Left-Garside categories, self-distributivity, and braids

Patrick Dehornoy[1]

  • [1] Laboratoire de Mathématiques Nicolas Oresme Université de Caen 14032 Caen France

Annales mathématiques Blaise Pascal (2009)

  • Volume: 16, Issue: 2, page 189-244
  • ISSN: 1259-1734

Abstract

top
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].

How to cite

top

Dehornoy, Patrick. "Left-Garside categories, self-distributivity, and braids." Annales mathématiques Blaise Pascal 16.2 (2009): 189-244. <http://eudml.org/doc/10576>.

@article{Dehornoy2009,
abstract = {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].},
affiliation = {Laboratoire de Mathématiques Nicolas Oresme Université de Caen 14032 Caen France},
author = {Dehornoy, Patrick},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Garside category; Garside monoid; self-distributivity; braid; greedy normal form; least common multiple; LD-expansion},
language = {eng},
month = {7},
number = {2},
pages = {189-244},
publisher = {Annales mathématiques Blaise Pascal},
title = {Left-Garside categories, self-distributivity, and braids},
url = {http://eudml.org/doc/10576},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Dehornoy, Patrick
TI - Left-Garside categories, self-distributivity, and braids
JO - Annales mathématiques Blaise Pascal
DA - 2009/7//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 2
SP - 189
EP - 244
AB - 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].
LA - eng
KW - Garside category; Garside monoid; self-distributivity; braid; greedy normal form; least common multiple; LD-expansion
UR - http://eudml.org/doc/10576
ER -

References

top
  1. S.I. Adyan, Fragments of the word Delta in a braid group, Mat. Zametki Acad. Sci. SSSR 36 (1984), 25-34 Zbl0599.20044MR757642
  2. D. Bessis, Garside categories, periodic loops and cyclic sets 
  3. D. Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. 36 (2003), 647-683 Zbl1064.20039MR2032983
  4. D. Bessis, A dual braid monoid for the free group, J. Algebra 302 (2006), 55-69 Zbl1181.20049MR2236594
  5. D. Bessis, Ruth Corran, Garside structure for the braid group of G(e,e,r) Zbl1128.20024
  6. J. Birman, V. Gebhardt, J. González-Meneses, Conjugacy in Garside groups I: Cyclings, powers and rigidity, Groups Geom. Dyn. 1 (2007), 221-279 Zbl1160.20026MR2314045
  7. J. Birman, V. Gebhardt, J. González-Meneses, Conjugacy in Garside groups III: Periodic braids, J. Algebra 316 (2007), 746-776 Zbl1165.20031MR2358613
  8. J. Birman, V. Gebhardt, J. González-Meneses, Conjugacy in Garside groups II: Structure of the ultra summit set, Groups Geom. Dyn. 2 (2008), 16-31 Zbl1163.20023MR2367207
  9. J. Birman, K.H. Ko, S.J. Lee, A new approach to the word problem in the braid groups, Adv. Math. 139 (1998), 322-353 Zbl0937.20016MR1654165
  10. E. Brieskorn, K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245-271 Zbl0243.20037MR323910
  11. J.W. Cannon, W.J. Floyd, W.R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. 42 (1996), 215-257 Zbl0880.20027MR1426438
  12. R. Charney, Artin groups of finite type are biautomatic, Math. Ann. 292 (1992), 671-683 Zbl0736.57001MR1157320
  13. R. Charney, J. Meier, The language of geodesics for Garside groups, Math. Zeitschr. 248 (2004), 495-509 Zbl1062.57002MR2097371
  14. R. Charney, J. Meier, K. Whittlesey, Bestvina’s normal form complex and the homology of Garside groups, Geom. Dedicata 105 (2004), 171-188 Zbl1064.20044MR2057250
  15. J. Crisp, L. Paris, Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups, Pac. J. Maths 221 (2005), 1-27 Zbl1147.20033MR2194143
  16. P. Dehornoy, Π 1 1 -complete families of elementary sequences, Ann. P. Appl. Logic 38 (1988), 257-287 Zbl0646.03030MR942526
  17. P. Dehornoy, Free distributive groupoids, J. Pure Appl. Algebra 61 (1989), 123-146 Zbl0686.20041MR1025918
  18. P. Dehornoy, Braids and Self-Distributivity, 192 (2000), Birkhäuser Zbl0958.20033MR1778150
  19. P. Dehornoy, Groupes de Garside, Ann. Sci. École Norm. Sup. (4) 35 (2002), 267-306 Zbl1017.20031MR1914933
  20. P. Dehornoy, Study of an identity, Algebra Universalis 48 (2002), 223-248 Zbl1058.03043MR1929906
  21. P. Dehornoy, Complete positive group presentations, J. Algebra 268 (2003), 156-197 Zbl1067.20035MR2004483
  22. P. Dehornoy, Geometric presentations of Thompson’s groups, J. Pure Appl. Algebra 203 (2005), 1-44 Zbl1150.20016MR2176650
  23. P. Dehornoy, L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc. 79 (1999), 569-604 Zbl1030.20021MR1710165
  24. P. Dehornoy, with I. Dynnikov, D. Rolfsen, and B. Wiest, Ordering braids, (2008), Math. Surveys and Monographs vol. 148, Amer. Math. Soc. Zbl1163.20024MR2463428
  25. P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103-161 Zbl0336.20029MR393266
  26. F. Digne, Présentations duales pour les groupes de tresses de type affine A ˜ , Comm. Math. Helvetici 8 (2008), 23-47 Zbl1143.20020MR2208796
  27. F. Digne, J. Michel, Garside and locally Garside categories Zbl1294.18003
  28. E.A. El-Rifai, H.R. Morton, Algorithms for positive braids, Quart. J. Math. Oxford Ser. 45 (1994), 479-497 Zbl0839.20051MR1315459
  29. D. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, W.P. Thurston, Word Processing in Groups, (1992), Jones and Bartlett Publ. Zbl0764.20017MR1161694
  30. R. Fenn, C.P. Rourke, Racks and links in codimension 2, J. Knot Theory Ramifications 1 (1992), 343-406 Zbl0787.57003MR1194995
  31. N. Franco, J. González-Meneses, Conjugacy problem for braid groups and Garside groups, J. Algebra 266 (2003), 112-132 Zbl1043.20019MR1994532
  32. F.A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. 20 (1969), 235-254 Zbl0194.03303MR248801
  33. V. Gebhardt, A new approach to the conjugacy problem in Garside groups, J. Algebra 292 (2005), 282-302 Zbl1105.20032MR2166805
  34. E. Godelle, Parabolic subgroups of Garside groups II Zbl1229.20032
  35. E. Godelle, Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groupes d’Artin-Tits, (2001) 
  36. E. Godelle, Parabolic subgroups of Garside groups, J. Algebra 317 (2007), 1-16 Zbl1173.20027MR2360138
  37. D. Joyce, A classifying invariant of knots: the knot quandle, J. Pure Appl. Algebra 23 (1982), 37-65 Zbl0474.57003MR638121
  38. C. Kassel, V. Turaev, Braid groups, (2008), Springer Verlag Zbl1208.20041MR2435235
  39. D. Krammer, A class of Garside groupoid structures on the pure braid group, Trans. Amer. Math. Soc. 360 (2008), 4029-4061 Zbl1194.20040MR2395163
  40. S. Mac Lane, Categories for the Working Mathematician, (1998), Springer Verlag Zbl0906.18001MR1712872
  41. R. Laver, The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math. 91 (1992), 209-231 Zbl0822.03030MR1149623
  42. E.K. Lee, S.J. Lee, A Garside-theoretic approach to the reducibility problem in braid groups, J. Algebra 320 (2008), 783-820 Zbl1191.20034MR2422316
  43. S.J. Lee, Garside groups are strongly translation discrete, J. Algebra 309 (2007), 594-609 Zbl1155.20038MR2303195
  44. S.V. Matveev, Distributive groupoids in knot theory, Sb. Math. 119 (1982), 78-88 Zbl0523.57006MR672410
  45. J. McCammond, An introduction to Garside structures, (2005) 
  46. M. Picantin, Garside monoids vs. divisibility monoids, Math. Struct. in Comp. Sci. 15 (2005), 231-242 Zbl1067.20074MR2132017
  47. H. Sibert, Tame Garside monoids, J. Algebra 281 (2004), 487-501 Zbl1065.20049MR2098379
  48. W. Thurston, Finite state algorithms for the braid group, (1988) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.