Boundary of the action of Thompson group F on dyadic numbers

Abstract

We prove that the Poisson boundary of a simple random walk on the Schreier graph of action of $F$ on $\mathbb{D}$, where $\mathbb{D}$ is the set of dyadic numbers in $[0,1]$, is non-trivial. This gives a new proof of the result of Kaimanovich: Thompson's group $F$ doesn't have Liouville property. In addition, we compute growth function of the Schreier graph of the action of $F$ on $\mathbb{D}$.

9 pages, 1 figure

Cited by in corpus (2)

• Infinitely supported Liouville measures of Schreier graphs
• Non-triviality of the Poisson boundary of random walks on the group $H(\mathbb{Z})$ of Monod