Sharp regularity for certain nilpotent group actions on the interval

TL;DR

This paper investigates the regularity of nilpotent group actions on the interval, establishing sharp conditions on the differentiability class for such actions, and introduces new techniques involving random paths on Heisenberg-like groups.

Contribution

It provides a sharp regularity threshold for nilpotent group actions on the interval, extending previous constructions to optimal Hölder regularity.

Findings

Constructs faithful $C^1$-diffeomorphisms for non-Abelian nilpotent groups

Identifies sharp Hölder regularity conditions for derivatives

Uses novel results on random paths in Heisenberg-like groups

Abstract

According to the classical Plante-Thurston Theorem, all nilpotent groups of $C^2$-diffeomorphisms of the closed interval are Abelian. Using techniques coming from the works of Denjoy and Pixton, Farb and Franks constructed a faithful action by $C^1$-diffeomorphisms of $[0,1]$ for every finitely-generated, torsion-free, non-Abelian nilpotent group. In this work, we give a version of this construction that is sharp in what concerns the H\"older regularity of the derivatives. Half of the proof relies on results on random paths on Heisenberg-like groups that are interesting by themselves.