Consider a finite collection of particles lying on a finite graph. The configuration space of these particles is the collection of all possible ways the particles can be arranged on the graph with no two particles at the same point. As we move through the configuration space, the particles move along the graph, without colliding. The braid group on our graph is then defined to be the fundamental group of this configuration space. By discretising the motion of the particles, we obtain a combinatorial version of the configuration space, which can be shown to be a special cube complex. Moreover, this cube complex deformation retracts onto the original configuration space, meaning the braid group is unchanged. In particular, this implies graph braid groups are hierarchically hyperbolic groups.

The cubical and hierarchically hyperbolic structures can be used to give simple characterisations of when graph braid groups are Gromov hyperbolic or acylindrically hyperbolic, expressed in terms of properties of the graph. However, I show that relative hyperbolicity requires more nuance. Speaker(s): Daniel Berlyne (University of Bristol)