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

TL;DR

This paper proves that for manifolds of dimension at least 2, any group homomorphism from the identity component of compactly supported diffeomorphisms to the group of $C^1$ diffeomorphisms of the line or circle is trivial, highlighting rigidity in group actions.

Contribution

It establishes a rigidity result showing no nontrivial actions exist of the diffeomorphism group on 1-dimensional manifolds under certain smoothness conditions.

Findings

Any homomorphism from $ ext{Diff}_c^r(M)_0$ to ${ m Diff}^1( extbf{R})$ or ${ m Diff}^1(S^1)$ is trivial for $ ext{dim}(M) eq 1$.

The result holds for $r eq ext{dim}(M)+1$, indicating smoothness constraints are crucial.

The theorem extends understanding of group actions and rigidity phenomena in geometric topology.

Abstract

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.