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

[摘要] We will give a general criterion—the existence of an FFF-obstruction—for showing that a subgroup of PL+I\mathrm{PL}_{+} IPL+​I does not embed into Thompson’s group FFF. An immediate consequence is that Cleary’s “golden ratio” group FτF_\tauFτ​ does not embed into FFF, answering a question of Burillo, Nucinkis, and Reves. Our results also yield a new proof that Stein’s groups Fp,qF_{p,q}Fp,q​ do not embed into FFF, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of FFF-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\mathrm{PL}_{+} IPL+​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\mathrm{PL}_{+} IPL+​I and also Brin’s ubiquity theorem.

[学科分类] 神经科学

[关键词] F -obstruction;piecewise linear;rotation number;Thompson’s group;topological conjugacy