Subexponential growth and C1 actions on one-manifolds

The uploaded paper proves exactly the statement invoked by the model: Theorem 1.1 asserts that if G is countable with no finitely generated subgroup of exponential growth, then every action of G on a countable linearly (resp. circularly) ordered set is realized as the restriction of an action by C1 diffeomorphisms on an interval (resp. the circle), with a G-equivariant injective order-preserving map i: X → M. This is stated explicitly and proved via a more general Theorem 3.1 using a moderate ℓ1-function on X and an equivariant Yoccoz family construction, ensuring ρ(g) has derivative 1 at interval endpoints and extends C1 across accumulation points of i(X) (see Theorem 1.1 and Theorem 3.1 with properties (C1)–(C3) and the moderate ℓ1-function construction). The model’s solution simply cites this result and sketches the same construction ideas, so both are correct and essentially the same proof approach .