On the Andreadakis problem
| Autor: | Darné , Jacques |
|---|---|
| Přispěvatelé: | Laboratoire Paul Painlevé - UMR 8524 ( LPP ), Université de Lille-Centre National de la Recherche Scientifique ( CNRS ), Université de Lille, Antoine Touzé, Aurélien Djament, Laboratoire Paul Painlevé - UMR 8524 (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP) |
| Jazyk: | francouzština |
| Rok vydání: | 2018 |
| Předmět: |
théorie combinatoire des groupes
[ MATH.MATH-GR ] Mathematics [math]/Group Theory [math.GR] automorphismes des groupes libres [ MATH.MATH-AT ] Mathematics [math]/Algebraic Topology [math.AT] Johnson morphism automorphisms of free groups groupes filtrés [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] calcul différentiel libre category theory algebraic topology combinatorial group theory théorie des catégories free differential calculus [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT] Topologie algébrique filtered groups braid groups groupes de tresses morphismes de Johnson |
| Zdroj: | Topologie algébrique [math.AT]. Université de Lille, 2018. Français Topologie algébrique [math.AT]. Université de Lille, 2018. Français. ⟨NNT : ⟩ |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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