González-Meneses, Juan
Basic results on braid groups  [ Resultats basiques dans les groupes de tresses. ]
Annales mathématiques Blaise Pascal, Tome 18 (2011) no. 1 , p. 15-59
MR 2830088 | Zbl pre05903953
doi : 10.5802/ambp.293
URL stable : http://www.numdam.org/item?id=AMBP_2011__18_1_15_0

Classification:  20F36
Mots clés: Tresses, groupes d’Artin-Tits
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.

Bibliographie

[1] Alexander, J. W. On the Deformation of an n-Cell, Proc. of the Nat. Acad. of Sci. of the USA., 9 (12) (1923), p. 406-407 Article  Zbl 49.0407.01

[2] Artin, E. Theorie der Zöpfe, Abh. Math. Sem. Hamburgischen Univ., 4 (1925), p. 47-72 Article  Zbl 51.0450.01

[3] Artin, E. The theory of braids, Annals of Math., 48 (1947), p. 101-126 Article  MR 19087 | Zbl 0030.17703

[4] Bacardit, L.; Dicks, W. Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue, Groups - Complexity - Criptology, 1 (2009), p. 77-129 Article  MR 2502938 | Zbl 1195.20041

[5] Baumslag, G. Automorphisms groups of residually finite groups, J. London Math. Soc., 38 (1963), p. 117-118 Article  MR 146271 | Zbl 0124.26003

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

[7] Bessis, D.; Digne, F.; Michel, J. Springer theory in braid groups and the Birman-Ko-Lee monoid, Pacific J. Math., 205 (2) (2002), p. 287-309 Article  MR 1922736 | Zbl 1056.20023

[8] Bigelow, S. J. Braid groups are linear, J. Amer. Math. Soc., 14 (2) (2001), p. 471-486 Article  MR 1815219 | Zbl 0988.20021

[9] Birman, J. S. braids, links and mapping class groups. Annals of Mathematics Studies, No. 82., Princeton University Press, Princeton, N.J. (1974) MR 375281 | Zbl 0305.57013

[10] Birman, J. S.; Gebhardt, V.; González-Meneses, J. Conjugacy in Garside groups. I. Cyclings, powers and rigidity, Groups Geom. Dyn., 1 (3) (2007), p. 221-279 Article  MR 2314045 | Zbl 1160.20026

[11] Birman, J. S.; Gebhardt, V.; González-Meneses, J. Conjugacy in Garside groups. III. Periodic braids, J. Algebra, 316 (2) (2007), p. 746-776 Article  MR 2358613 | Zbl 1165.20031

[12] Birman, J. S.; Ko, K.-H.; Lee, S. J. A new approach to the word and conjugacy problems in the braid groups, Adv. Math., 139 (2) (1998), p. 322-353 Article  MR 1654165 | Zbl 0937.20016

[13] Birman, J. S.; Lubotzky, A.; Mccarthy, J. Abelian and solvable subgroups of the mapping class groups, Duke Math. J., 50 (4) (1983), p. 1107-1120 Article  MR 726319 | Zbl 0551.57004

[14] Bohnenblust, F. The algebraical braid group, Ann. of Math. (2), 48 (1947), p. 127-136 Article  MR 19088 | Zbl 0030.17801

[15] Bosma, Wieb; Cannon, John; Playoust, Catherine The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997) no. 3-4, p. 235–265 (Computational algebra and number theory (London, 1993)) Article  MR 1484478 | Zbl 0898.68039

[16] Brieskorn, E.; Saito, K. Artin-Gruppen und Coxeter-Gruppen, Invent. Math., 17 (1972), p. 245-271 Article  MR 323910 | Zbl 0243.20037

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

[18] Charney, R. Artin groups of finite type are biautomatic, Math. Ann., 292 (4) (1992), p. 671-683 Article  MR 1157320 | Zbl 0736.57001

[19] Chow, W.-L. On the algebraical braid group, Ann. of Math. (2), 49 (1948), p. 654-658 Article  MR 26050 | Zbl 0033.01002

[20] Cohen, A. M.; Wales, D. B. Linearity of Artin groups of finite type, Israel J. Math., 131 (2002), p. 101-123 Article  MR 1942303 | Zbl 1078.20038

[21] Constantin, A.; Kolev, B. The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, L’Enseign. Math., 40 (1994), p. 193-204 MR 1309126 | Zbl 0852.57012

[22] Dehornoy, P. Braid groups and left distributive operations, Trans. Amer. Math. Soc., 345 (1) (1994), p. 115-150 Article  MR 1214782 | Zbl 0837.20048

[23] Dehornoy, P. Left-Garside categories, self-distributivity, and braids, Ann. Math. Blaise Pascal, 16 (2009), p. 189-244 Article  Numdam | MR 2568862 | Zbl 1183.18004

[24] Dehornoy, P.; Dynnikov, I.; Rolfsen, D.; Wiest, B. Why are braids orderable?, Panoramas et Synthèses 14. Société Mathématique de France, Paris (2002) MR 1988550 | Zbl 1048.20021

[25] Dehornoy, P.; Dynnikov, I.; Rolfsen, D.; Wiest, B. Ordering braids, Mathematical Surveys and Monographs, 148. American Mathematical Society, Providence, RI (2008) MR 2463428 | Zbl 1163.20024

[26] Dehornoy, P.; Paris, L. Gaussian groups and Garside groups, two generalisations of Artin groups., Proc. London Math. Soc. (3), 79 (3) (1999), p. 569-604 Article  MR 1710165 | Zbl 1030.20021

[27] Digne, F. On the linearity of Artin braid groups, J. Algebra, 268 (1) (2003), p. 39-57 Article  MR 2004479 | Zbl 1066.20044

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

[29] Eilenberg, S. Sur les transformations périodiques de la surface de la sphère, Fund. Math., 22 (1934), p. 28-44 Zbl 0008.37109

[30] El-Rifai, E. A.; Morton, H. R. Algorithms for positive braids, Quart. J. Math. Oxford Ser. (2), 45 (180) (1994), p. 479-497 Article  MR 1315459 | Zbl 0839.20051

[31] Epstein, D. B. A.; Cannon, J. W.; Holt, D. F.; Levy, S. V. F.; Paterson, M. S.; Thurston, W. P. Word processing in groups, Jones and Bartlett Publishers, Boston, MA (1992) MR 1161694 | Zbl 0764.20017

[32] Fadell, E.; Neuwirth, L. Configuration spaces, Math. Scand., 10 (1962), p. 111-118 MR 141126 | Zbl 0136.44104

[33] Fadell, E.; Van Buskirk, J. The braid groups of E 2 and S 2 , Duke Math. J., 29 (1962), p. 243-257 Article  MR 141128 | Zbl 0122.17804

[34] Fenn, R.; Greene, M. T.; Rolfsen, D.; Rourke, C.; Wiest, B. Ordering the braid groups, Pacific J. of Math., 191 (1) (1999), p. 49-74 Article  MR 1725462 | Zbl 1009.20042

[35] Fox, R.; Neuwirth, L. The braid groups, Math. Scand., 10 (1962), p. 119-126 MR 150755 | Zbl 0117.41101

[36] Franco, N.; González-Meneses, J. Conjugacy problem for braid groups and Garside groups, J. Algebra, 266 (1) (2003), p. 112-132 Article  MR 1994532 | Zbl 1043.20019

[37] Garside, F. A. The braid group and other groups, Quart. J. Math. Oxford Ser. (2), 20 (1969), p. 235-254 Article  MR 248801 | Zbl 0194.03303

[38] Gebhardt, V. A new approach to the conjugacy problem in Garside groups, J. Algebra, 292 (1) (2005), p. 282-302 Article  MR 2166805 | Zbl 1105.20032

[39] Gebhardt, Volker; González-Meneses, Juan The cyclic sliding operation in Garside groups, Math. Z., 265 (2010) no. 1, p. 85–114 Article  MR 2606950 | Zbl pre05700566

[40] Gebhardt, Volker; González-Meneses, Juan Solving the conjugacy problem in Garside groups by cyclic sliding, Journal of Symbolic Computation, 45 (2010) no. 6, p. 629 - 656 Article  MR 2639308 | Zbl pre05710825

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

[42] González-Meneses, J. Personal web page,

[43] González-Meneses, J. The n-th root of a braid is unique up to conjugacy, Alg. and Geom. Topology, 3 (2003), p. 1103-1118 Article  MR 2012967 | Zbl 1063.20041

