Manifold models for hyperbolic graph braid groups on three strands

Given a finite graph $\mathrm{\Gamma }$, the associated graph braid group $B_n(\mathrm{\Gamma })$ is the fundamental group of the unordered $n$-point configuration space of $\mathrm{\Gamma }$. Genevois classified which graph braid groups are Gromov hyperbolic and asked the question: When do these groups arise as $3$-manifold groups? In this paper, we give a partial answer for $B_3({\mathrm{\Theta }}_m)$, where ${\mathrm{\Theta }}_m$ is the generalized $\mathrm{\Theta }$-graph, a suspension of $m$-points. We show that $B_3({\mathrm{\Theta }}_5)$ is a $3$-manifold group while $B_3({\mathrm{\Theta }}_m)$ is not even quasi-isometric to a $3$-manifold group for $m\ge 7$.