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 ⊂ R / Z X \subset \mathbb {R}/\mathbb {Z} consisting of rotation numbers θ \theta which can be forced by finitely presented groups is an infinitely generated Q \mathbb {Q} –module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number θ \theta is forced by a pair ( G θ , α ) (G_\theta ,\alpha ) , where G θ G_\theta is a finitely presented group G θ G_\theta and α ∈ G θ \alpha \in G_\theta is some element, if the set of rotation numbers of ρ ( α ) \rho (\alpha ) as ρ \rho varies over ρ ∈ Hom ⁡ ( G θ , Homeo + ⁡ ( S 1 ) ) \rho \in \operatorname {Hom}(G_\theta ,\operatorname {Homeo}^+(S^1)) is precisely the set { 0 , ± θ } \lbrace 0, \pm \theta \rbrace . We show that the set of subsets of R / Z \mathbb {R}/\mathbb {Z} which are of the form \[ rot ⁡ ( X ( G , α ) ) = { r ∈ R / Z | r = rot ⁡ ( ρ ( α ) ) , ρ ∈ Hom ⁡ ( G , Homeo + ⁡ ( S 1 ) ) } , \operatorname {rot}(X(G,\alpha )) = \lbrace r \in \mathbb {R}/\mathbb {Z} \; | \; r = \operatorname {rot}(\rho (\alpha )), \rho \in \operatorname {Hom}(G,\operatorname {Homeo}^+(S^1)) \rbrace , \] where G G varies over countable groups, are exactly the set of closed subsets which contain 0 0 and are invariant under x → − x x \to -x . Moreover, we show that every such subset can be approximated from above by rot ⁡ ( X ( G i , α i ) ) \operatorname {rot}(X(G_i,\alpha _i)) for finitely presented G i G_i . As another application, we construct a finitely generated group Γ \Gamma which acts faithfully on the circle, but which does not admit any faithful C 1 C^1 action, thus answering in the negative a question of John Franks.