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

[45] González-Meneses, J.; Wiest, B. On the structure of the centralizer of a braid, Ann. Sci. École Norm. Sup. (4), 37 (5) (2004), p. 729-757 Numdam | MR 2103472 | Zbl 1082.20024

[46] Hall, M. Subgroups of finite index in free groups, Canadian J. of Math., 1 (1949), p. 187-190 Article  MR 28836 | Zbl 0031.34001

[47] Hée, Jean-Yves Une démonstration simple de la fidélité de la représentation de Lawrence-Krammer-Paris, J. Algebra, 321 (2009) no. 3, p. 1039–1048 Article  MR 2488566 | Zbl 1163.20025

[48] Hurwitz, A. Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., 39 (1) (1891), p. 1–60 Article  MR 1510692

[49] Ivanov, N. V. Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115. American Mathematical Society, Providence, RI (1992) MR 1195787 | Zbl 0776.57001

[50] Kassel, C; Turaev, V. Braid groups, Graduate Texts in Mathematics, 247. Springer, New York (2008) MR 2435235 | Zbl pre05268073

[51] Kerékjártó, B. Von Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., 80 (1919-1920), p. 36–38 Article  MR 1511945

[52] Krammer, D. The braid group B 4 is linear, Invent. Math., 142 (3) (2000), p. 451-486 Article  MR 1804157 | Zbl 0988.20023

[53] Krammer, D. Braid groups are linear, Ann. of Math. (2), 155 (1) (2002), p. 131-156 Article  MR 1888796 | Zbl 1020.20025

[54] Krammer, D. A class of Garside groupoid structures on the pure braid group (2005) (arxiv.org/abs/math/0509165) Zbl 1194.20040

[55] Lee, E.-K.; Lee, S. J. A Garside-theoretic approach to the reducibility problem in braid groups, J. Algebra, 320 (2) (2008), p. 783-820 Article  MR 2422316 | Zbl 1191.20034

[56] Levi, F. Über die Untergruppen der freien gruppen II, Math. Z., 37 (1933), p. 90-97 Article  MR 1545385

[57] Magnus, W. Über Automorphismen von Fundamentalgruppen berandeter Flächen., Math. Ann., 109 (1934), p. 617-646 Article  MR 1512913

[58] Magnus, W. Residually finite groups, Bull. Amer. Math. Soc., 75 (1969), p. 305-316 Article  MR 241525 | Zbl 0196.04704

[59] Magnus, W.; Karrass, A.; Solitar, D. Combinatorial group theory, Interscience Publishers (John Wiley & Sons, Inc.), New York-London-Sydney (1966) MR 207802 | Zbl 0138.25604

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

[61] Marin, I. On the residual nilpotence of pure Artin groups, J. Group Theory, 9 (4) (2006), p. 483-485 Article  MR 2243240 | Zbl 1103.20035

[62] Markoff, A. Foundations of the algebraic theory of tresses. (Russian), Trav. Inst. Math. Stekloff, 16 (1945), p. 53 pp. MR 17279 | Zbl 0061.02507

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

[64] Nielsen, J. Abbildungsklassen endlicher Ordnung, Acta Math., 75 (1943), p. 23-115 Article  MR 13306 | Zbl 0027.26601

[65] Ore, O. Linear equations in non-commutative fields, Ann. of Math. (2), 32 (3) (1931), p. 463-477 Article  MR 1503010

[66] Orlik, P.; Terao, H. Arrangements of hyperplanes., Grundlehren der Mathematischen Wissenschaften, 300. Springer-Verlag, Berlin (1992) MR 1217488 | Zbl 0757.55001

[67] Paris, L. Artin monoids inject in their groups, Commen. Math. Helv., 77 (3) (2002), p. 609-637 Article  MR 1933791 | Zbl 1020.20026

[68] Paris, L.; Papadopoulos., A. Braid groups and Artin groups, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., 13. Eur. Math. Soc. (2009), p. 389-451 MR 2497781 | Zbl pre05560291

[69] Thurston, W. P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (2) (1988), p. 417-431 Article  MR 956596 | Zbl 0674.57008

[70] Zariski, O. On the Poincaré group of rational plane curves, Amer. J. of Math., 58 (3) (1936), p. 607-619 Article  MR 1507185