There are no exotic actions of diffeomorphism groups on 1-manifolds

Abstract

Let $M$ be a manifold and $N$ a $1$-dimensional manifold. Assuming that $r\ne \dim (M)+1$, we show that any nontrivial homomorphism $\rho :{\mathsf{Diff}}_c^r(M)\to \mathsf{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are countably many embeddings ${\varphi }_i:M\to N$ with disjoint images such that the action of $\rho$ is conjugate (via the product of the ${\varphi }_i$) to the diagonal action of ${\mathsf{Diff}}_c^r(M)$ on $M\times M\times \cdots$ on ${\bigcup }_i{\varphi }_i(M)$, and trivial elsewhere. This solves a conjecture of Matsumoto. We also show that the groups ${\mathsf{Diff}}_c^r(M)$ have no countable index subgroups.