Braid pictures for Artin groups

Allcock, Daniel

doi:10.1090/S0002-9947-02-02944-6

Braid pictures for Artin groups
HTML articles powered by AMS MathViewer

by Daniel Allcock PDF

Trans. Amer. Math. Soc. 354 (2002), 3455-3474 Request permission

Abstract:

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams $A_n$, $B_n=C_n$ and $D_n$ and the affine diagrams $\tilde {A}_n$, $\tilde {B}_n$, $\tilde {C}_n$ and $\tilde {D}_n$ as subgroups of the braid groups of various simple orbifolds. The cases $D_n$, $\tilde {B}_n$, $\tilde {C}_n$ and $\tilde {D}_n$ are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except $\tilde {A}_n$ the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type $D_n$.

References

E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 279–284. MR 0437798
E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 57–61 (German). MR 293615, DOI 10.1007/BF01389827
Egbert Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol′d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 21–44 (French). MR 0422674
Egbert Brieskorn and Kyoji Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271 (German). MR 323910, DOI 10.1007/BF01406235
Ruth Charney, Artin groups of finite type are biautomatic, Math. Ann. 292 (1992), no. 4, 671–683. MR 1157320, DOI 10.1007/BF01444642
Ruth Charney and Michael W. Davis, Finite $K(\pi , 1)$s for Artin groups, Prospects in topology (Princeton, NJ, 1994) Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 110–124. MR 1368655
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
Pierre Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302 (French). MR 422673, DOI 10.1007/BF01406236
David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694
R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119–126. MR 150755, DOI 10.7146/math.scand.a-10518
S. Lambropoulou. A study of braids in 3-manifolds. Ph.D. thesis, Warwick University, 1993.
Sofia S. F. Lambropoulou, Solid torus links and Hecke algebras of ${\scr B}$-type, Proceedings of the Conference on Quantum Topology (Manhattan, KS, 1993) World Sci. Publ., River Edge, NJ, 1994, pp. 225–245. MR 1309934
Sofia Lambropoulou, Knot theory related to generalized and cyclotomic Hecke algebras of type $B$, J. Knot Theory Ramifications 8 (1999), no. 5, 621–658. MR 1693556, DOI 10.1142/S0218216599000419
José María Montesinos, Classical tessellations and three-manifolds, Universitext, Springer-Verlag, Berlin, 1987. MR 915761, DOI 10.1007/978-3-642-61572-6
Nguyễn Viêt Dũng, The fundamental groups of the spaces of regular orbits of the affine Weyl groups, Topology 22 (1983), no. 4, 425–435. MR 715248, DOI 10.1016/0040-9383(83)90035-6
Mario Salvetti, The homotopy type of Artin groups, Math. Res. Lett. 1 (1994), no. 5, 565–577. MR 1295551, DOI 10.4310/MRL.1994.v1.n5.a5
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
Tammo tom Dieck, Symmetrische Brücken und Knotentheorie zu den Dynkin-Diagrammen vom Typ $B$, J. Reine Angew. Math. 451 (1994), 71–88 (German). MR 1277295, DOI 10.1515/crll.1994.451.71
Tammo tom Dieck, Categories of rooted cylinder ribbons and their representations, J. Reine Angew. Math. 494 (1998), 35–63. Dedicated to Martin Kneser on the occasion of his 70th birthday. MR 1604452, DOI 10.1515/crll.1998.010
Harm van der Lek, Extended Artin groups, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 117–121. MR 713240
H. van der Lek. The Homotopy Type of Complex Hyperplane Arrangements. Ph.D. thesis, Katholieke Universiteit te Nijmegen, 1983.