On the critical regularity of nilpotent groups acting on the interval: the metabelian case
Maximiliano Escayola, Cristóbal Rivas
TL;DR
The article identifies and sharpens the critical regularity for actions of torsion-free finitely generated nilpotent metabelian groups on the interval. By constructing a controlled action of $G$ on $\mathbb{Z}^{k+1}$ that preserves the lexicographic order and then smoothing it via Pixton–Tsuboi techniques, the authors prove that $G$ embeds into $\text{Diff}_+^{1+\alpha}([0,1])$ for all $\alpha<1/k$, yielding a lower bound of $1+1/k$ on the critical regularity. They compute exact critical regularities for key families such as the $(2n+1)$-dimensional Heisenberg groups, obtaining $\text{Crit}_{[0,1]}(\mathscr{H}_n)=1+1/n$, and present constructions showing this threshold is sharp in many cases, while also highlighting scenarios where the bound is not optimal (e.g., certain products). The results illuminate how the algebraic structure—specifically the rank $k$ of $G/A$ with $A$ maximal abelian containing $[G,G]$—controls dynamical regularity on the interval, and they demonstrate that higher regularity can sometimes be achieved through indirect group decompositions, with implications for understanding rigidity and smoothability in nilpotent dynamics on one-dimensional manifolds.
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 $C^{1+α}$-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 equals $1+1/n$.