Left-Garside categories, self-distributivity, and braids

Dehornoy, Patrick

Annales mathématiques Blaise Pascal, Tome 16 (2009), p. 189-244 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé

In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the relationship between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This approach leads to a realistic program for establishing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhaüser (2000), Chap. IX].

Publié le : 2009-01-01

MR 2568862 | Zbl 1183.18004 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/ambp.263
Classification: 18B40, 20N02, 20F36

@article{AMBP_2009__16_2_189_0,
     author = {Dehornoy, Patrick},
     title = {Left-Garside categories, self-distributivity, and braids},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     year = {2009},
     pages = {189-244},
     doi = {10.5802/ambp.263},
     zbl = {1183.18004},
     mrnumber = {2568862},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2009__16_2_189_0}
}

Dehornoy, Patrick. Left-Garside categories, self-distributivity, and braids. Annales mathématiques Blaise Pascal, Tome 16 (2009) pp. 189-244. doi : 10.5802/ambp.263. http://gdmltest.u-ga.fr/item/AMBP_2009__16_2_189_0/

Bibliographie

[1] Adyan, S.I. Fragments of the word Delta in a braid group, Mat. Zametki Acad. Sci. SSSR, Tome 36 (1984) no. 1, pp. 25-34 ((Russian); English translation in Math. Notes of the Acad. Sci. USSR 36 (1984), no. 1, p. 505–510) | MR 757642 | Zbl 0599.20044

[2] Bessis, D. Garside categories, periodic loops and cyclic sets (math.GR/0610778)

[3] Bessis, D. The dual braid monoid, Ann. Sci. École Norm. Sup., Tome 36 (2003), pp. 647-683 | Numdam | MR 2032983 | Zbl 1064.20039

[4] Bessis, D. A dual braid monoid for the free group, J. Algebra, Tome 302 (2006), pp. 55-69 | Article | MR 2236594 | Zbl pre05072877

[5] Bessis, D.; Corran, Ruth Garside structure for the braid group of G(e,e,r) (math.GR/0306186)

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

[7] Birman, J.; Gebhardt, V.; González-Meneses, J. Conjugacy in Garside groups III: Periodic braids, J. Algebra, Tome 316 (2007), pp. 746-776 | Article | MR 2358613 | Zbl pre05222091

[8] Birman, J.; Gebhardt, V.; González-Meneses, J. Conjugacy in Garside groups II: Structure of the ultra summit set, Groups Geom. Dyn., Tome 2 (2008), pp. 16-31 | MR 2367207 | Zbl 1163.20023

[9] Birman, J.; Ko, K.H.; Lee, S.J. A new approach to the word problem in the braid groups, Adv. Math., Tome 139 (1998) no. 2, pp. 322-353 | Article | MR 1654165 | Zbl 0937.20016

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

[11] Cannon, J.W.; Floyd, W.J.; Parry, W.R. Introductory notes on Richard Thompson’s groups, Enseign. Math., Tome 42 (1996), pp. 215-257 | MR 1426438 | Zbl 0880.20027

[12] Charney, R. Artin groups of finite type are biautomatic, Math. Ann., Tome 292 (1992) no. 4, pp. 671-683 | Article | MR 1157320 | Zbl 0736.57001

[13] Charney, R.; Meier, J. The language of geodesics for Garside groups, Math. Zeitschr., Tome 248 (2004), pp. 495-509 | Article | MR 2097371 | Zbl 1062.57002

[14] Charney, R.; Meier, J.; Whittlesey, K. Bestvina’s normal form complex and the homology of Garside groups, Geom. Dedicata, Tome 105 (2004), pp. 171-188 | Article | MR 2057250 | Zbl 1064.20044

[15] Crisp, J.; Paris, L. Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups, Pac. J. Maths, Tome 221 (2005), pp. 1-27 | Article | MR 2194143 | Zbl 1147.20033

[16] Dehornoy, P. $Π_{1}^{1}$ -complete families of elementary sequences, Ann. P. Appl. Logic, Tome 38 (1988), pp. 257-287 | Article | MR 942526 | Zbl 0646.03030

[17] Dehornoy, P. Free distributive groupoids, J. Pure Appl. Algebra, Tome 61 (1989), pp. 123-146 | Article | MR 1025918 | Zbl 0686.20041

[18] Dehornoy, P. Braids and Self-Distributivity, Birkhäuser, Progr. Math., Tome 192 (2000) | MR 1778150 | Zbl 0958.20033

[19] Dehornoy, P. Groupes de Garside, Ann. Sci. École Norm. Sup. (4), Tome 35 (2002), pp. 267-306 | Numdam | MR 1914933 | Zbl 1017.20031

