On the cohomology rings of tree braid groups

Let ΓΓ be a finite connected graph. The (unlabelled) configuration space UCnΓUCnΓ of nn points on ΓΓ is the space of nn-element subsets of ΓΓ. The nn-strand braid group of ΓΓ, denoted BnΓBnΓ, is the fundamental group of UCnΓUCnΓ.We use the methods and results of [Daniel Farley, Lucas Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005) 1075–1109. Electronic] to get a partial description of the cohomology rings H∗(BnT)H∗(BnT), where TT is a tree. Our results are then used to prove that BnTBnT is a right-angled Artin group if and only if TT is linear or n<4n<4. This gives a large number of counterexamples to Ghrist’s conjecture that braid groups of planar graphs are right-angled Artin groups.