Divergence is a quasi-isometry invariant of groups that measures how difficult it is to connect two elements in the Cayley graph of a group by a path that does not pass close to the identity element. It is related to the existence of cut points in the asymptotic cones of the group. In this talk, we will explore divergence in groups with microsupported actions, a class of groups that include interesting examples such as Grigorchuk’s groups and Thompson’s groups. This is joint work with Letizia Issini and Tatiana Nagnibeda.