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.
This video was produced by Newcastle University, Australia, as part of the Symmetries in Newcastle seminar series.
