Dynamical forcing of circular groups

In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties.

As an application, we show that the set $X \subset \mathbb{R} /\mathbb{Z}$ consisting of rotation numbers $\theta$ which can be forced by finitely presented groups is an infinitely generated $\mathbb{Q}$-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number $\theta$ is forced by a pair $(G_\theta,\alpha)$, where $G_\theta$ is a finitely presented group $G_\theta$ and $\alpha \in G_\theta$ is some element, if the set of rotation numbers of $\rho(\alpha)$ as $\rho$varies over $\rho \in \operatorname{Hom}(G_\theta,\operatorname{Homeo}^+(S^1))$ is precisely the set $\lbrace 0, \pm \theta \rbrace$.

We show that the set of subsets of $\mathbb{R} /\mathbb{Z}$ which are of the form

where $G$ varies over countable groups, are exactly the set of closed subsets which contain $0$ and are invariant under $x \to -x$. Moreover, we show that every such subset can be approximated from above by $\operatorname{rot}(X(G_i,\alpha_i))$ for finitely presented $G_i$.

As another application, we construct a finitely generated group $\Gamma$ which acts faithfully on the circle, but which does not admit any faithful $C^1$action, thus answering in the negative a question of John Franks.