Negative curvature in graph braid groups

In this article, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous article, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic, and acylindrically hyperbolic.
29 pages, 9 figures. Extracted from the first version of arXiv:1709.01258