The one-point Schreier Poisson boundary of Thompson's group $F$

Abstract

We identify the Poisson boundary of the one-point Schreier-chain random walk obtained by projecting the simple symmetric random walk on Thompson's group $F$ to the dyadic orbit point $1/2$. For the associated simple labelled-generator walk on the dyadic Schreier graph, the full Poisson boundary is the skeleton end boundary. The proof combines the known description of this Schreier graph as a binary-tree skeleton with recurrent one-dimensional ray attachments with an explicit trace computation. After tracing to the grey skeleton and deleting holding probabilities, the walk becomes a reversible nearest-neighbor walk on the rooted binary tree with two unequal classes of edge conductance. This reduces the boundary identification to standard Poisson--Martin theory for transient walks on trees and leaves a finite electrical-network calculation for the harmonic measure. Following Kaimanovich's coding of skeleton ends by odd 2-adic integers [{\emph{Groups, Graphs and Random Walks}}, London Math. Soc. Lecture Note Ser.~436, pp.~300--342, 2017], the hitting measure is a biased Bernoulli product measure with explicitly computed bias. It is singular with respect to Haar measure, has full topological support, and is exact-dimensional; these properties and the exact constants are proved here.