Products of free groups inside graph braid groups

Abstract

Given a graph $\Gamma$ and a number $n$, the associated $n^{th}$ graph braid group $B_n(\Gamma )$ is the fundamental group of the unordered configuration space of $n$ points on $\Gamma$. Świątkowski showed that for a given $\Gamma$ and $n$ large enough, there is a free abelian subgroup of $B_n(\Gamma )$ of rank equal to the cohomological dimension of $B_n(\Gamma )$. 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.

10 pages, 3 figures. Comments are welcome!