The algebraic structure of hyperbolic graph braid groups

Genevois recently classified which graph braid groups on $\ge 3$ 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: sun and pulsar graphs. We prove that $3$-strand braid groups of sun graphs are free. On the other hand, it was known to experts that $3$-strand braid groups of most pulsar graphs contain surface subgroups. We provide a simple proof of this and prove an additional structure theorem for these groups.