Graph braid groups and right-angled Artin groups

Kim, Jee Hyoun; Ko, Ki Hyoung; Park, Hyo Won

doi:10.1090/S0002-9947-2011-05399-7

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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 -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.

1. Aaron David Abrams, Configuration spaces and braid groups of graphs, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Ph.D.)–University of California, Berkeley. MR 2701024
2. Aaron Abrams and Robert Ghrist, Finding topology in a factory: configuration spaces, Amer. Math. Monthly 109 (2002), no. 2, 140–150. MR 1903151 (2003c:68222), http://dx.doi.org/10.2307/2695326
3. W. Bruns and J. Gubeladze, Combinatorial invariance of Stanley-Reisner rings, Georgian Math. J. 3 (1996), no. 4, 315–318. MR 1397814 (97d:13025), http://dx.doi.org/10.1007/BF02256722
4. Ruth Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007), 141–158. MR 2322545 (2008f:20076), http://dx.doi.org/10.1007/s10711-007-9148-6
5. Ruth Charney and Michael W. Davis, Finite 𝐾(𝜋,1)s for Artin groups, Prospects in topology (Princeton, NJ, 1994) Ann. of Math. Stud., vol. 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. John Crisp and Bert 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), http://dx.doi.org/10.2140/agt.2004.4.439
8. Daniel Farley, Homology of tree braid groups, Topological and asymptotic aspects of group theory, Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 101–112. MR 2216709 (2007c:20093), http://dx.doi.org/10.1090/conm/394/07437
9. Daniel Farley and Lucas Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005), 1075–1109. MR 2171804 (2006f:20051), http://dx.doi.org/10.2140/agt.2005.5.1075
10. Daniel Farley and Lucas Sabalka, On the cohomology rings of tree braid groups, J. Pure Appl. Algebra 212 (2008), no. 1, 53–71. MR 2355034 (2008j:20107), http://dx.doi.org/10.1016/j.jpaa.2007.04.011
11. Robin Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90–145. MR 1612391 (99b:57050), http://dx.doi.org/10.1006/aima.1997.1650
12. Robert Ghrist, Configuration spaces and braid groups on graphs in robotics, Birman (New York, 1998) AMS/IP Stud. Adv. Math., vol. 24, Amer. Math. Soc., Providence, RI, 2001, pp. 29–40. MR 1873106 (2002j:55015)
13. Daniel Matei and Alexander I. Suciu, Cohomology rings and nilpotent quotients of real and complex arrangements, Arrangements—Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Kinokuniya, Tokyo, 2000, pp. 185–215. MR 1796900 (2002b:32045)
14. Lucas Sabalka, Embedding right-angled Artin groups into graph braid groups, Geom. Dedicata 124 (2007), 191–198. MR 2318544 (2008e:20057), http://dx.doi.org/10.1007/s10711-006-9101-0

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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.