On the cohomology rings of tree braid groups

Let Γ be a finite connected graph. The (unlabelled) configuration space image of n points on Γ is the space of n-element subsets of Γ. The n-strand braid group of Γ, denoted BnΓ, is the fundamental group of image. 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), where T is a tree. Our results are then used to prove that BnT is a right-angled Artin group if and only if T is linear or n<4. This gives a large number of counterexamples to Ghrist’s conjecture that braid groups of planar graphs are right-angled Artin groups.