Schreier graphs of actions of Thompson's group F on the unit interval and on the Cantor set

Abstract

Schreier graphs of the actions of Thompson's group $F$ on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set $\{x_0,x_1\}$ are explicitly constructed. The closure of the space of pointed Schreier graphs of the action of $F$ on the orbits of dyadic rational numbers and corresponding Schreier dynamical system are described. In particular, we answer the question of Grigorchuk on the Cantor-Bendixson rank of the underlying space of the Schreier dynamical system in the context of $F$. As applications we prove that the pointed Schreier graphs of points from $(0,1)$ are amenable, have infinitely many ends, and are pairwise non-isomorphic. Moreover, we prove that points $x,y\in(0,1)$ have isomorphic non-pointed Schreier graphs if and only if they belong to the same orbit of $F$.