Annales Mathématiques
Blaise Pascal
no. 1
Basic results on braid groups
[Resultats basiques dans les groupes de tresses.]
Juan González-Meneses1
1 Departamento de Álgebra Facultad de Matemáticas Universidad de Sevilla Apdo. 1160 41080 - Sevilla SPAIN
Annales mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 15-59.

Cet article contient les notes d’un course donné par l’auteur à l’Ecole Franco-Espagnole Tresses in Pau, qui a eu lieu à Pau (France) en Octobre 2009. Il s’agit essentiellement d’une introduction aux différents points des vue et techniques qui peuvent être utilisées pour montrer des résultats dans les groupes de tresses. En utilisant ces techniques on montre quelques résultats bien connus dans les groupes de tresses, à savoir l’exactitude de la presentation d’Artin, le fait que les groupes de tresses sont sans torsion, ou que son centre est engendré par le full twist. On rappelle quelques solutions des problèmes du mot et de la conjugaison, et aussi que les racines d’une tresse sont toutes conjuguées. On décrit aussi le centralisateur d’une tresse donnée. La plupart des preuves sont classiques, en utilisant de la terminologie moderne. J’ai choisi celles qui je trouve plus simples ou plus jolies.

These are Lecture Notes of a course given by the author at the French-Spanish School Tresses in Pau, held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin’s presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. We also describe the centralizer of a given braid. Most proofs are classical ones, using modern terminology. I have chosen those which I find simpler or more beautiful.

DOI : 10.5802/ambp.293
Classification : 20F36
Keywords: Braids, torsion-free, presentation, Garside, Nielsen-Thurston theory
Mot clés : Tresses, groupes d’Artin-Tits

Juan González-Meneses 1

1 Departamento de Álgebra Facultad de Matemáticas Universidad de Sevilla Apdo. 1160 41080 - Sevilla SPAIN 
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Juan González-Meneses. Basic results on braid groups. Annales mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 15-59. doi : 10.5802/ambp.293. https://ambp.centre-mersenne.org/articles/10.5802/ambp.293/

[1] J. W. Alexander On the Deformation of an n-Cell, Proc. of the Nat. Acad. of Sci. of the USA., Volume 9 (12) (1923), pp. 406-407 | DOI | JFM

[2] E. Artin Theorie der Zöpfe, Abh. Math. Sem. Hamburgischen Univ., Volume 4 (1925), pp. 47-72 | DOI | JFM

[3] E. Artin The theory of braids, Annals of Math., Volume 48 (1947), pp. 101-126 | DOI | MR | Zbl

[4] L. Bacardit; W. Dicks Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue, Groups - Complexity - Criptology, Volume 1 (2009), pp. 77-129 | DOI | MR | Zbl

[5] G. Baumslag Automorphisms groups of residually finite groups, J. London Math. Soc., Volume 38 (1963), pp. 117-118 | DOI | MR | Zbl

[6] D. Bessis Garside categories, periodic loops and cyclic sets (2006) (arxiv.org/abs/math.GR/0610778)

[7] D. Bessis; F. Digne; J. Michel Springer theory in braid groups and the Birman-Ko-Lee monoid, Pacific J. Math., Volume 205 (2) (2002), pp. 287-309 | DOI | MR | Zbl

[8] S. J. Bigelow Braid groups are linear, J. Amer. Math. Soc., Volume 14 (2) (2001), pp. 471-486 | DOI | MR | Zbl

[9] J. S. Birman braids, links and mapping class groups. Annals of Mathematics Studies, No. 82., Princeton University Press, Princeton, N.J., 1974 | MR | Zbl

[10] J. S. Birman; V. Gebhardt; J. González-Meneses Conjugacy in Garside groups. I. Cyclings, powers and rigidity, Groups Geom. Dyn., Volume 1 (3) (2007), pp. 221-279 | DOI | MR | Zbl

[11] J. S. Birman; V. Gebhardt; J. González-Meneses Conjugacy in Garside groups. III. Periodic braids, J. Algebra, Volume 316 (2) (2007), pp. 746-776 | DOI | MR | Zbl

[12] J. S. Birman; K.-H. Ko; S. J. Lee A new approach to the word and conjugacy problems in the braid groups, Adv. Math., Volume 139 (2) (1998), pp. 322-353 | DOI | MR | Zbl

[13] J. S. Birman; A. Lubotzky; J. McCarthy Abelian and solvable subgroups of the mapping class groups, Duke Math. J., Volume 50 (4) (1983), pp. 1107-1120 | DOI | MR | Zbl

[14] F. Bohnenblust The algebraical braid group, Ann. of Math. (2), Volume 48 (1947), pp. 127-136 | DOI | MR | Zbl

