THOMPSON'S GROUP F IS NOT LIOUVILLE

We prove that random walks on Thompson's group F driven by strictly non-degenerate finitely supported probability measures mu have a nontrivial Poisson boundary. The proof consists in an explicit construction of two non-trivial mu-boundaries. Both of them are described in terms of the `canonical' Schreier graph Gamma on the dyadic-rational orbit of the canonical action of F on the unit interval (in fact, we consider a natural embedding of F into the group PLF(R) of piecewise linear homeomorphisms of the real line, and realize Gamma on the dyadic-rational orbit in R). However, the definitions of these mu-boundaries are quite different (in perfect keeping with the ambivalence concerning amenability of the group F). The first mu-boundary is similar to the boundaries of the lamplighter groups: it consists of Z-valued configurations on F arising from the stabilization of logarithmic increments of slopes along the sample paths of the random walk. The second mu-boundary is more similar to the boundaries of the groups with hyperbolic properties as it consists of sections ('end fields') of the end bundle of the graph Gamma, i.e., of the collections of the limit ends of the induced random walk on Gamma parameterized by all possible starting points. The latter construction is more general than the former one, and is actually applicable to any group which has a transient Schreier graph with a non-trivial space of ends.