JournalsjcaVol. 5 , No. 1pp. 1–58

Complexity among the finitely generated subgroups of Thompson's group

  • Collin Bleak

    University of St. Andrews, UK
  • Matthew G. Brin

    Binghamton University, USA
  • Justin Tatch Moore

    Cornell University, Ithaca, USA
Complexity among the finitely generated subgroups of Thompson's group cover
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Abstract

We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group FF which is strictly well-ordered by the embeddability relation of type ϵ0+1\epsilon_0 +1. All except the maximum element of this family (which is FF itself) are elementary amenable groups. In fact we also obtain, for each α<ϵ0\alpha < \epsilon_0, a finitely generated elementary amenable subgroup of FF whose EA-class is α+2\alpha + 2. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with Z+Z\mathbf Z + \mathbf Z, ZZ\mathbf Z \wr \mathbf Z, and the Brin–Navas group BB. We also give an example of a pair of finitely generated elementary amenable subgroups of FF with the property that neither is embeddable into the other.

Cite this article

Collin Bleak, Matthew G. Brin, Justin Tatch Moore, Complexity among the finitely generated subgroups of Thompson's group. J. Comb. Algebra 5 (2021), no. 1 pp. 1–58

DOI 10.4171/JCA/49