Joseph MacManus

Coarse characterisations of planarity in Cayley graphs

Recall a graph is said to be planar if it can be drawn in the plane without edges crossing. Let's call a finitely generated group G (virtually) planar if (some finite-index subgroup of) G admits a planar Cayley graph.

In this talk I will present several results characterising virtually planar groups in terms of their coarse geometry, illustrating the philosophy that this class of groups is geometrically “very rigid”. For example, we will see how any finitely generated group which is QI to some planar graph must itself be virtually planar. Time permitting, I will also advertise some fun open problems in this direction.