There are no exotic actions of diffeomorphism groups on 1-manifolds (Q2694825)

Summary: Let \(M\) be a manifold and \(N\) a \(1\)-dimensional manifold. Assuming that \(r \neq \dim (M) + 1\), 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 \(\phi_i : M \to N\) with disjoint images such that the action of \(\rho\) is conjugate (via the product of the \(\phi_i\)) to the diagonal action of \(\mathsf{Diff}_c^r (M)\) on \(M \times M \times \cdots\) on \(\bigcup_i \phi_i (M)\), 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.