[15] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symbolic Comput., Volume 24 (1997) no. 3-4, pp. 235-265 Computational algebra and number theory (London, 1993) | DOI | MR | Zbl

[16] E. Brieskorn; K. Saito Artin-Gruppen und Coxeter-Gruppen, Invent. Math., Volume 17 (1972), pp. 245-271 | DOI | MR | Zbl

[17] J. C. Cha; C. Livingstone; M. Durbin Braid group calculator

[18] R. Charney Artin groups of finite type are biautomatic, Math. Ann., Volume 292 (4) (1992), pp. 671-683 | DOI | MR | Zbl

[19] W.-L. Chow On the algebraical braid group, Ann. of Math. (2), Volume 49 (1948), pp. 654-658 | DOI | MR | Zbl

[20] A. M. Cohen; D. B. Wales Linearity of Artin groups of finite type, Israel J. Math., Volume 131 (2002), pp. 101-123 | DOI | MR | Zbl

[21] A. Constantin; B. Kolev The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, L’Enseign. Math., Volume 40 (1994), pp. 193-204 | MR | Zbl

[22] P. Dehornoy Braid groups and left distributive operations, Trans. Amer. Math. Soc., Volume 345 (1) (1994), pp. 115-150 | DOI | MR | Zbl

[23] P. Dehornoy Left-Garside categories, self-distributivity, and braids, Ann. Math. Blaise Pascal, Volume 16 (2009), pp. 189-244 | DOI | Numdam | MR | Zbl

[24] P. Dehornoy; I. Dynnikov; D. Rolfsen; B. Wiest Why are braids orderable?, Panoramas et Synthèses 14. Société Mathématique de France, Paris, 2002 | MR | Zbl

[25] P. Dehornoy; I. Dynnikov; D. Rolfsen; B. Wiest Ordering braids, Mathematical Surveys and Monographs, 148. American Mathematical Society, Providence, RI, 2008 | MR | Zbl

[26] P. Dehornoy; L. Paris Gaussian groups and Garside groups, two generalisations of Artin groups., Proc. London Math. Soc. (3), Volume 79 (3) (1999), pp. 569-604 | DOI | MR | Zbl

[27] F. Digne On the linearity of Artin braid groups, J. Algebra, Volume 268 (1) (2003), pp. 39-57 | DOI | MR | Zbl

[28] F. Digne; J. Michel Garside and locally Garside categories (2006) (arxiv.org/abs/math/0612652)

[29] S. Eilenberg Sur les transformations périodiques de la surface de la sphère, Fund. Math., Volume 22 (1934), pp. 28-44 | Zbl

[30] E. A. El-Rifai; H. R. Morton Algorithms for positive braids, Quart. J. Math. Oxford Ser. (2), Volume 45 (180) (1994), pp. 479-497 | DOI | MR | Zbl

[31] D. B. A. Epstein; J. W. Cannon; D. F. Holt; S. V. F. Levy; M. S. Paterson; W. P. Thurston Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992 | MR | Zbl

[32] E. Fadell; L. Neuwirth Configuration spaces, Math. Scand., Volume 10 (1962), pp. 111-118 | MR | Zbl

[33] E. Fadell; J. Van Buskirk The braid groups of E 2 and S 2 , Duke Math. J., Volume 29 (1962), pp. 243-257 | DOI | MR | Zbl

[34] R. Fenn; M. T. Greene; D. Rolfsen; C. Rourke; B. Wiest Ordering the braid groups, Pacific J. of Math., Volume 191 (1) (1999), pp. 49-74 | DOI | MR | Zbl

[35] R. Fox; L. Neuwirth The braid groups, Math. Scand., Volume 10 (1962), pp. 119-126 | MR | Zbl

[36] N. Franco; J. González-Meneses Conjugacy problem for braid groups and Garside groups, J. Algebra, Volume 266 (1) (2003), pp. 112-132 | DOI | MR | Zbl

[37] F. A. Garside The braid group and other groups, Quart. J. Math. Oxford Ser. (2), Volume 20 (1969), pp. 235-254 | DOI | MR | Zbl

[38] V. Gebhardt A new approach to the conjugacy problem in Garside groups, J. Algebra, Volume 292 (1) (2005), pp. 282-302 | DOI | MR | Zbl

[39] Volker Gebhardt; Juan González-Meneses The cyclic sliding operation in Garside groups, Math. Z., Volume 265 (2010) no. 1, pp. 85-114 | DOI | MR

[40] Volker Gebhardt; Juan González-Meneses Solving the conjugacy problem in Garside groups by cyclic sliding, Journal of Symbolic Computation, Volume 45 (2010) no. 6, pp. 629 -656 http://www.sciencedirect.com/science/article/B6WM7-4Y9CF87-1/2/5586e319f008d37a633c1f164a76aede | DOI | MR

