Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20F36

Retrieve articles in all journals with MSC (2000): 20F36


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
Email: allcock@math.harvard.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02944-6
PII: S 0002-9947(02)02944-6
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