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A class of Garside groupoid structures on the pure braid group
Author(s):
Daan
Krammer
Journal:
Trans. Amer. Math. Soc.
360
(2008),
4029-4061.
MSC (2000):
Primary 20F36;
Secondary 20F05, 20F60, 57M07
Posted:
March 20, 2008
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Abstract:
We construct a class of Garside groupoid structures on the pure braid groups, one for each function (called labelling) from the punctures to the integers greater than 1. The object set of the groupoid is the set of ball decompositions of the punctured disk; the labels are the perimeters of the regions. Our construction generalises Garside's original Garside structure, but not the one by Birman-Ko-Lee. As a consequence, we generalise the Tamari lattice ordering on the set of vertices of the associahedron.
References:
-
- [BKL98]
- Birman, Joan; Ko, Ki Hyoung; Lee, Sang Jin. A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139 (1998), no. 2, 322-353. MR 1654165 (99m:20082)
- [Deh00]
- Dehornoy, Patrick. Chapter 2 in Braids and self-distributivity. Progress in Mathematics, 192. Birkhäuser-Verlag, Basel, 2000. MR 1778150 (2001j:20057)
- [Deh02]
- Dehornoy, Patrick. Groupes de Garside. Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 267-306. MR 1914933 (2003f:20068)
- [DehPar99]
- Dehornoy, Patrick; Paris, Luis. Gaussian groups and Garside groups, two generalisations of Artin groups. Proc. London Math. Soc. (3) 79 (1999), no. 3, 569-604. MR 1710165 (2001f:20061)
- [Eps92]
- Epstein, D.B.A.; Cannon, J.W.; Holt, D.F.; Levy, S.V.F.; Paterson, M.S.; Thurston, W.P. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694 (93i:20036)
- [FriTam67]
- Friedman, Haya; Tamari, Dov. Problèmes d'associativité: Une structure de treillis finis induite par une loi demi-associative. J. Combinatorial Theory 2 (1967), 215-242. MR 0238984 (39:344)
- [Gar69]
- Garside, F. A. The braid group and other groups. Quart. J. Math. Oxford Ser. (2) 20 (1969), 235-254. MR 0248801 (40:2051)
- [Grä78]
- Grätzer, George. General lattice theory. Second edition. Birkhäuser-Verlag, Basel, 1998. (First edition published 1978). MR 1670580 (2000b:06001)
- [Lee89]
- Lee, Carl W. The associahedron and triangulations of the
 -gon. European J. Combin. 10 (1989), no. 6, 551-560. MR 1022776 (90i:52010) - [Par05]
- Paris, Luis. From braid groups to mapping class groups. Proc. Sympos. Pure Math. 74 (2006), 355-371.
- [Sta63]
- Stasheff, James. Homotopy associativity of
 -spaces I. Trans. Amer. Math. Soc. 108 (1963), 275-292. MR 0158400 (28:1623) - [Tam51]
- Tamari, Dov. Monoïdes préordonnés et chaines de Malcev. Thesis, Paris, 1951. MR 0051833 (14:532b)
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Additional Information:
Daan
Krammer
Affiliation:
Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom
Email:
daan@maths.warwick.ac.uk
DOI:
10.1090/S0002-9947-08-04313-4
PII:
S 0002-9947(08)04313-4
Received by editor(s):
September 28, 2005
Received by editor(s) in revised form:
March 27, 2006
Posted:
March 20, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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