Subgroups of which do not embed into Thompson’s group

  • James Hyde

    Cornell University, Ithaca, USA; University of Copenhagen, Denmark
  • Justin Tatch Moore

    Cornell University, Ithaca, USA
Subgroups of $\mathrm{PL}_{+} I$ which do not embed into Thompson’s group $F$ cover
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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.

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James Hyde, Justin Tatch Moore, Subgroups of which do not embed into Thompson’s group . Groups Geom. Dyn. 17 (2023), no. 2, pp. 533–554

DOI 10.4171/GGD/708