Mathematics – Group Theory
Scientific paper
2006-02-20
Journal of Pure and Applied Algebra, 212(1) (2007) 53-71
Mathematics
Group Theory
25 pages, 7 figures. Revised version, accepted by the Journal of Pure and Applied Algebra
Scientific paper
10.1016/j.jpaa.2007.04.011
Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration space $UC^n \Gamma$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $B_n\Gamma$, is the fundamental group of $UC^n \Gamma$. 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 $T$ is a tree. Our results are then used to prove that $B_n T$ 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.
Farley Daniel
Sabalka Lucas
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