Convergence towards the end space for random walks on Schreier graphs

Bogdan Stankov

Published 2019-05-24Version 1

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ it defines. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the induced random walk is transient, it converges towards the space of ends of $\Gamma$. As a corollary we obtain that for a probability measure on Thompson's group $F$ with a finite first moment, the support of which generates $F$ as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.