Abstract

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+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 is equal to 1+1/n.