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Characters of the Grothendieck-Teichmüller group through rigidity of the Burau representation
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 4, pp. 765-780.

We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a “rigidity” approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to Grothendieck-Teichmüller groups.

We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a “rigidity” approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to Grothendieck-Teichmüller groups.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1202
Ivan Marin 1

1 Institut de Mathématiques de Jussieu, Université Paris 7, 175 rue du Chevaleret, F-75013 Paris
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     title = {Characters of the {Grothendieck-Teichm\"uller} group through rigidity of the {Burau} representation},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {765--780},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 17},
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Ivan Marin. Characters of the Grothendieck-Teichmüller group through rigidity of the Burau representation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 4, pp. 765-780. doi : 10.5802/afst.1202. https://afst.centre-mersenne.org/articles/10.5802/afst.1202/

[Bo] Bourbaki (N.).— Groupes et algèbres de Lie, chapitre 2, Hermann (1972). | MR: 573068

[De] Deligne (P.).— Le groupe fondamental de la droite projective moins trois points, in Galois Groups over , Math. Sci. Res. Inst. Publ. 16, Springer (1989). | MR: 1012168 | Zbl: 0742.14022

[Dr] Drinfeld (V.G.).— On quasitriangular quasi-Hopf algebras and a group closely connected with Gal ( ¯/), Leningrad Math. J. 2, p. 829-860 (1991). | MR: 1080203 | Zbl: 0728.16021

[Go] González-Lorca (J.).— Série de Drinfeld, monodromie et algèbres de Hecke, Thèse de l’université Paris XI-Orsay (1998).

[IS] Ichimura (H.), Sakaguchi (K.).— The non-vanishing of a certain Kummer character χ m (after C. Soulé), and some related topics, in Galois representations and arithmetic algebraic geometry, Proc. Symp., Kyoto 1985 and Tokyo 1986, Adv. Stud. Pure Math. 12, p. 53-64 (1987). | MR: 948236 | Zbl: 0647.12007

[Ka] Katz (N.).— Rigid local systems, Annals of Math. Studies 139, Princeton University Press (1996). | MR: 1366651 | Zbl: 0864.14013

[Ma] Marin (I.).— Caractères de rigidité du groupe de Grothendieck-Teichmüller, Compos. Math. 142, p. 657-678 (2006). | MR: 2231196 | Zbl: 1133.14027

[Na] Nakamura (H.).— Limits of Galois representations in fundamental groups along maximal degeneration of marked curves I, Amer. J. Math 121, p. 315-358 (1999). | MR: 1680325 | Zbl: 1006.12001

[NS] Nakamura (H.), Schneps (L.).— On a subgroup of the Grothendieck-Teichmüller group acting on the tower of profinite Teichmüller modular groups, Invent. Math. 141, p. 503-560 (2000). | MR: 1779619 | Zbl: 1077.14030

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