Flat braid groups, right-angled Artin groups, and commensurability

Abstract

For every $n\ge 1$, the flat braid group ${\mathrm{FB}}_n$ is an analogue of the braid group $B_n$ that can be described as the fundamental group of the configuration space $$\left\{\{x_1,\dots ,x_n\}\in {\mathbb{R}}^n/\mathrm{Sym}(n)\mid \text{there exist at most two indices i,j such that }{x}_i=x_j\right\}.$$ Alternatively, ${\mathrm{FB}}_n$ can also be described as the right-angled Coxeter group $C(P_{n-2}^{\mathrm{opp}})$, where $P_{n-2}^{\mathrm{opp}}$ denotes the opposite graph of the path $P_{n-2}$ of length $n-2$. In this article, we prove that, for every $n=7$ or $\ge 11$, ${\mathrm{PFB}}_n$ is not virtually a right-angled Artin group, disproving a conjecture of Naik, Nanda, and Singh. In the opposite direction, we observe that ${\mathrm{FB}}_7$ turns out to be commensurable to the right-angled Artin group $A(P_4)$.

arXiv comment27 pages, 11 figures. Comments are welcome!

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