Emmanuel Breuillard - Nilprogressions and finite groups with moderate growth

I will discuss some applications of the structure theorem for approximate groups proved some years ago by Hurshovski and Breuillard-Green-Tao. We describe the structure of finite groups with very large diameter (larger than a power of the size of the group). In particular we obtain a universal upper bound on the diameter of Cayley graphs of finite simple groups, which is independent of CFSG. Using the structure of nilprogressions, we show that these groups (which we call almost flat) are precisely the groups of moderate growth introduced by Diaconis and Saloff-Coste in the 90’s in connection with mixing time estimates for random walks (joint work with Matt Tointon).