Coherent actions by homeomorphisms on the real line or an interval

Abstract: We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non elementary amenable, but satisfies that every proper quotient is solvable. As a first application, we demonstrate that any coherent group action $G<Homeo^+(\mathbf{R})$ that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompson's group $F$. This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms. As a second application, we establish that the Brown-Stein-Thompson groups $F(2,p_1,...,p_n)$ for $n\geq 1$ and $p_1,...,p_n$ distinct odd primes, do not embed into Thompson's group $F$. This answers a question recently raised by C. Bleak, M. Brin, and J. Moore. Our tools also allow us to prove additional non embeddability results for Brown-Stein-Thompson and Bieri-Strebel groups.