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
$\Cal A$. 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
Ã
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.