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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Braid pictures for Artin groups

Author: Daniel Allcock
Journal: Trans. Amer. Math. Soc. 354 (2002), 3455-3474
MSC (2000): Primary 20F36
Published electronically: April 30, 2002
MathSciNet review: 1911508
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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 $D_n$.

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Additional Information

Daniel Allcock
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712

Keywords: Braid group, Artin group, orbifold, Garside element
Received by editor(s): July 29, 2000
Received by editor(s) in revised form: September 10, 2001
Published electronically: April 30, 2002
Additional Notes: Supported by an NSF postdoctoral fellowship.
Article copyright: © Copyright 2002 American Mathematical Society