no. 10
Finite extensions and unipotent shadows of affine crystallographic groups
[Extensions finies et ombres unipotentes des groupes affines cristallographiques]
Baues, Oliver1
1 Departement Mathematik, ETHZ, CH-8092 Zürich, Switzerland
Comptes Rendus. Mathématique, Tome 335 (2002) no. 10, pp. 785-788.

Soit Γ un groupe virtuellement polycyclique tel que le sous-groupe de Fitting soit sans torsion et contienne son centralisateur. Nous montrons qu'une extension effective de Γ par un groupe fini μ est isomorphe à un groupe affine cristallographique si et seulement si μ laisse fixe un point dans l'espace des déformations des actions affines cristallographiques de Γ. Nous associons à Γ un groupe nilpotent sans torsion et de type fini Θ que nous appelons l'ombre unipotente de Γ. Ensuite nous relions l'espace des déformations de Γ à l'espace des déformations de Θ. Comme application nous montrons que Γ est isomorphe à un groupe affine cristallographique si, par exemple, Θ est de classe de nilpotence ⩽3, ou si le rang polycyclique de Γ est ⩽5, ainsi que dans certains autres cas.

Let Γ be a virtually polycyclic group so that the Fitting subgroup is torsion-free and contains its centralizer. We prove that an effective extension of Γ by a finite group μ is isomorphic to an affine crystallographic group if and only if there exists a fixed point for the action of μ on the deformation space of affine crystallographic actions of Γ. We associate to Γ a finitely generated torsion-free nilpotent group Θ which is called the unipotent shadow of Γ, and we relate the deformation space of Γ to the deformation space of Θ. As an application, we show that Γ is isomorphic to an affine crystallographic group if, e.g., Θ has nilpotency class ⩽3, or if the polycylic rank of Γ is ⩽5, and also in some other cases.

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DOI : 10.1016/S1631-073X(02)02562-1
Baues, Oliver 1

1 Departement Mathematik, ETHZ, CH-8092 Zürich, Switzerland 
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Baues, Oliver. Finite extensions and unipotent shadows of affine crystallographic groups. Comptes Rendus. Mathématique, Tome 335 (2002) no. 10, pp. 785-788. doi : 10.1016/S1631-073X(02)02562-1. http://www.numdam.org/articles/10.1016/S1631-073X(02)02562-1/

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