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 non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $\mu$-boundaries. Both of them are defined in terms of the Schreier graph $\Gamma$ on the dyadic-rational orbit of the canonical action of $F$ on the unit interval (actually, we consider a natural embedding of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of the real line, and realize $\Gamma$ on the dyadic-rational orbit in ${\mathbb R}$). However, the behaviours at infinity described by 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 ${\mathbb Z}$-valued configurations on $\Gamma$ arising from the stabilization of the logarithmic increments of slopes along the sample paths of the random walk. The second $\mu$-boundary is more similar to the boundaries of groups with hyperbolic properties as it consists of the sections of the end bundle of the graph $\Gamma$: these are the collections of the limit ends of the induced random walk on $\Gamma$ parameterized by all possible starting points.