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

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