Complexity among the finitely generated subgroups of Thompson's group

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.