Some graphs related to Thompson's group F

TL;DR

This paper explicitly describes Schreier graphs related to Thompson's group F, investigates the coamenability of certain stabilizers, and constructs a non-amenable subgraph of the Cayley graph, advancing understanding of F's geometric properties.

Contribution

It provides explicit descriptions of Schreier graphs and proves non-amenability of a specific subgraph of Thompson's group F, offering new insights into its structure.

Findings

Schreier graphs of F are explicitly described

Coamenability of stabilizers of dyadic rationals is established

A non-amenable subgraph of the Cayley graph is constructed

Abstract

The Schreier graphs of Thompson's group F with respect to the stabilizer of 1/2 and generators x_0 and x_1, and of its unitary representation in L_2([0,1]) induced by the standard action on the interval [0,1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x_nv, where n>=0 and v is any word over the alphabet {x_0, x_1}, is constructed. It is proved that the latter graph is non-amenable.