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

TL;DR

This paper introduces a criterion called the $F$-obstruction to determine when subgroups of $ ext{PL}_+ I$ cannot embed into Thompson's group $F$, providing new proofs and insights into subgroup structures.

Contribution

The paper develops the theory of $F$-obstructions, offering a novel criterion for non-embeddability into $F$, and proves a dichotomy theorem with broad implications.

Findings

$F$-obstructions prevent certain subgroups from embedding into $F$

The criterion applies to Cleary's $F_ au$ and Stein's $F_{p,q}$ groups

The dichotomy theorem generalizes Rubin's and Brin's theorems

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 $F$. 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.