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 $\Cal A$. 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 à _n and its associated Artin group, the affine braid group $\tilde{\Cal A}$. Using the fact that $\tilde{\Cal A}$ 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 $\tilde{\Cal A}$, 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.