The $\mathcal{K}(\pi ,1)$-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 ${\mathbb{C}}^n$ is known to be a $\mathcal{K}(\pi ,1)$ space for the corresponding Artin group $\mathcal{A}$. 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 ${\tilde{\mathcal{A}}}_n$ and its associated Artin group, the affine braid group $\tilde{\mathcal{A}}$. Using the fact that $\tilde{\mathcal{A}}$ 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 $\mathcal{K}(\pi ,1)$-space for $\tilde{\mathcal{A}}$, and use it to prove the $\mathcal{K}(\pi ,1)$-conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.