Actions of groups of diffeomorphisms on one-manifolds by $C^{1}$ diffeomorphisms

Abstract

Denote by $\mathrm{Diff}_{c}^{r}(M)_{0}$ the identity component of the group of the compactly supported $C^{r}$ diffeomorphisms of a connected $C^{\infty}$ manifold $M$. We show that if $\dim(M)\geq2$ and $r\neq \dim(M)+1$, then any homomorphism from $\mathrm{Diff}_{c}^{r}(M)_{0}$ to $\mathrm{Diff}^{1}(\mathbb{R})$ or $\mathrm{Diff}^{1}(S^{1})$ is trivial.