# zbMATH — the first resource for mathematics

Basic results on braid groups. (Resultats basiques dans les groupes de tresses.) (English. French summary) Zbl 1264.20035
Summary: 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.

##### MSC:
 20F36 Braid groups; Artin groups 20F05 Generators, relations, and presentations of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
Magma; CHEVIE
Full Text:
##### References:
 [1] Alexander, J. W., On the deformation of an n-cell, Proc. of the Nat. Acad. of Sci. of the USA., 9 (12), 406-407, (1923) · JFM 49.0407.01 [2] Artin, E., Theorie der Zöpfe, Abh. Math. Sem. Hamburgischen Univ., 4, 47-72, (1925) · JFM 51.0450.01 [3] Artin, E., The theory of braids, Annals of Math., 48, 101-126, (1947) · 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, 77-129, (2009) · Zbl 1195.20041 [5] Baumslag, G., Automorphisms groups of residually finite groups, J. London Math. Soc., 38, 117-118, (1963) · Zbl 0124.26003 [6] Bessis, D., Garside categories, periodic loops and cyclic sets, (2006) [7] Bessis, D.; Digne, F.; Michel, J., Springer theory in braid groups and the Birman-ko-Lee monoid, Pacific J. Math., 205 (2), 287-309, (2002) · Zbl 1056.20023 [8] Bigelow, S. J., Braid groups are linear, J. Amer. Math. Soc., 14 (2), 471-486, (2001) · Zbl 0988.20021 [9] Birman, J. S., braids, links and mapping class groups. Annals of Mathematics Studies, No. 82., (1974), Princeton University Press, Princeton, N.J. · 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), 221-279, (2007) · Zbl 1160.20026 [11] Birman, J. S.; Gebhardt, V.; González-Meneses, J., Conjugacy in garside groups. III. periodic braids, J. Algebra, 316 (2), 746-776, (2007) · 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), 322-353, (1998) · 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), 1107-1120, (1983) · Zbl 0551.57004 [14] Bohnenblust, F., The algebraical braid group, Ann. of Math. (2), 48, 127-136, (1947) · Zbl 0030.17801 [15] Bosma, Wieb; Cannon, John; Playoust, Catherine, The magma algebra system. I. the user language, J. Symbolic Comput., 24, 3-4, 235-265, (1997) · Zbl 0898.68039 [16] Brieskorn, E.; Saito, K., Artin-gruppen und Coxeter-gruppen, Invent. Math., 17, 245-271, (1972) · 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), 671-683, (1992) · Zbl 0736.57001 [19] Chow, W.-L., On the algebraical braid group, Ann. of Math. (2), 49, 654-658, (1948) · Zbl 0033.01002 [20] Cohen, A. M.; Wales, D. B., Linearity of Artin groups of finite type, Israel J. Math., 131, 101-123, (2002) · 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, 193-204, (1994) · Zbl 0852.57012 [22] Dehornoy, P., Braid groups and left distributive operations, Trans. Amer. Math. Soc., 345 (1), 115-150, (1994) · Zbl 0837.20048 [23] Dehornoy, P., Left-garside categories, self-distributivity, and braids, Ann. Math. Blaise Pascal, 16, 189-244, (2009) · Zbl 1183.18004 [24] Dehornoy, P.; Dynnikov, I.; Rolfsen, D.; Wiest, B., Why are braids orderable?, (2002), Panoramas et Synthèses 14. Société Mathématique de France, Paris · Zbl 1048.20021 [25] Dehornoy, P.; Dynnikov, I.; Rolfsen, D.; Wiest, B., Ordering braids, (2008), Mathematical Surveys and Monographs, 148. American Mathematical Society, Providence, RI · 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), 569-604, (1999) · Zbl 1030.20021 [27] Digne, F., On the linearity of Artin braid groups, J. Algebra, 268 (1), 39-57, (2003) · Zbl 1066.20044 [28] Digne, F.; Michel, J., Garside and locally Garside categories, (2006) [29] Eilenberg, S., Sur LES transformations périodiques de la surface de la sphère, Fund. Math., 22, 28-44, (1934) · Zbl 0008.37109 [30] El-Rifai, E. A.; Morton, H. R., Algorithms for positive braids, Quart. J. Math. Oxford Ser. (2), 45 (180), 479-497, (1994) · 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, (1992), Jones and Bartlett Publishers, Boston, MA · Zbl 0764.20017 [32] Fadell, E.; Neuwirth, L., Configuration spaces, Math. Scand., 10, 111-118, (1962) · Zbl 0136.44104 [33] Fadell, E.; Van Buskirk, J., The braid groups of $$E^2$$ and $$S^2,$$ Duke Math. J., 29, 243-257, (1962) · 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), 49-74, (1999) · Zbl 1009.20042 [35] Fox, R.; Neuwirth, L., The braid groups, Math. Scand., 10, 119-126, (1962) · Zbl 0117.41101 [36] Franco, N.; González-Meneses, J., Conjugacy problem