On the critical regularity of nilpotent groups acting on the interval: the metabelian case

Let G be a torsion-free, finitely generated, nilpotent and metabelian group. In this work, we show that G embeds into the group of orientation-preserving C1+α -diffeomorphisms of the compact interval for all α < 1/k , where k is the torsion-free rank of $G/A$ and A is a maximal abelian subgroup. We show that, in many situations, the corresponding $1/k$ is critical in the sense that there is no embedding of G with higher regularity. A particularly nice family where this happens is the family of $(2n+1)$ -dimensional Heisenberg groups, for which we can show that the critical regularity is equal to $1+1/n$ .