The \(K(\pi,1)\)-conjecture for the affine braid groups. (Q1417430)

The authors prove that the complement of the hyperplane arrangement associated to the complexified action of a Euclidean reflection group of type \(\widetilde A_n\) is a \(K(\pi,1)\) space. This is done using the fact that the associated affine braid group \(\widetilde{\mathcal A}\) can be embedded in a finite type Artin braid group. The authors also prove a number of other conjectures about the braid groups \(\widetilde{\mathcal A}\). E.g., they show that \(\widetilde{\mathcal A}\) is equipped with a biautomatic structure.