Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order

TL;DR

This paper investigates the spectral properties of Laplacians on Schreier graphs from Grigorchuk's group, revealing a Cantor spectrum of measure zero and absence of eigenvalues, with methods applicable to broader classes of graphs.

Contribution

It establishes a novel connection between Schreier graph Laplacians and aperiodic Schrödinger operators, proving Cantor spectrum and eigenvalue absence.

Findings

Spectrum is a Cantor set of Lebesgue measure zero.

Eigenvalues are absent almost-surely and in specific cases.

Methods apply to a broad class of Schreier graphs.

Abstract

We study spectral properties of the Laplacians on Schreier graphs arising from Grigorchuk's group acting on the boundary of the infinite binary tree. We establish a connection between the action of $G$ on its space of Schreier graphs and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger operators with aperiodic order. We use this relation to prove that the spectrum of the anisotropic Laplacians is a Cantor set of Lebesgue measure zero. We also use it to show absence of eigenvalues both almost-surely and for certain specific graphs. The methods developed here apply to a large class of examples.