Graph braid groups and right-angled Artin groups

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 ≥ 5 \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 n -braid groups as a conjecture along with a few other conjectures about graphs whose braid groups of index ≤ 4 \le 4 are right-angled Artin groups.