Making use of the Nielsen fixed point theory, we study a conjugacy invariant of braids, which we call the level index function. We present a simple algorithm for computing it for positive permutation cyclic braids.
Let be a surface, let be a subsurface, and let be two positive integers. The inclusion of in gives rise to a homomorphism from the braid group with strings on to the braid group with strings on . We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of in . Then we calculate the commensurator, the normalizer and the centralizer of in for large surface braid...
Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincaré polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups.
In the paper we give a survey of (co)homologies of braid groups and groups connected with them. Among these groups are pure braid groups and generalized braid groups. We present explicit formulations of some theorems of V. I. Arnold, E. Brieskorn, D. B. Fuks, F. Cohen, V. V. Goryunov and others. The ideas of some proofs are outlined. As an application of (co)homologies of braid groups we study the Thom spectra of these groups.
We describe new combinatorial methods for constructing explicit free resolutions of by -modules when is a group of fractions of a monoid where enough lest common multiples exist (“locally Gaussian monoid”), and therefore, for computing the homology of . Our constructions apply in particular to all Artin-Tits groups of finite Coexter type. Technically, the proofs rely on the properties of least common multiples in a monoid.
Nous étudions le problème de l’irréductibilité du produit tensoriel de deux représentations irréductibles d’un groupe fondamental , quand est le complémentaire d’hypersurfaces dans un espace projectif. Nous mettons en place un formalisme adapté et utilisons une approche par monodromie pour définir une classe de représentations irréductibles de dont les produits tensoriels restent irréductibles pour des valeurs génériques de paramètres de définition. Ceci est appliqué au groupe de tresses pures...
The purpose of this paper is to put together a large amount of results on the conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts, is as...