Titre :

Autour du problème d'Andreadakis

Titre en anglais :

On the Andreadakis problem

Auteur(s) :

Darné, Jacques [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Directeur(s) de thèse :

Antoine Touzé
Aurélien Djament

Date de soutenance :

2018-03-20

Organisme de délivrance :

Université de Lille

Mot(s)-clé(s) :

Topologie algébrique
théorie combinatoire des groupes
théorie des catégories
groupes filtrés
automorphismes des groupes libres
groupes de tresses
calcul différentiel libre
morphismes de Johnson

Mot(s)-clé(s) en anglais :

algebraic topology
combinatorial group theory
category theory
filtered groups
automorphisms of free groups
braid groups
free differential calculus
Johnson morphism

Discipline(s) HAL :

Mathématiques [math]/Topologie algébrique [math.AT]
Mathématiques [math]/Théorie des groupes [math.GR]

Résumé en anglais : [en]

Let $F_n$ be the free group on $n$ generators. Consider the group $IA_n$ of automorpisms of $F_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA_n$: the first one is its lower central ...
Lire la suite >Let $F_n$ be the free group on $n$ generators. Consider the group $IA_n$ of automorpisms of $F_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA_n$: the first one is its lower central series $\Gamma_*$; the second one is the Andreadakis filtration $\mathcal A_*$, defined from the action on $F_n$. Andreadakis asked if and how these filtrations were different. We begin by describing a framework adapted to the study of such filtrations and their counterparts on group algebras. We then study several versions of the problem. In particular, we look at its restriction to some subgroups of $IA_n$ : we show that the two filtration coïncide when restricted to the triangular subroups and to braid groups. We also consider a stable version of the problem : we establish that the canonical morphism between the associated graded Lie rings is surjective when $n$ is big enough compared to a fixed degree. We also investigate a $p$-restricted version of the Andreadakis problem, and provide a calculation of the Lie algebra of the classical congruence group.Our methods are algebraic in nature. The tools come from combinatorial group theory and the study of mapping class groups; we often introduce some categorical langage to reformulate them.Lire moins >

Langue :

Français

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL