Growth of groups and diffeomorphisms of the interval

Abstract

We prove that, for all α > 0, every finitely generated group of C1+α diffeomorphisms of the interval with sub-exponential growth is almost nilpotent. Consequently, there is no group of C1+α interval diffeomorphisms having intermediate growth. In addition, we show that the C1+α regularity hypothesis for this assertion is essential by giving a C1 counter-example.