# 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

## Abstract

We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation of type $\epsilon_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $\alpha < \epsilon_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $\alpha + 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf Z + \mathbf Z$, $\mathbf Z \wr \mathbf Z$, and the Brin–Navas group $B$. We also give an example of a pair of finitely generated elementary amenable subgroups of $F$ 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