[20] Dehornoy, P. Study of an identity, Algebra Universalis, Tome 48 (2002), pp. 223-248 | Article | MR 1929906 | Zbl 1058.03043

[21] Dehornoy, P. Complete positive group presentations, J. Algebra, Tome 268 (2003), pp. 156-197 | Article | MR 2004483 | Zbl 1067.20035

[22] Dehornoy, P. Geometric presentations of Thompson’s groups, J. Pure Appl. Algebra, Tome 203 (2005), pp. 1-44 | Article | MR 2176650 | Zbl 1150.20016

[23] Dehornoy, P.; Paris, L. Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc., Tome 79 (1999) no. 3, pp. 569-604 | Article | MR 1710165 | Zbl 1030.20021

[24] Dehornoy, With I. Dynnikov, D. Rolfsen, And B. Wiest, P. Ordering braids, Math. Surveys and Monographs vol. 148, Amer. Math. Soc. (2008) | MR 2463428 | Zbl 1163.20024

[25] Deligne, P.; Lusztig, G. Representations of reductive groups over finite fields, Ann. of Math., Tome 103 (1976), pp. 103-161 | Article | MR 393266 | Zbl 0336.20029

[26] Digne, F. Présentations duales pour les groupes de tresses de type affine $\tilde{A}$ , Comm. Math. Helvetici, Tome 8 (2008), pp. 23-47 | MR 2208796 | Zbl 1143.20020

[27] Digne, F.; Michel, J. Garside and locally Garside categories (arXiv: math.GR/0612652)

[28] El-Rifai, E.A.; Morton, H.R. Algorithms for positive braids, Quart. J. Math. Oxford Ser., Tome 45 (1994) no. 2, pp. 479-497 | Article | MR 1315459 | Zbl 0839.20051

[29] Epstein, D.; Cannon, J.W.; Holt, D.F.; Levy, S.V.F.; Paterson, M.S.; Thurston, W.P. Word Processing in Groups, Jones and Bartlett Publ. (1992) | MR 1161694 | Zbl 0764.20017

[30] Fenn, R.; Rourke, C.P. Racks and links in codimension 2, J. Knot Theory Ramifications, Tome 1 (1992), pp. 343-406 | Article | MR 1194995 | Zbl 0787.57003

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

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

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

[34] Godelle, E. Parabolic subgroups of Garside groups II (math.GR/0811.0751)

[35] Godelle, E. Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groupes d’Artin-Tits, Université d’Amiens (2001) (PhD. Thesis)

[36] Godelle, E. Parabolic subgroups of Garside groups, J. Algebra, Tome 317 (2007), pp. 1-16 | Article | MR 2360138 | Zbl pre05226979

[37] Joyce, D. A classifying invariant of knots: the knot quandle, J. Pure Appl. Algebra, Tome 23 (1982), pp. 37-65 | Article | MR 638121 | Zbl 0474.57003

[38] Kassel, C.; Turaev, V. Braid groups, Springer Verlag, Grad. Texts in Math. (2008) | MR 2435235 | Zbl pre05268073

[39] Krammer, D. A class of Garside groupoid structures on the pure braid group, Trans. Amer. Math. Soc., Tome 360 (2008), pp. 4029-4061 | Article | MR 2395163 | Zbl pre05308755

[40] Lane, S. Mac Categories for the Working Mathematician, Springer Verlag, Grad. Texts in Math. (1998) | MR 1712872 | Zbl 0906.18001

[41] Laver, R. The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math., Tome 91 (1992) no. 2, pp. 209-231 | Article | MR 1149623 | Zbl 0822.03030

[42] Lee, E.K.; Lee, S.J. A Garside-theoretic approach to the reducibility problem in braid groups, J. Algebra, Tome 320 (2008), pp. 783-820 | Article | MR 2422316 | Zbl pre05349031

[43] Lee, S.J. Garside groups are strongly translation discrete, J. Algebra, Tome 309 (2007), pp. 594-609 | Article | MR 2303195 | Zbl 1155.20038

[44] Matveev, S.V. Distributive groupoids in knot theory, Sb. Math., Tome 119 (1982) no. 1-2, pp. 78-88 | MR 672410 | Zbl 0523.57006

[45] Mccammond, J. An introduction to Garside structures (2005) (circulated notes)

[46] Picantin, M. Garside monoids vs. divisibility monoids, Math. Struct. in Comp. Sci., Tome 15 (2005) no. 2, pp. 231-242 | Article | MR 2132017 | Zbl 1067.20074

[47] Sibert, H. Tame Garside monoids, J. Algebra, Tome 281 (2004), pp. 487-501 | Article | MR 2098379 | Zbl 1065.20049

[48] Thurston, W. Finite state algorithms for the braid group (1988) (circulated notes)