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

TL;DR

This paper explicitly constructs Schreier graphs for Thompson's group F acting on the unit interval and Cantor set, analyzing their structure, dynamical properties, and isomorphism classes, with implications for amenability and graph ends.

Contribution

It provides explicit constructions of Schreier graphs for F's actions, describes the dynamical system of these graphs, and answers a question about the Cantor-Bendixson rank, revealing new structural insights.

Findings

Schreier graphs of F on (0,1) are amenable

They have infinitely many ends

Pointwise non-isomorphic graphs for different orbits

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 $F$.