Subexponential Growth and C1 Actions on One-Manifolds

Abstract Let $G$ be a countable group with no finitely generated subgroup of exponential growth. We show that every action of $G$ on a countable set preserving a linear (respectively, circular) order can be realised as the restriction of some action by $C^{1}$ diffeomorphisms on an interval (respectively, the circle) to an invariant subset. As a consequence, every action of $G$ by homeomorphisms on a compact connected one-manifold can be made $C^{1}$ upon passing to a semi-conjugate action. The proof is based on a functional characterisation of groups of local subexponential growth.