Commentarii Mathematici Helvetici
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The ${\Cal K}(\pi ,1)$-conjecture for the affine braid groups
Ruth Charney (1) and David Peifer (2)
(1) Mathematics Department, Brandeis University, Mail Stop 050, MA 02454-9110, WALTHAM, UNITED STATES(2) Department of Mathematics, University of North Carolina at Asheville, NC 28804-3299, ASHEVILLE, UNITED STATES
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.
Keywords: Braid groups, Artin groups, hyperplane arrangements