The group F introduced by Richard Thompson in 1965 is the group of the orientation preserving piecewise linear dyadic self-homeomorphisms of the closed unit interval. Arguably, the most important open question about it is the one about its amenability as, due to the plethora of rather unusual properties of this group either answer would imply very interesting consequences. This problem has attracted a lot of attention, with an impressive number of failed attempts to prove either amenability or non-amenability of the group F.
In view of the solution of Furstenberg's conjecture by Rosenblatt and by Kaimanovich--Vershik, a possible approach to proving non-amenability of a given group consists in showing that there are no non-degenerate Liouville random walks on it. Here we make the first step in this direction for Thompson's group F by showing that all reasonable finitely supported random walks on it are non-Liouville.
We shall also discuss more recent developments due to Yuschenko and Zheng as well as the ensuing notion of L-amenability for group actions.