The ${\Cal K}(\pi ,1)$-conjecture for the affine braid groups

• Autores: Ruth Charney, David Peifer
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 78, Nº 3, 2003, págs. 584-600
• Idioma: inglés
• DOI: 10.1007/s00014-003-0764-y
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• Resumen

  • 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.