The bounded complex of a uniform affine oriented matroid is a ball

Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.     

On the classifying space of Artin monoids

A theorem proved by Dobrinskaya [9 Dobrinskaya, N. E. (2006). Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes. Proc. Steklov Inst. Math. 252(1):30–46.[Crossref] , [Google Scholar]] shows that there is a strong connection between the K(π,1) conjecture for Artin groups and the classifying spaces of Artin monoids. More recently Ozornova obtained a different proof of Dobrinskaya’s theorem based on the application of discrete Morse theory to the standard CW model of the classifying space of an Artin monoid. In Ozornova’s work, there are hints at some deeper connections between the above-mentioned CW model and the Salvetti complex, a CW complex which arises in the combinatorial study of Artin groups. In this work we show that such connections actually exist, and as a consequence, we derive yet another proof of Dobrinskaya’s theorem.     

Presentations of subgroups of the braid group generated by powers of band generators

According to the Tits conjecture proved by Crisp and Paris (2001) [4], the subgroups of the braid group generated by proper powers of the Artin elements σi 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 subgroups generated by powers of the band generators aij is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with exponent 1 or 2. Such presentations are distinctively more complicated than those of right-angled Artin groups.     

Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the -norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of and for all . As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the group . Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3-manifold.

     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.      

Bestvina's Normal Form Complex and the Homology of Garside Groups

A Garside group is a group admitting a finite lattice generating set . 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 , and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvina's weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually Abelian.     

The braid group for a quiver with superpotential

We survey various generalizations of braid groups for quivers with superpotential and focus on the cluster braid groups, which are introduced in a joint work with King(2018). Our motivations come from the study of cluster algebras, Calabi-Yau categories and Bridgeland stability conditions.     

The braid group of a necklace

We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of (unlinked) circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will call braid group of a necklace) is isomorphic to the braid group over an annulus quotiented by the square of the center. We then define braid groups of necklaces and affine braid groups of type \(\mathcal {A}\) in terms of automorphisms of free groups and characterize these automorphisms among all automorphisms of free groups. In the case of affine braid groups of type \(\mathcal {A}\) such a representation is faithful.     

Universal Upper Bound for the Growth of Artin Monoids

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.     

Relative Hyperbolicity and Artin Groups

Let G=⟨ a₁,&ldots; , a _n | a _i a _j a _i &ldots; = a_ja_ia_j,&ldots; ,i>j⟩ be an Artin group and let m _ij =m _ji be the length of each of the sides of the defining relation involving a _i and a _j . We show if all m_ij ⩾ 7 then G is relatively hyperbolic in the sense of Farb with respect to the collection of its two-generator subgroups a _i , a _j for which m_ij >&infty;.     

The cohomology ring of the colored braid group

The cohomology ring is obtained for the space of ordered sets of n different points of a plane.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 227–231, February, 1969.The author thanks V. P. Palamodov and D. B. Fuks for useful discussions.     

Multiplier ideals of hyperplane arrangements

In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustata (2004).

     

The D n generalized pure braid group

A corepresentation for the generalized pure braid group ID _n of the Coxeter group D _n is constructed. The lower central series of ID _n is investigated. It is proved that ID _n is approximable by torsion-free nilpotent groups, so R. Hain's obstruction to the solvability of the generalized Riemann-Hilbert problem is trivial for ID _n.     

Towards generating secure keys for braid cryptography

Braid cryptosystem was proposed in CRYPTO 2000 as an alternate public-key cryptosystem. The security of this system is based upon the conjugacy problem in braid groups. Since then, there have been several attempts to break the braid cryptosystem by solving the conjugacy problem in braid groups. In this article, we first survey all the major attacks on the braid cryptosystem and conclude that the attacks were successful because the current ways of random key generation almost always result in weaker instances of the conjugacy problem. We then propose several alternate ways of generating hard instances of the conjugacy problem for use braid cryptography.      

The pure virtual braid group is quadratic

If an augmented algebra $K$ over $\mathbb Q $ is filtered by powers of its augmentation ideal $I$ , the associated graded algebra $gr_I K$ need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper, we give a sufficient criterion (called the PVH Criterion) for $gr_I K$ to be quadratic. When $K$ is the group algebra of a group $G$ , quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for $G$ . Thus, the PVH Criterion also implies the existence of such a universal finite type invariant for the group $G$ . We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.     

A note on Mustaţă's computation of multiplier ideals of hyperplane arrangements

In 2006, M. Mustaţă used jet schemes to compute the multiplier ideals of reduced hyperplane arrangements. We give a simpler proof using a log resolution and generalize to non-reduced arrangements. By applying the idea of wonderful models introduced by De Concini-Procesi in 1995, we also simplify the result. Indeed, Mustaţă's result expresses the multiplier ideal as an intersection, and our result uses (generally) fewer terms in the intersection.

     

The braid group B4 is linear

We study a linear representation ρ:B _n ? GL _m (Z[q ^±1,t ^±1]) with m=n(n-1)/2. We will show that for n=4, this representation is faithful. We prove a relation with the new Charney length function. We formulate a conjecture implying that ρ is faithful for all n. Oblatum 15-VI-1999 & 24-II-2000?Published online: 18 September 2000