Smoothing nilpotent actions on 1-manifolds

TL;DR

This paper proves that any action of a finitely-generated, torsion-free nilpotent group on a 1-manifold by homeomorphisms can be smoothly conjugated to an action by $C^1$ diffeomorphisms, strengthening previous results.

Contribution

It shows that all such group actions are topologically conjugate to smooth actions, extending prior work that only established the existence of smooth actions.

Findings

Any nilpotent group action on 1-manifolds by homeomorphisms can be smoothed via conjugation.

The result applies to all finitely-generated, torsion-free nilpotent groups.

It enhances the understanding of group actions on 1-manifolds by connecting topological and smooth dynamics.

Abstract

Let $M$ be a connected 1-manifold, i.e., $M = \R \cong (0, 1), [0, 1), [0, 1]$, or $S^1$, and let $\Homeo_+(M)$ (resp. $\Diff_+^1(M)$) be the group of orientation-preserving homeomorphisms (resp. $C^1$ diffeomorphisms) of $M$. It is a classical result that if $N$ is a finitely-generated, torsion-free nilpotent group, then there exist 1-1 homomorphisms $\phi\colon N \to \Homeo_+(M)$. Farb and Franks show that, in fact, there exists a 1-1 homomorphism $N \to \Diff_+^1(M)$. In this paper we obtain a stronger result: every action $\phi\colon N \to \Homeo_+(M)$ is topologically conjugate to an action $\tilde{\phi}\colon N \to \Diff_+^1(M)$.