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

Abstract

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 $F$, 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 ${\mathrm{PL}}_{+}I$. 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.