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

Graph braid groups and right-angled Artin groups
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by Jee Hyoun Kim, Ki Hyoung Ko and Hyo Won Park

Trans. Amer. Math. Soc. 364 (2012), 309-360

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

Published electronically: August 2, 2011

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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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