Accessibility, QI-rigidity, and planar graphs

A common pastime of geometric group theorists is to try and derive algebraic information about a group from the geometric properties of its Cayley graphs. One of the most classical demonstrations of this can be seen in the work of Maschke (1896) in characterising those finite groups with planar Cayley graphs. Since then, much work has been done on this topic. In this talk, I will attempt to survey some results in this area, before presenting some of my own work towards showing that the class group with planar Cayley graphs is QI-rigid.