Coherent actions by homeomorphisms on the real line or an interval

We study actions of groups by orientation preserving homeomorphisms on R (or an interval) that are minimal, have solvable germs at +/-infinity and contain a pair of elements of a certain dynamical 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. The structure theory we develop allows us to prove a plethora of non-embeddability statements concerning groups of piecewise linear and piecewise projective homeomorphisms. For instance, we demonstrate that any coherent group action G < Horneo(+) (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. We also establish that the Brown-Stein-Thompson groups F(2, pi, horizontal ellipsis ,p(n)) for n >= 1 and p(1), horizontal ellipsis ,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.