Subgroups of $\operatorname{Diff}^{\infty}_+ (\mathbb S^1)$ acting transitively on $4$-tuples

We consider subgroups of $C^{\infty}$-diffeomorphisms of the circle $\mathbb S^1$which act transitively on $4$-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of $\mathbb S^1$. A stronger result concerning $C^{\infty}$-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category.