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Filtrations of free groups arising from the lower central series

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Abstract

We make a systematic study of filtrations of a free group F defined as products of powers of the lower central series of F. Under some assumptions on the exponents, we characterize these filtrations in terms of the group algebra, the Magnus algebra of non-commutative power series, and linear representations by upper-triangular unipotent matrices. These characterizations generalize classical results of Grun, Magnus, Witt, and Zassenhaus from the 1930's, as well as later results on the lower p-central filtration and the p-Zassenhaus filtrations. We derive alternative recursive definitions of such filtrations, extending results of Lazard. Finally, we relate these filtrations to Massey products in group cohomology.

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... IV.1.9]. We also refer to [CE16] for a nice discussion of filtrations defined by induction. This description has a nice consequence, similar to Proposition 1.9 (we abbreviate L(Γ ...
Preprint
Let F_nF\_n be the free group on n generators. Consider the group IA_nIA\_n of automorpisms of F_nF\_n acting trivially on its abelianization. There are two canonical filtrations on IA_nIA\_n: the first one is its lower central series Γ_\Gamma\_*; the second one is the Andreadakis filtration A_\mathcal A\_*, defined from the action on F_nF\_n. In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ_){\mathcal L}(\Gamma\_*) and L(A_){\mathcal L}(\mathcal A\_*) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included.
... Then, for a more general profinite group G (such as G F ), one takes a profinite presentation, i.e., a continuous epimorphism π : S → G, where S is a free profinite group, and transfers the equality (1.1) from S to G. The first part is purely group-theoretic, and is usually proved using Magnus theory, i.e., by viewing the elements of G = S as formal power series. A general machinery to obtain such results in the free profinite case is given in [Efr14b] (see also [CE16]). ...
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For a list L\cal{L} of finite groups and for a profinite group G, we consider the intersection T(G) of all open normal subgroups N of G with G/N in L\cal{L}. We give a cohomological characterization of the epimorphisms π ⁣:SG\pi\colon S\to G of profinite groups (satisfying some additional requirements) such that π[T(S)]=T(G)\pi[T(S)]=T(G). For p prime, this is used to describe cohomologically the profinite groups G whose nth term G(n,p)G_{(n,p)} (resp., G(n,p)G^{(n,p)}) in the p-Zassenhaus filtration (resp., lower p-central filtration) is an intersection of this form. When G=GFG=G_F is the absolute Galois group of a field F containing a root of unity of order p, we recover as special cases results by Minac, Spira and the author, describing G(3,p)G_{(3,p)} and G(3,p)G^{(3,p)} as T(G) for appropriate lists L\cal{L}.
... [p] * et on l'appelle suite centrale descendante p-restreinte. Le lecteur pourra consulter[CE16] pour une discussion des suites fortement centrales définies par récurrence.La description explicite donnée dans la proposition 2.1.14 a pour conséquence un résultat similaire à la proposition 1.2.5 (on écrit L [p] (G) pour L(Γ[p] * G)) : Proposition 2.5.1. ...
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Let FnF_n be the free group on n generators. Consider the group IAnIA_n of automorpisms of FnF_n acting trivially on its abelianization. There are two canonical filtrations on IAnIA_n: the first one is its lower central series Γ\Gamma_*; the second one is the Andreadakis filtration A\mathcal A_*, defined from the action on FnF_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 IAnIA_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.
... IV.1.9]. We also refer to [CE16] for a nice discussion of filtrations defined by induction. This description has a nice consequence, similar to Proposition 1.9 (we abbreviate L(Γ ...
Article
Full-text available
Let F_nF\_n be the free group on n generators. Consider the group IA_nIA\_n of automorpisms of F_nF\_n acting trivially on its abelianization. There are two canonical filtrations on IA_nIA\_n: the first one is its lower central series Γ_\Gamma\_*; the second one is the Andreadakis filtration A_\mathcal A\_*, defined from the action on F_nF\_n. In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ_){\mathcal L}(\Gamma\_*) and L(A_){\mathcal L}(\mathcal A\_*) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included.
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