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arXiv · 1403.7781

Smoothing nilpotent actions on 1-manifolds

Kiran Parkhe
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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 $ϕ\colon N \to \Homeo_+(M)$. 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 $\tildeϕ\colon N \to \Diff_+^1(M)$.

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Kiran Parkhe. 2014-09-26. Smoothing nilpotent actions on 1-manifolds. https://arxiv.org/abs/1403.7781

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