The algebraic structure of hyperbolic graph braid groups

Genevois has recently classified which graph braid groups on ≥ 3 strands are word hyperbolic. In the $3$-strand case, he asked whether all such word hyperbolic groups are actually free; this reduced to checking two infinite classes of graphs. We give positive and negative answers to this question: one class of graphs always has free $3$-strand braid group, while the braid groups in the other class usually contain surface subgroups.