An extension criterion for lattice actions on the circle

Let Γ < G be a lattice in a locally compact second countable group G. The aim of this paper is to establish a necessary and sufficient condition for a Γ-action by homeomorphisms of the circle to extend continuously to G. This condition will be expressed in terms of the real bounded Euler class of this action. Combined with classical vanishing theorems in bounded cohomology, one recovers rigidity results of Ghys, Witte-Zimmer, Navas and Bader-Furman-Shaker in a unified manner. For a survey of various approaches to the problem of classifying lattice actions on the circle we refer to the article of Witte Morris [26] in this volume. Let Homeo(S1) be the group of orientation preserving homeomorphisms of the circle and e ∈ H2(Homeo(S1),Z) the Euler class; recall that e corresponds to the central extension defined by the universal covering of Homeo(S1). The Euler class admits a representing cocycle which is bounded and this defines a bounded class e ∈ H2 b (Homeo (S1),Z) called the bounded Euler class. The relevance of bounded cohomology to the study of group actions on the circle comes from a result of Ghys [14], namely that the bounded Euler class ρ∗(eb) ∈ H2 b (Γ,Z) of an action ρ : Γ → Homeo(S1) determines ρ up to quasi-conjugation; a quasi-conjugation is a self map of the circle which is weakly cyclic order preserving and in particular not necessarily continuous, see Section 3 for details. If e R denotes then the bounded class obtained by considering the bounded cocycle defining e as real valued, we call the invariant ρ∗(eb R ) ∈ H2 b (Γ,R) the real bounded Euler class of ρ. From this point of view we have the following dichotomy (see Proposition 3.2): (E) ρ∗(eb R ) = 0: in this case, ρ is quasi-conjugated to an action of Γ by rotations; as far as the extension problem is concerned, it reduces to the properties of the restriction map Homc(G,R/Z) → Hom(Γ,R/Z) . (NE) ρ∗(eb R ) 6= 0: in this case, ρ is quasi-conjugated to a minimal unbounded action, that is, every orbit is dense and the group of homeomorphisms ρ(Γ) is not equicontinuous.