Manifold models for hyperbolic graph braid groups on three strands

Given a finite graph $\Gamma$, the associated graph braid group $B_n(\Gamma)$ is the fundamental group of the unordered $n$-point configuration space of $\Gamma$. 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 $B_3(\Theta_m)$, where $\Theta_m$ is the generalized $\Theta$-graph, a suspension of $m$-points. We show that $B_3(\Theta_5)$ is a $3$-manifold group while $B_3(\Theta_m)$ is not even quasi-isometric to a $3$-manifold group for $m \geq 7$.