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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05399-7

Published electronically:
August 2, 2011

MathSciNet review:
2833585

Full-text PDF

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.

**1.**A. Abrams,*Configuration space of braid groups of graphs*, Ph.D. thesis in UC Berkeley, ProQuest LLC, Ann Arbor, MI, 2000. MR**2701024****2.**A. Abrams and R. Ghrist,*Finding topology in factory: Configuration spaces*, Amer. Math. Monthly**109**(2) (2002), 140-150. MR**1903151 (2003c:68222)****3.**W. Bruns and J. Gubeladze,*Combinatorial invariance of Stanley-Reisner rings*, Georgian Math. J.**3**(4) (1996), 315-318. MR**1397814 (97d:13025)****4.**R. Charney,*An introduction to right-angled Artin groups*, Geom. Dedicate**125**(2007), 141-158. MR**2322545 (2008f:20076)****5.**R. Charney and M. Davis,*Finite s for Artin groups*, Prospects in topology, Ann. of Math. Stud.**138**, Princeton Univ. Press, Princeton, NJ, 1995, pp. 110-124. MR**1368655 (97a:57001)****6.**F. Connelly and M. Doig,*Braid groups and right-angled Artin groups*, arXiv:math.GT/0411368.**7.**J. Crisp and B. Wiest,*Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups*, Algebr. Geom. Topol.**4**(2004), 439-472. MR**2077673 (2005e:20052)****8.**D. Farley,*Homology of tree braid groups*, Contemp. Math.**394**(2006), 102-112. MR**2216709 (2007c:20093)****9.**D. Farley and L. Sabalka,*Discrete Morse theory and graph braid groups*, Algebr. Geom. Topol.**5**(2005), 1075-1109 (electronic). MR**2171804 (2006f:20051)****10.**D. Farley and L. Sabalka,*On the cohomology rings of tree braid groups*, J. Pure Appl. Algebra**212**(1) (2007), 53-71. MR**2355034 (2008j:20107)****11.**R. Forman,*Morse theory for cell complexes*, Adv. Math.**143**(1) (1998), 90-145. MR**1612391 (99b:57050)****12.**R. Ghrist,*Configuration spaces and braid groups on graphs in robotics*, Knots, braids, and mapping class groups--papers dedicated to Joan S. Birman (New York, 1998), pp. 29-40, AMS/IP Stud. Adv. Math.**24**, Amer. Math. Soc., Providence, RI, 2001. MR**1873106 (2002j:55015)****13.**D. Matei and A. Suciu,*Cohomology rings and nilpotent quotients of real and complex arrangements*, Arrangements--Tokyo 1998, Adv. Stud. Pure Math.**27**, Kinokuniya, Tokyo, 2000, pp. 185-215. MR**1796900 (2002b:32045)****14.**L. Sabalka,*Embedding right-angled Artin groups into graph braid groups*, Geom. Dedicata**124**(2007), 191-198. MR**2318544 (2008e:20057)**

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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:
https://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.