Braids, inner automorphisms and the Andreadakis problem
[Tresses, automorphismes intérieurs et le problème d’Andreadakis]
Darné, Jacques1
1 UCLouvain - IRMP Chemin du Cyclotron 2 1348 Ottignies-Louvain-la-Neuve (Belgique)
Annales de l'Institut Fourier, Online first, 43 p.

Nous généralisons les outils introduits dans [13] pour étudier le problème d’Andreadakis pour les sous-groupes de IA n . En particulier, nous étudions comment la réponse au problème d’Andreadakis varie lorsque les automorphismes intérieurs sont ajoutés à un sous-groupe donné. Nous utilisons les résultats obtenus pour montrer notamment que l’égalité d’Andreadakis est vraie pour le groupe de tresses pures à n brins modulo son centre agissant sur le groupe libre F n-1 . Cette action est celle du groupe de difféotopie (pur, pointé) de la sphère avec n points marqués sur le groupe fondamental de la sphère privée de n points.

In this paper, we generalize the tools that were introduced in [13] in order to study the Andreadakis problem for subgroups of IA n . In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IA n . We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group F n-1 , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.

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DOI : 10.5802/aif.3662
Classification : 20F12, 20F14, 20F36, 57K20, 20E05, 20F28, 17B01, 17B40
Keywords: Lower central series, Central filtrations, Lie algebras, Automorphisms of free groups, Braid groups
Mot clés : Suite centrale descendante, filtrations centrales, algèbres de Lie, Automorphismes des groupes libres, groupes de tresses
Darné, Jacques 1

1 UCLouvain - IRMP Chemin du Cyclotron 2 1348 Ottignies-Louvain-la-Neuve (Belgique) 
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Darné, Jacques. Braids, inner automorphisms and the Andreadakis problem. Annales de l'Institut Fourier, Online first, 43 p.

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