On the cohomology rings of tree braid groups

Let $\mathrm{\Gamma }$ be a finite connected graph. The (unlabelled) configuration space $U{C}^n\mathrm{\Gamma }$ of $n$ points on $\mathrm{\Gamma }$ is the space of $n$-element subsets of $\mathrm{\Gamma }$. The $n$-strand braid group of $\mathrm{\Gamma }$, denoted $B_n\mathrm{\Gamma }$, is the fundamental group of $U{C}^n\mathrm{\Gamma }$. We use the methods and results of our paper “Discrete Morse theory and graph braid groups” to get a partial description of the cohomology rings $H^*(B_n T)$,where$T$isatree\mathrm{.}Ourresultsarethenusedtoprovethat$B_n T$ 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.