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

### Ruth Charney

Brandeis University, Waltham, USA### David Peifer

University of North Carolina at Asheville, USA

## Abstract

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on $C_{n}$ 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 $A~_{n}$ and its associated Artin group, the affine braid group $A~$. Using the fact that $A~$ 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 $A~$, 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.

## Cite this article

Ruth Charney, David Peifer, The $K(π,1)$-conjecture for the affine braid groups. Comment. Math. Helv. 78 (2003), no. 3, pp. 584–600

DOI 10.1007/S00014-003-0764-Y