Comptes Rendus
Group Theory/Topology
Finiteness properties for a subgroup of the pure symmetric automorphism group
[Propriétés de finitude pour un sous-groupe du groupe des automorphismes symétriques et purs]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 127-130.

Soit Fn le groupe libre engendré par n éléments, et soit PΣn le groupe des automorphismes de Fn qui envoient chaque générateur sur un conjugué. Le noyau Kn de l'homomorphisme PΣnPΣn1, obtenu en envoyant un des générateurs du groupe libre sur l'identité, est de type fini. On démontre que Kn est de dimension cohomologique n1, est que Hi(Kn;Z) n'est pas de type fini pour 2in1. Par conséquent Kn n'est pas de présentation finie pour n3.

Let Fn be the free group on n generators, and let PΣn be the group of automorphisms of Fn that send each generator to a conjugate of itself. The kernel Kn of the homomorphism PΣnPΣn1, induced by mapping one of the free group generators to the identity, is finitely generated. We show that Kn has cohomological dimension n1, and that Hi(Kn;Z) is not finitely generated for 2in1. It follows that Kn is not finitely presentable for n3.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2009.12.011

Alexandra Pettet 1

1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
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Alexandra Pettet. Finiteness properties for a subgroup of the pure symmetric automorphism group. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 127-130. doi : 10.1016/j.crma.2009.12.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.011/

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