The divergence of a group is a quasi-isometry invariant that measures how difficult it is to connect two points while avoiding a ball around the identity. It is easy to see that it is linear for direct products and deduce the same result for branch groups (a class of groups acting on rooted trees, for example the Grigorchuk group). I will discuss divergence for weakly branch groups, in particular the Basilica group. I will also present a generalisation for certain groups admitting a micro-supported action on a Hausdorff topological space, i.e containing elements with arbitrarily small support.
Joint work in progress with D. Francoeur and T. Nagnibeda