Symmetry in Newcastle

3:00 pm — 5:30 pm

Friday, 7th Aug 2020

Prof Anthony Guttmann

(Department of Mathematics and Statistics, The University of Melbourne)

On the amenability of Thompson's Group $F$

In 1967 Richard Thompson introduced the group $F$, hoping that it was non-amenable, since then it would disprove the von Neumann conjecture. Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including mention of the many incorrect "proofs" of amenability or non-amenability. Then I will give details of a new, efficient algorithm for obtaining terms of the co-growth sequence. Finally I will describe a number of numerical methods to analyse the co-growth sequences of a number of infinite, finitely-generated groups, and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an alternative route to a rigorous proof. (This is joint work with Andrew Elvey Price).

Dr Collin Bleak

(University of St Andrews)

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 $\Gamma$ 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 $\varepsilon_{0}+1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In this way, for each $\alpha<\varepsilon_{0}$, we obtain a finitely generated elementary amenable subgroup of F whose EA-class is $\alpha+2$. 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.