We will give a general criterion—the existence of an F-obstruction—for showing that a subgroup of PL+I does not embed into Thompson’s group F. An immediate consequence is that Cleary’s “golden ratio” group Fτ does not embed into F, answering a question of Burillo, Nucinkis, and Reves. Our results also yield a new proof that Stein’s groups Fp,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+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 PL+I and also Brin’s ubiquity theorem.
1
Cornell University, Ithaca, USA; University of Copenhagen, Denmark
2
Cornell University, Ithaca, USA
James Hyde; Justin Tatch Moore. Subgroups of $\mathrm{PL}_{+} I$ which do not embed into Thompson’s group $F$. Groups, geometry, and dynamics, Tome 17 (2023) no. 2, pp. 533-554. doi: 10.4171/ggd/708
@article{10_4171_ggd_708,
author = {James Hyde and Justin Tatch Moore},
title = {Subgroups of $\mathrm{PL}_{+} I$ which do not embed into {Thompson{\textquoteright}s} group $F$},
journal = {Groups, geometry, and dynamics},
pages = {533--554},
year = {2023},
volume = {17},
number = {2},
doi = {10.4171/ggd/708},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/708/}
}
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AU - Justin Tatch Moore
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