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.     

Two-generator subgroups of the pure braid group

We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group actions on trees.     

Finite quotients of the pure symplectic braid group

We show how the finite symplectic groups arise as quotients of the pure symplectic braid group. Via [SV] certain of these groups — in particular, all groups Sp _n (2) — occur as Galois groups over ℚ. Supported by NSF grant DMS-9306479.     

Tree groups and the 4-string pure braid group

Given a graph Г, undirected, with no loops or multiple edges, we define the graph group on Г, F_Г, as the group generated by the vertices of Г, with one relation xy = xy for each pair x and y of adjacent vertices of Г.

In this paper we will show that the unpermuted braid group on four strings is an HNN-extension of the graph group F_s, where

S =

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The form of the extension will resolve a conjecture of Tits for the 4-string braid group. We will conclude, by analyzing the subgroup structure of graph groups in the case of trees, that for any tree T on a countable vertex set, F_t is a subgroup of the 4-string braid group.

We will also show that this uncountable collection of subgroups of the 4-string braid group is linear, that is, each subgroup embeds in GL(3, ), as well as embedding in Aut(F), where F is the free group of rank 2.     

A class of Garside groupoid structures on the pure braid group

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.

     

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 faithfulness criterion for the Gassner representation of the pure braid group

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, . Long and Paton proved that if a Burau matrix has ones on the diagonal and zeros below the diagonal then is the identity matrix. In this paper, a generalization of Long and Paton's result will be proved. Our main theorem is that if the trace of the image of an element of under the reduced Gassner representation is , then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.

     

Complex specializations of the reduced Gassner representation of the pure braid group

We will give a necessary and sufficient condition for the specialization of the reduced Gassner representation to be irreducible. It will be shown that for , is irreducible if and only if .

     

A rational sextic associated with a Desargues configuration

In this paper we consider a Desargues configuration in the projective plane, i.e. ten points and ten lines, on each line we have three of the points and through each point we have three of the lines. We construct a rational curve of order 6 which has a node at each of the ten points. We have never seen this kind of curve in the literature, but it is well known that for anyn there exists a rational curve of ordern which has [(n–1)(n–2)]/2 nodes and ifn=6 we find a sextic with ten nodes. The purpose of this paper is to obtain a sextic of this kind as a locus of points in connection with special projectivities of the plane associated with the Desargues configuration and to find a rational parametric representation of it. A large part of this paper is done with MACSYMA: it is an application of computer algebra in algebraic geometry. Special cases, where we find a quintic, a quartic or a cubic, are given in the last section.     

Placement of the Desargues configuration on a cubic curve

Hilbert and Cohn-Vossen once declared that the configurations of Desargues and Pappus are by far the most important projective configurations. These two are very similar in many respects: both are regular and self-dual, both could be constructed with ruler alone and hence exist over the rational plane, the final collinearity in both instances are automatic and both could be regarded as self-inscribed and self-circumscribed p9lygons (see [1, p. 128]). Nevertheless, there is one fundamental difference between these two configurations, viz. while the Pappus-Brianchon configuration can be realized as nine points on a non-singular cubic curve over the complex plane (in doubly infinite ways), it is impossible to get such a representation for the Desargues configuration. In fact, the configuration of Desargues can be placed in a projective plane in such a way that its vertices lie on a cubic curve over a field k if and only if k is of characteristic 2 and has at least 16 elements. Moreover, any cubic curve containing the vertices of this configuration must be singular.This research of all the three authors was supported by the HSERC of Canada.     

On the structure of surface pure braid groups

We describe some of the properties of the pure braid groups of surfaces different from and . In the case of compact, connected, orientable surfaces without boundary and of genus at least two, we give a necessary and sufficient condition for the splitting of the pure braid group exact sequence of Fadell and Neuwirth, thus answering completely a question of Birman.     

On the structure of surface pure braid groups

We describe some of the properties of the pure braid groups of surfaces different from and . In the case of compact, connected, orientable surfaces without boundary and of genus at least two, we give a necessary and sufficient condition for the splitting of the pure braid group exact sequence of Fadell and Neuwirth, thus answering completely a question of Birman.     

An acyclic extension of the braid group

We relate Artin's braid groupB _℞=lim_→B_n to a certain groupF′ ofpl-homeomorphisms of the interval. Namely, there exists a short exact sequence 1→B _∞→A→F′→1 whereH _kA=0,k≥1.     

On Krammer's representation of the braid group

A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found. Following a rescaling and change of parameters, the matrices are found to be identical to those of the Krammer representation. The two representations are thus the same, proving the irreducibility of one and the faithfulness of the other. Received: 16 April 2000 / Published online: 23 July 2001     

Generalised Magnus modules over the braid group

Supported byDeutsche Forschungsgemeinschaft     

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.     

Flat connections on configuration spaces and braid groups of surfaces

We construct an explicit bundle with flat connection on the configuration space of n points on a complex curve. This enables one to recover the ‘1-formality’ isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n   points on a surface and an explicitly presented Lie algebra, and to extend it to a morphism from the full braid group of the surface to the semidirect product of the associated group with the symmetric group _Sn$S_n$.     

The Deligne complex for the four-strand braid group

This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on . A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).