The K(π, 1)-conjecture for the affine braid groups

Abstract

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on ℂn is known to be a K(π, 1) space for the corresponding Artin group A. A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type Ãn and its associated Artin group, the affine braid group Ã. Using the fact that à can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, n-dimensional K(π, 1) -space for Ã, and use it to prove the K(π, 1) conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.