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

Shigenori Matsumoto

Published 2014-04-22, updated 2014-04-24Version 2

Denote by $\DC(M)_0$ 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 $\DC(M)_0$ to ${\Diff}^1(\R)$ or ${\Diff}^1(S^1)$ is trivial.