×

zbMATH — the first resource for mathematics

Left-Garside categories, self-distributivity, and braids. (English) Zbl 1183.18004
The notion of a Garside monoid emerged at the end of the 1990’s as a development of Garside’s theory of braids, and it led to many developments. More recently, Bessis, Digne-Michel and Krammer introduced the notion of a Garside category as a further extension, and they used it to capture new, nontrivial examples and improve our understanding of their algebraic structure.
In this paper the author describes and investigates a new example of a (left)-Garside category, namely a certain category associated with the left self-distributivity law \(\text{LD}: x(yz) = (xy)(xz)\). He also develops the notion of a locally left-Garside monoid.
In this framework, the connection 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.
The interest in this law originated in the discovery of several nontrivial structures that obey it, in particular in set theory and in low-dimensional topology.
Moreover this approach leads to a realistic program for establishing the Embedding Conjecture of P. Dehornoy [“Braids and self-distributivity”. Prog. Math. 192 (2000; Zbl 0958.20033), Chap. IX].

MSC:
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
20N02 Sets with a single binary operation (groupoids)
20F36 Braid groups; Artin groups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Adyan, S. I., Fragments of the word delta in a braid group, Mat. Zametki Acad. Sci. SSSR, 36, 1, 25-34, (1984) · Zbl 0599.20044
[2] Bessis, D., Garside categories, periodic loops and cyclic sets
[3] Bessis, D., The dual braid monoid, Ann. Sci. École Norm. Sup., 36, 647-683, (2003) · Zbl 1064.20039
[4] Bessis, D., A dual braid monoid for the free group, J. Algebra, 302, 55-69, (2006) · Zbl 1181.20049
[5] Bessis, D.; Corran, Ruth, Garside structure for the braid group of G(e,e,r) · Zbl 1128.20024
[6] Birman, J.; Gebhardt, V.; González-Meneses, J., Conjugacy in garside groups I: cyclings, powers and rigidity, Groups Geom. Dyn., 1, 221-279, (2007) · Zbl 1160.20026
[7] Birman, J.; Gebhardt, V.; González-Meneses, J., Conjugacy in garside groups III: periodic braids, J. Algebra, 316, 746-776, (2007) · Zbl 1165.20031
[8] Birman, J.; Gebhardt, V.; González-Meneses, J., Conjugacy in garside groups II: structure of the ultra summit set, Groups Geom. Dyn., 2, 16-31, (2008) · 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., 139, 2, 322-353, (1998) · Zbl 0937.20016
[10] Brieskorn, E.; Saito, K., Artin-gruppen und Coxeter-gruppen, Invent. Math., 17, 245-271, (1972) · Zbl 0243.20037
[11] Cannon, J. W.; Floyd, W. J.; Parry, W. R., Introductory notes on richard thompson’s groups, Enseign. Math., 42, 215-257, (1996) · Zbl 0880.20027
[12] Charney, R., Artin groups of finite type are biautomatic, Math. Ann., 292, 4, 671-683, (1992) · Zbl 0736.57001
[13] Charney, R.; Meier, J., The language of geodesics for garside groups, Math. Zeitschr., 248, 495-509, (2004) · Zbl 1062.57002
[14] Charney, R.; Meier, J.; Whittlesey, K., Bestvina’s normal form complex and the homology of garside groups, Geom. Dedicata, 105, 171-188, (2004) · 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, 221, 1-27, (2005) · Zbl 1147.20033
[16] Dehornoy, P.\(, {Π }_1^1\)-complete families of elementary sequences, Ann. P. Appl. Logic, 38, 257-287, (1988) · Zbl 0646.03030
[17] Dehornoy, P., Free distributive groupoids, J. Pure Appl. Algebra, 61, 123-146, (1989) · Zbl 0686.20041
[18] Dehornoy, P., Braids and Self-Distributivity, 192, (2000), Birkhäuser · Zbl 0958.20033
[19] Dehornoy, P., Groupes de garside, Ann. Sci. École Norm. Sup. (4), 35, 267-306, (2002) · Zbl 1017.20031
[20] Dehornoy, P., Study of an identity, Algebra Universalis, 48, 223-248, (2002) · Zbl 1058.03043
[21] Dehornoy, P., Complete positive group presentations, J. Algebra, 268, 156-197, (2003) · Zbl 1067.20035
[22] Dehornoy, P., Geometric presentations of thompson’s groups, J. Pure Appl. Algebra, 203, 1-44, (2005) · Zbl 1150.20016
[23] Dehornoy, P.; Paris, L., Gaussian groups and garside groups, two generalisations of Artin groups, Proc. London Math. Soc., 79, 3, 569-604, (1999) · Zbl 1030.20021
[24] Dehornoy, with I. Dynnikov, D. Rolfsen, and B. Wiest, P., Ordering braids, (2008), Math. Surveys and Monographs vol. 148, Amer. Math. Soc. · Zbl 1163.20024
[25] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. of Math., 103, 103-161, (1976) · Zbl 0336.20029
[26] Digne, F., Présentations duales pour LES groupes de tresses de type affine \(\widetilde{A},\) Comm. Math. Helvetici, 8, 23-47, (2008) · Zbl 1143.20020
[27] Digne, F.; Michel, J., Garside and locally Garside categories · Zbl 1294.18003
[28] El-Rifai, E. A.; Morton, H. R., Algorithms for positive braids, Quart. J. Math. Oxford Ser., 45, 2, 479-497, (1994) · 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, (1992), Jones and Bartlett Publ. · Zbl 0764.20017
[30] Fenn, R.; Rourke, C. P., Racks and links in codimension 2, J. Knot Theory Ramifications, 1, 343-406, (1992) · Zbl 0787.57003
[31] Franco, N.; González-Meneses, J., Conjugacy problem for braid groups and garside groups, J. Algebra, 266, 112-132, (2003) · Zbl 1043.20019
[32] Garside, F. A., The braid group and other groups, Quart. J. Math. Oxford Ser., 20, 235-254, (1969) · Zbl 0194.03303
[33] Gebhardt, V., A new approach to the conjugacy problem in garside groups, J. Algebra, 292, 282-302, (2005) · Zbl 1105.20032
[34] Godelle, E., Parabolic subgroups of Garside groups II · Zbl 1229.20032
[35] Godelle, E., Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groupes d’Artin-Tits, (2001)
[36] Godelle, E., Parabolic subgroups of garside groups, J. Algebra, 317, 1-16, (2007) · Zbl 1173.20027
[37] Joyce, D., A classifying invariant of knots: the knot quandle, J. Pure Appl. Algebra, 23, 37-65, (1982) · Zbl 0474.57003
[38] Kassel, C.; Turaev, V., Braid groups, (2008), Springer Verlag
[39] Krammer, D., A class of garside groupoid structures on the pure braid group, Trans. Amer. Math. Soc., 360, 4029-4061, (2008) · Zbl 1194.20040
[40] Lane, S. Mac, Categories for the Working Mathematician, (1998), Springer Verlag · Zbl 0906.18001
[41] Laver, R., The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math., 91, 2, 209-231, (1992) · Zbl 0822.03030
[42] Lee, E. K.; Lee, S. J., A garside-theoretic approach to the reducibility problem in braid groups, J. Algebra, 320, 783-820, (2008) · Zbl 1191.20034
[43] Lee, S. J., Garside groups are strongly translation discrete, J. Algebra, 309, 594-609, (2007) · Zbl 1155.20038
[44] Matveev, S. V., Distributive groupoids in knot theory, Sb. Math., 119, 1-2, 78-88, (1982) · Zbl 0523.57006
[45] McCammond, J., An introduction to Garside structures, (2005)
[46] Picantin, M., Garside monoids vs. divisibility monoids, Math. Struct. in Comp. Sci., 15, 2, 231-242, (2005) · Zbl 1067.20074
[47] Sibert, H., Tame garside monoids, J. Algebra, 281, 487-501, (2004) · Zbl 1065.20049
[48] Thurston, W., Finite state algorithms for the braid group, (1988)
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.