Semi-algebraic Geometry of Braid Groups

The braid group of n-strings is the group of homotopy types of movements of n distinct points in the 2-plane R. It was introduced by E. Artin [1] in 1926 in order to study knots in R. He gave a presentation of the braid group by generators and relations, which are, nowadays, called the Artin braid relations. Since then, not only in the study of knots, the braid groups appear in several contexts in mathematics, since it is the fundamental group of the configuration space of n-points in the plane. Early in 70’s the braid groups are generalized to a wider class of groups, the fundamental groups of the regular orbit spaces of finite reflection groups (Brieskorn [6]), which are called either the generalized braid group (Deligne [3]) or the Artin group (Brieskorn-Saito [2]). The regular orbit space turns out to be an Eilenberg-MacLane space (Deligne [3], c.f. BrieskornSaito [2]). Through the study of holonomic systems on the Eilenberg-Maclane spaces, representations of the generalized braid groups are studied (Kohno,...). Also through the braid relations, the actions of braid groups on triangulated categories are studied (Seidel-Thomas,...). Still, we are far from full understanding of their representations. As for the study of the Eilenberg-Maclane spaces, it was from the beginning a question raised by Deligne, Brieskorn, Saito,. . . to find the paths in the EilenbergMacLane spaces which give a generator system of the Artin groups satisfying the Artin braid relations. In this note (based on [4]), we will give two answers to this question. We approach the problem by the semi-algebraic geometry of the orbit space induced from the flat structure on it [7].