We will give a general criterion—the existence of an -obstruction—for showing that a subgroup of does not embed into Thompson’s group . An immediate consequence is that Cleary’s “golden ratio” group does not embed into , answering a question of Burillo, Nucinkis, and Reves. Our results also yield a new proof that Stein’s groups do not embed into , a result first established by Lodha using his theory of coherent actions. We develop the basic theory of -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 . 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 and also Brin’s ubiquity theorem.
Cite this article
James Hyde, Justin Tatch Moore, Subgroups of which do not embed into Thompson’s group . Groups Geom. Dyn. 17 (2023), no. 2, pp. 533–554DOI 10.4171/GGD/708