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

Abstract: For every $n\geq 1$, 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 $$\left{ {x_1, \ldots, x_n } \in \mathbb{R}^n / \mathrm{Sym}(n) \mid \text{there exist at most two indices $i,j$ such that } x_i=x_j \right}.$$ 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 $\geq 11$, $\mathrm{PFB}_n$ 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 $A(P_4)$.