Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive substitution

TL;DR

This paper explores the connection between the spectral properties of Schrödinger operators with aperiodic potentials and Laplacians on groups acting on rooted trees, focusing on Grigorchuk's group and associated subshifts.

Contribution

It provides an overview of spectral results related to Grigorchuk's group and introduces the combinatorial and dynamical tools, including subshifts from substitutions, used to establish these connections.

Findings

Different spectral types in isotropic and anisotropic cases

Cantor spectrum of Lebesgue measure zero

Absence of eigenvalues in certain cases

Abstract

There is a recently discovered connection between the spectral theory of Schr\"o-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators.