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Conjugacy in singular Artin monoids

Part of: Semigroups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Ruth Corran
Ruth Corran
Affiliation:
Institut de Géométrie, Algèbre et Topologie, Bâtiment BCH, École Polytechnique Fédérale de Lausanne, CH-1015, Switzerland, e-mail: [email protected]
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Abstract

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We define a notion of conjugacy in singular Artin moniods, and solve the corresponding conjugacy problem for finite types. We sgiw that this definition is appropriate to describe type (1) singular Markov moves on singular braids. Parabolic submonoids of singular Artin monoids are defined and, in finite type, are shown to be singular Artin monoids. Solutions to conjugacy-type problems of parabolic submonoids are described. Geometric objects defined by Fenn, Rolfsen and Zhu, called (j, k)-bands, are algebraically characterised, and a procedure is given which determines when a word represents a (j, k)-band.

MSC classification

Secondary: 20F36: Braid groups; Artin groups 20M05: Free semigroups, generators and relations, word problems 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 2 , October 2005 , pp. 183 - 212
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Antony, N., ‘On singular Artin monoids and contributions to Birman's conjecture’, Comm. Algebra, to appear.Google Scholar
[2]Armand-Ugon, D., Gambini, R. and Mora, P., ‘Intersecting braids and intersecting knot theory’, J. Knot Theory Ramifications 4 (1995), 1–12.CrossRefGoogle Scholar
[3]Artin, E., ‘Theorie der Zöpfe’, Hamburg Abh. 4 (1925), 47–72.CrossRefGoogle Scholar
[4]Baez, J., ‘Link invariants and perturbation theory’, Lett. Math. Phys. 2 (1992), 43–51.CrossRefGoogle Scholar
[5]Birman, J., ‘New points of view in knot theory’, Bull. Amer. Math. Soc. (N.S.) 28 (1993), 253–287.CrossRefGoogle Scholar
[6]Bourbaki, N., Groupes et algèbres de Lie (Hermann, Paris, 1968).Google Scholar
[7]Brieskorn, E. and Saito, K., ‘Artin-Gruppen und Coxeter-Gruppen’, Invent. Math. 17 (1972), 245–271.CrossRefGoogle Scholar
[8]Brink, B. and Howlett, R. B., ‘Normalizers of parabolic subgroups in Coxeter groups’, Invent. Math. 136 (1999), 323–351.CrossRefGoogle Scholar
[9]Corran, R., ‘A normal form for a class of monoids including the singular braid monoids’, J. Algebra 223 (2000), 256–282.CrossRefGoogle Scholar
[10]Deligne, P., ‘Les immeubles des groupes de tresses généralisés’, Invent. Math. 17 (1972), 273–302.CrossRefGoogle Scholar
[11]Van der Lek, H., The homotopy type of complex hyperplane complements (Ph.D. Thesis, Nijmegen, 1983).Google Scholar
[12]Fenn, R., Rolfsen, D. and Zhu, J., ‘Centralisers in the braid group and singular braid monoid’, l' Enseignement Mathématique 42 (1996), 75–96.Google Scholar
[13]Garside, F. A., ‘The braid group and other groups’, Quart. J. Math. Oxford (2) 20 (1969), 235–254.CrossRefGoogle Scholar
[14]Gemein, B., ‘Singular braids and Markov's theorem’, J. Knot Theory Ramifications 6 (1997), 441–454.CrossRefGoogle Scholar
[15]Godelle, E., ‘Normalisateur et groupe d'Artin de type spherique’, J. Algebra 269 (2003), 263–274.CrossRefGoogle Scholar
[16]Godelle, E., ‘Parabolic subgroups of Artin groups of type FC’, Pacific J. Math. 208 (2003), 243–254.CrossRefGoogle Scholar
[17]Godelle, E. and Paris, L., ‘On singular Artin monoids’, in: Proceedings of the special session Geometric Methods in Group Theory, First Joint Meeting RMSE and AMS, Sevilla, 2003, Contemporary Math., to appear.Google Scholar
[18]Howie, J. M., Fundamentals of semigroup theory, London Math. Soc. Monogr., New Series 12 (Oxford University Press, Oxford, 1995).CrossRefGoogle Scholar
[19]Howlett, R. B., ‘Normalizers of parabolic subgroups of reflection groups’, J. London Math. Soc. 21 (1980), 62–80.CrossRefGoogle Scholar
[20]Humphreys, J. E., Reflection groups and Coxeter groups, Cambridge Studies in Adv. Math. 29 (Cambridge Univ. Press, Cambridge, UK, 1990).CrossRefGoogle Scholar
[21]Paris, L., ‘The proof of Birman's conjective on singular braid monoids’, Geom. Topol. 8 (2004), 1281–1300.CrossRefGoogle Scholar
[22]Paris, L., ‘Parabolic subgroups of Artin groups’, J. Algebra 196 (1997), 369–399.CrossRefGoogle Scholar