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 | References | Similar Articles | Additional Information

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 . 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 -braid groups as a conjecture along with a few other conjectures about graphs whose braid groups of index 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

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.