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 $C^{1+\alpha}$-diffeomorphisms of the compact interval, for all $\alpha< 1/k$ 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 equals $1+1/n$.