Smoothing nilpotent actions on 1-manifolds

Abstract

Let $M$ be a connected 1-manifold, i.e., $M=\mathbb{R}\cong (0,1),[0,1),[0,1]$, 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 $ϕ\colon N \to \Homeo_+(M)$. 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 $ϕ\colon N \to \Homeo_+(M)$ is topologically conjugate to an action $\tildeϕ\colon N \to \Diff_+^1(M)$.

16 pages

Cited by in corpus (1)

• Fixed points of nilpotent actions on ${\mathbb{S}}^2$