On spectral properties of the Schreier graphs of the Thompson group $F$

TL;DR

This paper investigates the spectral properties of Schreier graphs related to the Thompson group $F$, analyzing spectra of Laplace operators, spectral measures, and random walk asymptotics on these graphs.

Contribution

It provides a detailed spectral analysis of Schreier graphs of the Thompson group $F$, including spectra, spectral measures, and random walk behavior, which are novel insights in this context.

Findings

Spectra of Laplace type operators on Schreier graphs are characterized.

Spectral measures for the Schreier graph of the orbit of 1/2 are computed.

Asymptotics of return probabilities for random walks on these graphs are derived.

Abstract

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group $F$ on the interval [0,1]. In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph $\Upsilon$ of the orbit of 1/2. As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on $\Upsilon$ starting at 1/2. In addition, given a Laplace type operator $L$ on a tree-like graph we study relations between the spectral measures of $L$ associated to delta functions of different vertices and the spectrum of $L$.