SYM lecture: The Thompson groups F, T, V and their C*- and von Neumann algebras

In this monthly 45 minute expository SYM lecture, Uffe Haagerup will give an introduction to the three Thompson groups F, T and V, and discuss some recent result obtained in collaboration with Kristian Knudsen Olesen, Søren Haagerup and Maria Ramirez-Solano.

It is a long standing open problem whether the Thompson group F is amenable, or equivalently whether its group von Neumann algebra L(F) isomorphic to the hyperfinite II-1 factor R. Paul Jolissaint has proved that F is inner amenable and L(F) has property Gamma. In a recent work with Kristian Knudsen Olesen, we prove that T and V are not inner amenable and L(T) and L(V) does not have property Gamma. We also prove that if the reduced C*-algebra C*_r(T)  of T is simple, then F is non-amenable.

In collaboration with Maria Ramirez-Solano and Søren Haagerup we use extensive numerical computations to test the amenability problem for F by estimating the norms of certain elements of C*_r(F). Numerical computations alone cannot detect whether or not F is amenable, but the results we have obtained suggest that the most likely outcome is that F is non-amenable.