## Advanced Studies in Pure Mathematics

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

Shigenori Matsumoto

#### 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.

#### Article information

Dates
Received: 22 April 2014
Revised: 13 September 2014
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1538671780

Digital Object Identifier
doi:10.2969/aspm/07210441

Mathematical Reviews number (MathSciNet)
MR3726723

Zentralblatt MATH identifier
1388.57027

#### Citation

Matsumoto, Shigenori. Actions of groups of diffeomorphisms on one-manifolds by $C^{1}$ diffeomorphisms. Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, 441--451, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07210441. https://projecteuclid.org/euclid.aspm/1538671780