Subgroups of $\mathrm{PL}_+ I$ which do not embed into Thompson's group $F$

We will give a general criterion - the existence of an $F$-obstruction - for showing that a subgroup of $\mathrm{PL}_+ I$ does not embed into Thompson's group $F$. An immediate consequence is that Cleary's "golden ratio" group $F_\tau$ does not embed into $F$. Our results also yield a new proof that Stein's groups $F_{p,q}$ do not embed into $F$, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of $F$-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the paper, we prove a dichotomy theorem for subgroups of $\mathrm{PL}_+ I$. In addition to playing a central role in our proof, it is strong enough to imply both Rubin's Reconstruction Theorem restricted to the class of subgroups of $\mathrm{PL}_+ I$ and also Brin's Ubiquity Theorem.