2016/02/09 by Kaimanovich, Vadim A.
#FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)
paper · doi:10.48550/arxiv.1602.02971
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.