Annales Mathématiques
Blaise Pascal
no. 2
Left-Garside categories, self-distributivity, and braids
Patrick Dehornoy1
1 Laboratoire de Mathématiques Nicolas Oresme Université de Caen 14032 Caen France
Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 189-244.

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].

DOI : 10.5802/ambp.263
Classification : 18B40, 20N02, 20F36
Mots clés : Garside category, Garside monoid, self-distributivity, braid, greedy normal form, least common multiple, LD-expansion

Patrick Dehornoy 1

1 Laboratoire de Mathématiques Nicolas Oresme Université de Caen 14032 Caen France 
@article{AMBP_2009__16_2_189_0,
     author = {Patrick Dehornoy},
     title = {Left-Garside categories, self-distributivity, and braids},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {189--244},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     number = {2},
     year = {2009},
     doi = {10.5802/ambp.263},
     mrnumber = {2568862},
     zbl = {1183.18004},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.263/}
}
TY  - JOUR
AU  - Patrick Dehornoy
TI  - Left-Garside categories, self-distributivity, and braids
JO  - Annales mathématiques Blaise Pascal
PY  - 2009
SP  - 189
EP  - 244
VL  - 16
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.263/
DO  - 10.5802/ambp.263
LA  - en
ID  - AMBP_2009__16_2_189_0
ER  - 
%0 Journal Article
%A Patrick Dehornoy
%T Left-Garside categories, self-distributivity, and braids
%J Annales mathématiques Blaise Pascal
%D 2009
%P 189-244
%V 16
%N 2
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.263/
%R 10.5802/ambp.263
%G en
%F AMBP_2009__16_2_189_0
Patrick Dehornoy. Left-Garside categories, self-distributivity, and braids. Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 189-244. doi : 10.5802/ambp.263. https://ambp.centre-mersenne.org/articles/10.5802/ambp.263/

[1] S.I. Adyan Fragments of the word Delta in a braid group, Mat. Zametki Acad. Sci. SSSR, Volume 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 | Zbl

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

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

[4] D. Bessis A dual braid monoid for the free group, J. Algebra, Volume 302 (2006), pp. 55-69 | DOI | MR

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

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

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

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

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

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

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

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

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

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

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

[16] P. Dehornoy Π 1 1 -complete families of elementary sequences, Ann. P. Appl. Logic, Volume 38 (1988), pp. 257-287 | DOI | MR | Zbl

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

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

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

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

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

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

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

[24] P. Dehornoy, with I. Dynnikov, D. Rolfsen, and B. Wiest Ordering braids, Math. Surveys and Monographs vol. 148, Amer. Math. Soc., 2008 | MR | Zbl

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

[26] F. Digne Présentations duales pour les groupes de tresses de type affine A ˜, Comm. Math. Helvetici, Volume 8 (2008), pp. 23-47 | MR | Zbl

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

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

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

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

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

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

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

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

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

[36] E. Godelle Parabolic subgroups of Garside groups, J. Algebra, Volume 317 (2007), pp. 1-16 | DOI | MR

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

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

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

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

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

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

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

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

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

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

[47] H. Sibert Tame Garside monoids, J. Algebra, Volume 281 (2004), pp. 487-501 | DOI | MR | Zbl

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

Cité par Sources :