Dynamical forcing of circular groups

Abstract:

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 \[ \operatorname {rot}(X(G,\alpha )) = \lbrace r \in \mathbb {R}/\mathbb {Z} \; | \; r = \operatorname {rot}(\rho (\alpha )), \rho \in \operatorname {Hom}(G,\operatorname {Homeo}^+(S^1)) \rbrace ,\] 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.