Complexity among the finitely generated subgroups of Thompson's group (Q2233682)

Summary: 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 \(\varepsilon_0 +1\). All except the maximum element of this family (which is \(F\) itself) are elementary amenable groups. In fact we also obtain, for each \(\alpha < \varepsilon_0\), 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.