Products of free groups inside graph braid groups

Given a graph Γ and a number n, the associated nᵗʰ graph braid group Bₙ(Γ) is the fundamental group of the unordered configuration space of n points on Γ. \'{S}wi{a}tkowski showed that for a given Γ and n large enough, there is a free abelian subgroup of Bₙ(Γ) of rank equal to the cohomological dimension of Bₙ(Γ). In this note, we observe that at the cost of possibly adding additional points, we can find a subgroup of the same cohomological dimension which is a direct product of non-abelian free groups, and give some applications.