For the Schrodinger equation, this has been proven in a joint work with S. Kuksin, and for the wave equation we shall report on a work in progress with B. Grebert and S. Kuksin.
In the first part, we will discuss two options for using groups to construct expander graphs (Cayley graphs and Schreier diagrams). Specifically, we will see how to construct monotone expanders in this way. As in recent works (e.g. of Bourgain and Gamburd), we will see that the proof consists of 3 different steps. We will shortly discuss these 3 steps.