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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Graph braid groups and right-angled Artin groups


Authors: Jee Hyoun Kim, Ki Hyoung Ko and Hyo Won Park
Journal: Trans. Amer. Math. Soc. 364 (2012), 309-360
MSC (2010): Primary 20F36, 20F65, 57M15
Published electronically: August 2, 2011
MathSciNet review: 2833585
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Abstract: We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $ \ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of the homological or cohomological characteristics of right-angled Artin groups can be applied. Finally we show that a given graph is planar iff the first homology of its 2-braid group is torsion-free, and we leave the corresponding statement for $ n$-braid groups as a conjecture along with a few other conjectures about graphs whose braid groups of index $ \le 4$ are right-angled Artin groups.


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Additional Information

Jee Hyoun Kim
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Email: kimjeehyoun@kaist.ac.kr

Ki Hyoung Ko
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Email: knot@kaist.ac.kr

Hyo Won Park
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Email: H.W.Park@kaist.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05399-7
PII: S 0002-9947(2011)05399-7
Keywords: Braid group, right-angled Artin group, discrete Morse theory, planar, graph
Received by editor(s): August 11, 2009
Received by editor(s) in revised form: May 24, 2010, June 6, 2010, and June 12, 2010
Published electronically: August 2, 2011
Additional Notes: This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R01-2006-000-10152-0)
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.