In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2sphere, constructed as a certain path groupoid of the universal ramified cover of the 2-sphere with finitely many marked-points. Our approach to ramified covers is based on cosheaf spaces, which are closely related to Fox’s complete spreads. A ...

We define an action of the braid group of a simple Lie algebra on the space of imaginary roots in the corresponding quantum affine algebra. We then use this action to determine an explicit condition for a tensor product of arbitrary irreducible finite–dimensional representations is cyclic. This allows us to determine the set of points at which the corresponding R–matrix has a zero. 0. Introduct...

The past several years have seen an explosion of interest in the cryptographic applications of non-commutative groups. Braid groups in particular are especially desirable, as they provide difficult computational problems and can be implemented quite efficiently. Several different groups of researchers have proposed numerous cryptographic protocols that make use of braid groups, but unfortunatel...

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic 13space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our...

Artin’s braid groups have been recently suggested as a new source for public-key cryptography. In this paper we propose the first group signature schemes based on the conjugacy problem, decomposition problem and root problem in the braid groups which are believed to be hard problems.

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from Cn, or complex hyperbolic space CHn, or the Hermitian symmetric space for O(2, n), and then takes the quotient by a discrete group PΓ. The classical example is the braid group, but there are many similar “braid-like” groups that arise in topol...

In this paper I give a similar method for finding the multivariable Alexander polynomial of a link L presented as the closure of a braid β. The main ingredient is a readily constructed multivariable version of the reduced Burau matrices. Other versions of ‘coloured’ Burau matrices have been developed, for example by Penne, [4], which can be interpreted as determining linear presentations of sui...

We describe a new algorithm for comparing braid words, and establish some order properties for the braid groups which are connected with left distributive operations. Abridged English Version. We apply to braids some tools used in the study of left distributive operations (c.f. [2]). The results deal with special decompositions of braid words. One describes a new quadratic algorithm for compari...

We show that for any given n, there exists a sequence of words {ak}k≥1 in the generators σ1, . . . , σn−1 of the braid group Bn, representing the identity element of Bn, such that the number of braid relations of the form σiσi+1σi = σi+1σiσi+1 needed to pass from ak to the empty word is quadratic with respect to the length of ak.