My talk will follow a joint work with Maximiliano Escayola, devoted to the study of critical regularities for nilpotent group actions. The questions of critical regularities have been studied by many authors in many different contexts: starting from the classical Denjoy theorem and example, there are works by M. Herman, J.-C. Yoccoz, N. Kopell, B. Deroin, A. Navas, C. Rivas, E. Jorquera, K. Parkhe, S.-H. Kim, T. Koberda, any many others.

Our main result allows to describe the critical regularity in algebraic terms; while proving it, we introduce some new technique for establishing the bounds. We also obtain a generalisation of Bass’ formula to the case of a relative growth of a nilpotent group with respect to its subgroup.