# The \( {\Cal K}(\pi ,1) \)-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 ${\Cal K}(\pi ,1)$ 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 $\tilde{\Cal A}_n$ 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 ${\Cal K}(\pi ,1)$-space for $\tilde{\Cal A}$, and use it to prove the ${\Cal K}(\pi ,1)$-conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.

## Cite this article

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

DOI 10.1007/S00014-003-0764-Y