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.

Reçu le :
Révisé le :
Accepté le :
Première publication :
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) 
@unpublished{AIF_0__0_0_A101_0,
     author = {Darn\'e, Jacques},
     title = {Braids, inner automorphisms and the {Andreadakis} problem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3662},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Darné, Jacques
TI  - Braids, inner automorphisms and the Andreadakis problem
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3662
LA  - en
ID  - AIF_0__0_0_A101_0
ER  - 
%0 Unpublished Work
%A Darné, Jacques
%T Braids, inner automorphisms and the Andreadakis problem
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3662
%G en
%F AIF_0__0_0_A101_0
Darné, Jacques. Braids, inner automorphisms and the Andreadakis problem. Annales de l'Institut Fourier, Online first, 43 p.

[1] Alekseev, Anton; Torossian, Charles The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. Math., Volume 175 (2008), pp. 415-463 | DOI | Zbl

[2] Andreadakis, Stylianos On the automorphisms of free groups and free nilpotent groups, Proc. Lond. Math. Soc., Volume 15 (1965), pp. 239-268 | DOI | MR | Zbl

[3] Artin, Emil Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamb., Volume 4 (1925) no. 1, pp. 47-72 | DOI | MR | Zbl

[4] Artin, Emil Theory of braids, Ann. Math., Volume 48 (1947), pp. 101-126 | DOI | MR

[5] Bardakov, Valeriy G.; Neshchadim, Mikhail V. Subgroups, automorphisms, and Lie algebras related to the basis-conjugating automorphism group, Algebra Logika, Volume 55 (2016) no. 6, pp. 670-703 | DOI | MR

[6] Bartholdi, Laurent Automorphisms of free groups. I—erratum [ MR3084710], New York J. Math., Volume 22 (2016), pp. 1135-1137 http://nyjm.albany.edu:8000/j/2016/22_1135.html | MR | Zbl

[7] Bellingeri, Paolo; Bodin, Arnaud The braid group of a necklace, Math. Z., Volume 283 (2016) no. 3-4, pp. 995-1010 | DOI | MR | Zbl

[8] Birman, Joan S. Braids, links, and mapping class groups, Annals of Mathematics Studies, 82, Princeton University Press; University of Tokyo Press, 1974, ix+228 pages | MR

[9] Chow, Wei-Liang On the algebraical braid group, Ann. Math., Volume 49 (1948), pp. 654-658 | DOI | MR | Zbl

[10] Cohen, Frederick R.; Pakianathan, Jonathan N.; Vershinin, Vladimir V.; Wu, Jie Basis-conjugating automorphisms of a free group and associated Lie algebras, Groups, homotopy and configuration spaces (Geometry and Topology Monographs), Volume 13, Geometry and Topology Publications, 2008, pp. 147-168 | DOI | MR | Zbl

[11] Darné, Jacques Milnor invariants of braids and welded braids up to homotopy (2019) (https://arxiv.org/abs/1904.10677)

[12] Darné, Jacques On the stable Andreadakis problem, J. Pure Appl. Algebra, Volume 223 (2019) no. 12, pp. 5484-5525 | DOI | MR | Zbl

[13] Darné, Jacques On the Andreadakis problem for subgroups of IA n , Int. Math. Res. Not. (2021) no. 19, pp. 14720-14742 | DOI | MR | Zbl

[14] Falk, Michael; Randell, Richard The lower central series of generalized pure braid groups, Geometry and topology (Athens, Ga., 1985) (Lecture Notes in Pure and Applied Mathematics), Volume 105, Marcel Dekker, 1987, pp. 103-108 | MR | Zbl

[15] Habegger, Nathan; Masbaum, Gregor The Kontsevich integral and Milnor’s invariants, Topology, Volume 39 (2000) no. 6, pp. 1253-1289 | DOI | MR | Zbl

[16] Ibrahim, Abdoulrahim On a subgroup of McCool group PΣ n (2019) (https://arxiv.org/abs/1902.10033)

[17] Ihara, Yasutaka The Galois representation arising from ℙ 1 -{0,1,∞} and Tate twists of even degree, Galois groups over ℚ (1989) | DOI

[18] Kaloujnine, Léo Sur quelques propriétés des groupes d’automorphismes d’un groupe abstrait. (Généralisation d’un theorème de M. Ph. Hall), C. R. Acad. Sci. Paris, Volume 231 (1950), pp. 400-402 | MR | Zbl

[19] Kohno, Toshitake Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math., Volume 82 (1985) no. 1, pp. 57-75 | DOI | MR | Zbl

[20] Lazard, Michel Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Éc. Norm. Supér., Volume 71 (1954), pp. 101-190 | DOI | Numdam | MR | Zbl

[21] Magnus, Wilhelm Über Automorphismen von Fundamentalgruppen berandeter Flächen, Math. Ann., Volume 109 (1934) no. 1, pp. 617-646 | DOI | MR | Zbl

[22] Mostovoy, Jacob; Willerton, Simon Free groups and finite-type invariants of pure braids, Math. Proc. Camb. Philos. Soc., Volume 132 (2002) no. 1, pp. 117-130 | DOI | MR | Zbl

[23] Orr, Kent E. Homotopy invariants of links, Invent. Math., Volume 95 (1989) no. 2, pp. 379-394 | DOI | MR | Zbl

[24] Satoh, Takao On the Andreadakis conjecture restricted to the “lower-triangular” automorphism groups of free groups, J. Algebra Appl., Volume 16 (2017) no. 5, 1750099, 31 pages | DOI | MR | Zbl

[25] Serre, Jean-Pierre Lie algebras and Lie groups, Lecture Notes in Mathematics, 1500, Springer, 2006, viii+168 pages 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition | MR

Cité par Sources :