for braid groups and garside groups, J. Algebra, 266 (1), 112-132, (2003) · Zbl 1043.20019 [37] Garside, F. A., The braid group and other groups, Quart. J. Math. Oxford Ser. (2), 20, 235-254, (1969) · Zbl 0194.03303 [38] Gebhardt, V., A new approach to the conjugacy problem in garside groups, J. Algebra, 292 (1), 282-302, (2005) · Zbl 1105.20032 [39] Gebhardt, Volker; González-Meneses, Juan, The cyclic sliding operation in garside groups, Math. Z., 265, 1, 85-114, (2010) · Zbl 1253.20034 [40] Gebhardt, Volker; González-Meneses, Juan, Solving the conjugacy problem in garside groups by cyclic sliding, Journal of Symbolic Computation, 45, 6, 629 -656, (2010) · Zbl 1235.20032 [41] Geck, M.; Hiß, G.; Lübeck, F.; Malle, G.; Michel, J.; Pfeiffer, G., CHEVIE: computer algebra package for GAP3. [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, 1103-1118, (2003) · Zbl 1063.20041 [44] González-Meneses, J., On reduction curves and garside properties of braids, Contemporary Mathematics, 538, 227-244, (2011) · Zbl 1255.20035 [45] González-Meneses, J.; Wiest, B., On the structure of the centralizer of a braid, Ann. Sci. École Norm. Sup. (4), 37 (5), 729-757, (2004) · Zbl 1082.20024 [46] Hall, M., Subgroups of finite index in free groups, Canadian J. of Math., 1, 187-190, (1949) · Zbl 0031.34001 [47] Hée, Jean-Yves, Une démonstration simple de la fidélité de la représentation de Lawrence-krammer-Paris, J. Algebra, 321, 3, 1039-1048, (2009) · Zbl 1163.20025 [48] Hurwitz, A., Über riemannsche flächen mit gegebenen verzweigungspunkten, Math. Ann., 39 (1), 1-60, (1891) · JFM 23.0429.01 [49] Ivanov, N. V., Subgroups of Teichmüller modular groups, (1992), Translations of Mathematical Monographs, 115. American Mathematical Society, Providence, RI · Zbl 0776.57001 [50] Kassel, C; Turaev, V., Braid groups, (2008), Graduate Texts in Mathematics, 247. Springer, New York [51] Kerékjártó, B. von, Über die periodischen transformationen der kreisscheibe und der kugelfläche, Math. Ann., 80, 36-38, (19191920) · JFM 47.0526.05 [52] Krammer, D., The braid group $$B_4$$ is linear, Invent. Math., 142 (3), 451-486, (2000) · Zbl 0988.20023 [53] Krammer, D., Braid groups are linear, Ann. of Math. (2), 155 (1), 131-156, (2002) · Zbl 1020.20025 [54] Krammer, D., A class of Garside groupoid structures on the pure braid group, (2005) · 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), 783-820, (2008) · Zbl 1191.20034 [56] Levi, F., Über die untergruppen der freien gruppen II, Math. Z., 37, 90-97, (1933) · Zbl 0006.24601 [57] Magnus, W., Über automorphismen von fundamentalgruppen berandeter flächen., Math. Ann., 109, 617-646, (1934) · Zbl 0009.03901 [58] Magnus, W., Residually finite groups, Bull. Amer. Math. Soc., 75, 305-316, (1969) · Zbl 0196.04704 [59] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory, (1966), Interscience Publishers (John Wiley & Sons, Inc.), New York-London-Sydney · Zbl 0138.25604 [60] Mal’cev, A. I., On isomorphic matrix representations of infinite groups, Mat. Sb., 182, 142-149, (1940) [61] Marin, I., On the residual nilpotence of pure Artin groups, J. Group Theory, 9 (4), 483-485, (2006) · Zbl 1103.20035 [62] Markoff, A., Foundations of the algebraic theory of tresses. (Russian), Trav. Inst. Math. Stekloff, 16, 53 pp. pp., (1945) · Zbl 0061.02507 [63] McCarthy, J. D., Normalizers and Centralizers of pseudo-Anosov mapping classes, (1982) [64] Nielsen, J., Abbildungsklassen endlicher ordnung, Acta Math., 75, 23-115, (1943) · Zbl 0027.26601 [65] Ore, O., Linear equations in non-commutative fields, Ann. of Math. (2), 32 (3), 463-477, (1931) · JFM 57.0166.01 [66] Orlik, P.; Terao, H., Arrangements of hyperplanes., (1992), Grundlehren der Mathematischen Wissenschaften, 300. Springer-Verlag, Berlin · Zbl 0757.55001 [67] Paris, L., Artin monoids inject in their groups, Commen. Math. Helv., 77 (3), 609-637, (2002) · Zbl 1020.20026 [68] Paris, L.; Papadopoulos., A., Handbook of Teichmüller theory. Vol. II, Braid groups and Artin groups, 389-451, (2009), IRMA Lect. Math. Theor. Phys., 13. Eur. Math. Soc. [69] Thurston, W. P., On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (2), 417-431, (1988) · Zbl 0674.57008 [70] Zariski, O., On the Poincaré group of rational plane curves, Amer. J. of Math., 58 (3), 607-619, (1936) · JFM 62.0758.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.