[41] M. Geck; G. Hiß; F. Lübeck; G. Malle; J. Michel; G. Pfeiffer CHEVIE: computer algebra package for GAP3. (http://people.math.jussieu.fr/~jmichel/chevie/chevie.html)

[42] J. González-Meneses Personal web page, http://personal.us.es/meneses

[43] J. González-Meneses The n-th root of a braid is unique up to conjugacy, Alg. and Geom. Topology, Volume 3 (2003), pp. 1103-1118 | DOI | MR | Zbl

[44] J. González-Meneses On reduction curves and Garside properties of braids, Contemporary Mathematics, Volume 538 (2011), pp. 227-244

[45] J. González-Meneses; B. Wiest On the structure of the centralizer of a braid, Ann. Sci. École Norm. Sup. (4), Volume 37 (5) (2004), pp. 729-757 | Numdam | MR | Zbl

[46] M. Hall Subgroups of finite index in free groups, Canadian J. of Math., Volume 1 (1949), pp. 187-190 | DOI | MR | Zbl

[47] Jean-Yves Hée Une démonstration simple de la fidélité de la représentation de Lawrence-Krammer-Paris, J. Algebra, Volume 321 (2009) no. 3, pp. 1039-1048 | DOI | MR | Zbl

[48] A. Hurwitz Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1) (1891), pp. 1-60 | DOI | MR

[49] N. V. Ivanov Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115. American Mathematical Society, Providence, RI, 1992 | MR | Zbl

[50] C Kassel; V. Turaev Braid groups, Graduate Texts in Mathematics, 247. Springer, New York, 2008 | MR

[51] B. von Kerékjártó Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., Volume 80 (1919-1920), pp. 36-38 | DOI | MR

[52] D. Krammer The braid group B 4 is linear, Invent. Math., Volume 142 (3) (2000), pp. 451-486 | DOI | MR | Zbl

[53] D. Krammer Braid groups are linear, Ann. of Math. (2), Volume 155 (1) (2002), pp. 131-156 | DOI | MR | Zbl

[54] D. Krammer A class of Garside groupoid structures on the pure braid group (2005) (arxiv.org/abs/math/0509165) | Zbl

[55] E.-K. Lee; S. J. Lee A Garside-theoretic approach to the reducibility problem in braid groups, J. Algebra, Volume 320 (2) (2008), pp. 783-820 | DOI | MR | Zbl

[56] F. Levi Über die Untergruppen der freien gruppen II, Math. Z., Volume 37 (1933), pp. 90-97 | DOI | MR

[57] W. Magnus Über Automorphismen von Fundamentalgruppen berandeter Flächen., Math. Ann., Volume 109 (1934), pp. 617-646 | DOI | MR

[58] W. Magnus Residually finite groups, Bull. Amer. Math. Soc., Volume 75 (1969), pp. 305-316 | DOI | MR | Zbl

[59] W. Magnus; A. Karrass; D. Solitar Combinatorial group theory, Interscience Publishers (John Wiley & Sons, Inc.), New York-London-Sydney, 1966 | MR | Zbl

[60] A. I. Mal’cev On isomorphic matrix representations of infinite groups, Mat. Sb., Volume 182 (1940), pp. 142-149

[61] I. Marin On the residual nilpotence of pure Artin groups, J. Group Theory, Volume 9 (4) (2006), pp. 483-485 | DOI | MR | Zbl

[62] A. Markoff Foundations of the algebraic theory of tresses. (Russian), Trav. Inst. Math. Stekloff, Volume 16 (1945), pp. 53 pp. | MR | Zbl

[63] J. D. McCarthy Normalizers and Centralizers of pseudo-Anosov mapping classes (1982) (Preprint)

[64] J. Nielsen Abbildungsklassen endlicher Ordnung, Acta Math., Volume 75 (1943), pp. 23-115 | DOI | MR | Zbl

[65] O. Ore Linear equations in non-commutative fields, Ann. of Math. (2), Volume 32 (3) (1931), pp. 463-477 | DOI | MR

[66] P. Orlik; H. Terao Arrangements of hyperplanes., Grundlehren der Mathematischen Wissenschaften, 300. Springer-Verlag, Berlin, 1992 | MR | Zbl

[67] L. Paris Artin monoids inject in their groups, Commen. Math. Helv., Volume 77 (3) (2002), pp. 609-637 | DOI | MR | Zbl

[68] L. Paris; A. Papadopoulos. Braid groups and Artin groups, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., 13. Eur. Math. Soc., 2009, pp. 389-451 | MR

[69] W. P. Thurston On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., Volume 19 (2) (1988), pp. 417-431 | DOI | MR | Zbl

[70] O. Zariski On the Poincaré group of rational plane curves, Amer. J. of Math., Volume 58 (3) (1936), pp. 607-619 | DOI | MR

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