The Thompson group is the group of homeomorphisms of [0,1] that are piecewise linear with slopes a power of 2 and breakpoints a dyadic rational. It is one of the most studied discrete group which still remains very mysterious.
We will present a very general process introduced by Jones that produces a unitary representation of the Thompson group from very few data such as an isometry between Hilbert spaces or a planar algebra together with a certain element. We will present concrete examples.
This is a joint work with Jones.