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

Let $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $ρ: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are countably many embeddings $ϕ_i: M\to N$ with disjoint images such that the action of $ρ$ is conjugate (via the product of the $ϕ_i$) to the diagonal action of $\text{Diff}^r_c(M)$ on $M \times M \times ...$ on $\bigcup_i ϕ_i(M)$, 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.

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