Subgroups of PL_CI 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_CI does not embed into Thompson’s group F . An immediate consequence is that Cleary’s “golden ratio” group F does not embed into F, answering a question of Burillo, Nucinkis, and Reves. 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 PL_CI . 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_CI and also Brin’s ubiquity theorem.