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The K(?,1)-conjecture for the affine braid groups

2003; European Mathematical Society; Volume: 78; Issue: 3 Linguagem: Inglês

10.1007/s00014-003-0764-y

1420-8946

Ruth Charney, David Peifer,

Homotopy and Cohomology in Algebraic Topology

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on \mathbb C^n is known to be a {\mathcal K}(\pi ,1) space for the corresponding Artin group \mathcal 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{\mathcal A}_n and its associated Artin group, the affine braid group \tilde{\mathcal A} . Using the fact that \tilde{\mathcal 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 {\mathcal K}(\pi ,1) -space for \tilde{\mathcal A} , and use it to prove the {\mathcal 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.