Collin Bleak

On the complexity of elementary amenable subgroups of R. Thompson's group F

The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group Γ is amenable but not elementary amenable. In this talk we report on work where 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 in type ε_0 + 1. All except the maximum element of this family (which is F itself) are elementary amenable groups. In this way, for each α < ε_0, we obtain a finitely generated elementary amenable subgroup of F whose EA-class is α + 2. The talk will be pitched for an algebraically inclined audience, but little background knowledge will be assumed. Joint work with Matthew Brin and Justin Moore.