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 consisting of rotation numbers θ which can be forced by finitely presented groups is an infinitely generated Q-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number θ is forced by a pair (G_θ, α), where G_θ is a finitely presented group G_θ and α ∈ G_θ is some element, if the set of rotation numbers of ρ(α) as ρ ∈ Hom(G_θ, Homeo^(+)(S^1)) is precisely the set {0,±θ}. We show that the set of subsets of R/Z which are of the form rot(X(G, α)) = {r ∈ R/Z | r = rot(ρ(α)), ρ ∈ Hom(G, Homeo^(+)(S^1))}, where G varies over countable groups, are exactly the set of closed subsets which contain 0 and are invariant under x→−x. Moreover, we show that every such subset can be approximated from above by rot(X(G_i, α_i)) for finitely presented G_i. As another application, we construct a finitely generated group Γ 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.