Subgroups of PLCI 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 PL+I does not embed into Thompson's group F. An immediate consequence is that Cleary's "golden ratio" group Fr does not embed into F, answering a question of Burillo, Nucinkis, and Reves. Our results also yield a new proof that Stein's groups Fp,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 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 PL+I and also Brin's ubiquity theorem.