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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 $μ$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $μ$-boundaries. Both of them are defined in terms of the Schreier graph $Γ$ 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 $Γ$ on the dyadic-rational orbit in ${\mathbb R}$). However, the behaviours at infinity described by these $μ$-boundaries are quite different (in perfect keeping with the ambivalence concerning amenability of the group $F$). The first $μ$-boundary is similar to the boundaries of the lamplighter groups: it consists of ${\mathbb Z}$-valued configurations on $Γ$ arising from the stabilization of the logarithmic increments of slopes along the sample paths of the random walk. The second $μ$-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 $Γ$: these are the collections of the limit ends of the induced random walk on $Γ$ parameterized by all possible starting points.