Irreducible Pythagorean representations of Thompson's groups

Tuesday, 20-June-2023 

Abstract

Richard Thompson’s groups F, T and V are one of the most fascinating discrete groups for their several unusual properties and their analytical properties have been challenging experts for many decades. Most notably, it was conjectured by Ross Geoghegan in 1979 that F is not amenable and thus another rare counterexample to the von Neumann problem. Surprisingly, these discrete groups were recently discovered by Vaughn Jones while working on the very continuous structures of conformal nets and subfactors.

We will explain how using the novel technology of Jones, a so-called Pythagorean unitary representation of Thompson’s groups can be constructed given any isometry from a Hilbert space to its square. These representations are particularly amenable to a diagrammatic calculus which we use to develop powerful techniques to study their properties. For a large class of Pythagorean representations we introduce necessary and sufficient conditions for irreducibility and pair-wise equivalence. This provides a new rich class of irreducible representations of F that extends many previously known classes of irreducible representations of F.