arXiv research

Classifies braids with positive Artin presentations and their fundamental groups.

The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.

In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.

In this article we construct a piecewise Euclidean, non-positively curved 2-complex for the 3-generator Artin groups of large type. As a consequence we show that these groups are biautomatic. A slight modification of the proof shows that many other Artin groups are also biautomatic. The general question (whether all Ar…

New method finds large curved subcomplexes, proving conjectures for specific groups.

We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatorics of tilings of the plane we describe precisely the building blocks for quasiflat…

Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F, and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h: F-->F which pointwise fix the boundary …

Proves a conjecture for 3D Artin groups using new combinatorial curvature.

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits monoid of spherical type (hence of each braid monoid) with respect to the standard g…

Characterizes periodic elements in Artin-Tits groups via stability conditions.

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 satis…

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^{…

The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.

The paper finds Artin presentations for the trivial group and identifies hyperbolic 3-braids.

According to the Tits conjecture proved by Crisp and Paris, [CP], the subgroups of the braid group generated by proper powers of the Artin elements are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups. The case of subgroup…

This paper characterizes a specific type of twisted Artin groups embedded in knot groups.

The paper proves spaces associated to certain cubical presentations are aspherical.

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 …

We construct a family of morphisms between Artin-Tits groups which generalise the ones constructed by J. Crisp in [Injective maps between Artin groups, Proceedings of the Special Year in Geometric Group Theory, Berlin, (1999), 119 -- 138]. We show that their restrictions to the positive Artin monoids respect normal for…

Proof of $K(π, 1)$ conjecture for affine Artin groups.

Introduces a reduction system for Artin-Tits groups, improving algorithms and proving periodicity results.

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…

Study a specific line arrangement and compute its fundamental group via braid monodromy.

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and h…

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).

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

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of graph-manifolds with boundary in terms of certain finite two-colored graphs. A corollary i…

Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.

Affine Artin groups have a finite classifying space.

The minimal standardizer of a curve system on a punctured disk is the minimal braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin-Tits groups of spherical type and we s…

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…

New right-angled Artin subgroups found in Artin groups.

Graphs with specific spanning trees yield RAAGs, with applications to BBGs.

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…

The computation of the fundamental group of the complement of an algebraic plane curve has been theoretically solved since Zariski-van Kampen, but actual computations are usually cumbersome. In this work, we describe the notion of Wirtinger presentation of such a group relying on the real picture of the curve and with …

An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.

The paper introduces a Deligne complex for Artin monoids and studies its properties.

Artin groups not free of infinity are shown to have finite centers.

The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk an…

Reduces conjecture to tree-based Artin groups.

Artin groups get $K(π,1)$-conjecture proof for tree and cyclic diagrams.

The study proves conjecture for specific Artin groups.

Let X be a subcomplex of the standard CW-decomposition of the n-dimensional torus. We exhibit an explicit optimal motion planning algorithm for X. This construction is used to calculate the topological complexity of complements of general position arrangements and Eilenberg-Mac Lane spaces associated to right-angled Ar…

Even Artin groups generalize right-angled Artin groups by allowing the labels in the defining graph to be even. In this paper a complete characterization of quasi-projective even Artin groups is given in terms of their defining graphs. Also, it is shown that quasi-projective even Artin groups are realizable by K(pi,1) …

Many 2D Artin groups are residually finite.

Uniqueness of quasi-roots explored in right-angled Artin groups.

Equivalence proven between two torsion invariants for flat vector bundles.