to be \emph{periodic} if some power of $g$ lies in the cyclic group generated by $Δ$. 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 ce…

The cycling operation endows the super summit set $S_x$ of any element $x$ of a Garside group $G$ with the structure of a directed graph $Γ_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of $Γ_x$ can be used instead of $S_x$ for deciding conjugacy to $x$ in $G$, yielding a faster and more pr…

Let $G$ be a Garside group with Garside element $Δ$, and let $Δ^m$ be the minimal positive central power of $Δ$. An element $g\in G$ 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…

In this article, we introduce the notion of cycling operations of arbitrary order in Garside groups, which is a full generalization of the cycling and decycling operations. Theoretically, this notion together with other related concepts provides a context in which various definitions and arguments concerning Garside gr…

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 difficult…

Method produces faithful representations of Garside groups and torus knot groups.

We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).

Solves double coset problem for braid group H_n.

We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoret…

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown …

From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $ρ: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation shall be called the Artin type representation associated to the pair $(H,h)$. The goal …

We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and decycling operations by a new one, called cyclic sliding, which appears to be a …

When Daan Krammer and Stephen Bigelow independently proved that braid groups are linear, they used the Lawrence-Krammer-Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid group and plays a major role in Krammer's alge…

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 conugacy problem given by the authors in a previous paper, are two main steps for solving conjugacy systems, thus breaking recently discovered crypto…

Benardete, Gutierrez and Nitecki showed an important result which relates the geometrical properties of a braid, as a homeomorphism of the punctured disk, to its algebraic Garside-theoretical properties. Namely, they showed that if a braid sends a curve to another curve, then the image of this curve after each factor o…

The paper presents methods to write presentations for Dehn quandles.

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 convex geodesic bico…

Study positive 3-braids to compute Khovanov homology.

Recently, there have been several progresses for the conjugacy search problem (CSP) in Garside groups, especially in braid groups. All known algorithms for solving this problem use a sort of exhaustive search in a particular finite set such as the super summit set and the ultra summit set. Their complexities are propor…

New link groups are derived from torus necklaces, connecting braid groups to reflection groups.

Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomia…

We prove that an Artin-Tits group of type $\tilde C$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the "generated group" method. This answers, in this particular case, a general question on Artin-Tits groups, gives …

This paper is the second in a series in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups. The ultra summit set USS(X) of an element X in a Garside group G is a finite set of elements in G, introduced by the second author, which is a complete invariant…

The braid group $B_{n}$, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism ${\rm{rev}}: B_{n} \to B_{n}$, $v \mapsto \bar{v}$, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation $τ:x \mapsto Δ^{…

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 geometric…

Affine Artin groups have a finite classifying space.

Reduces conjecture to tree-based Artin groups.

We show that the span of the variable $q$ in the Lawrence-Krammer-Bigelow representation matrix of a braid is equal to the twice of the dual Garside length of the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow's geometric approach. The key observation is that the dual Garside length of a …

We give a computational algorithm which decides if a braid is quasipositive or not. A braid is quasipositive if it's a product of conjuguates of generators. For this, we use the theory of Garside and the combinatorials properties of the Artin monoid.

Garside's results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.

The paper proves that certain spaces are injective and Helly graphs.

The study explores continuous noncrossing partitions and their relation to weighted circular factorizations.

In [J.Birman, V.Gebhardt, J.Gonzalez-Meneses, Conjugacy in Garside groups I: cyclings, powers and rigidity] authors asked: (open question 2) is the size of USS of a rigid pseudo-Anosov braid is bounded above by some polynomial in the number of strands and the braid length? We answer this question in the negative.

The Burau representation of braid group B4 is shown to be faithful almost everywhere.

In this work we present a natural surjective map from rigid braids in B_3 (in Garside sense) to SL_2(N). This map provides an upper and a lower bound for the dilatation factor of a pseudo-Anosov 3-strand braid. These bounds only depend on the canonical length of the classical Garside structure of B_3.