Finiteness properties for a subgroup of the pure symmetric automorphism group

Pettet, Alexandra

doi:10.1016/j.crma.2009.12.011

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]

Présenté par : Michel Duflo

Pettet, Alexandra ¹

¹ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Comptes Rendus. Mathématique, Tome 348 (2010) no. 3-4, pp. 127-130.

Résumé
Abstract

Soit $F_{n}$ le groupe libre engendré par n éléments, et soit $P Σ_{n}$ le groupe des automorphismes de $F_{n}$ qui envoient chaque générateur sur un conjugué. Le noyau $K_{n}$ de l'homomorphisme $P Σ_{n} \to P Σ_{n - 1}$ , obtenu en envoyant un des générateurs du groupe libre sur l'identité, est de type fini. On démontre que $K_{n}$ est de dimension cohomologique $n - 1$ , est que $H_{i} (K_{n}; Z)$ n'est pas de type fini pour $2 ⩽ i ⩽ n - 1$ . Par conséquent $K_{n}$ n'est pas de présentation finie pour $n ⩾ 3$ .

Let $F_{n}$ be the free group on n generators, and let $P Σ_{n}$ be the group of automorphisms of $F_{n}$ that send each generator to a conjugate of itself. The kernel $K_{n}$ of the homomorphism $P Σ_{n} \to P Σ_{n - 1}$ , induced by mapping one of the free group generators to the identity, is finitely generated. We show that $K_{n}$ has cohomological dimension $n - 1$ , and that $H_{i} (K_{n}; Z)$ is not finitely generated for $2 ⩽ i ⩽ n - 1$ . It follows that $K_{n}$ is not finitely presentable for $n ⩾ 3$ .

Reçu le : 2008-03-12
Accepté le : 2009-12-10
Publié le : 2009-12-31

DOI : 10.1016/j.crma.2009.12.011

Affiliations des auteurs :

Pettet, Alexandra ¹

¹ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

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Pettet, Alexandra. Finiteness properties for a subgroup of the pure symmetric automorphism group. Comptes Rendus. Mathématique, Tome 348 (2010) no. 3-4, pp. 127-130. doi : 10.1016/j.crma.2009.12.011. http://www.numdam.org/articles/10.1016/j.crma.2009.12.011/

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