Manifold models for hyperbolic graph braid groups on three strands

Abstract

Given a finite graph $\hat{I}“$, the associated graph braid group $B_n(\hat{I}“)$ is the fundamental group of the unordered $n$-point configuration space of $\hat{I}“$. 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 $B_3(\hat{I}{˜}_m)$, where $\hat{I}{˜}_m$ is the generalized $\hat{I}˜$-graph, a suspension of $m$-points. We show that $B_3(\hat{I}{˜}_5)$ is a $3$-manifold group while $B_3(\hat{I}{˜}_m)$ is not even quasi-isometric to a $3$-manifold group for $m\ge 7$.

12 pages, 10 figures